LONDON MATHEMATICAL SOCIETY STUDENT TEXTSpanchish/ETE LAMA 2018-AP/Haruzo... · 14 Combinatorial group theory: a topological approach, DANIEL E. COHEN 15 Presentations of groups,

LONDON MATHEMATICAL SOCIETY STUDENT TEXTS

Managing editor: Dr CM. Series, Mathematics InstituteUniversity of Warwick, Coventry CV4 7AL, United Kingdom

1 Introduction to combinators and A,-calculus, J.R. HINDLEY & J.P. SELDIN

2 Building models by games, WILFRID HODGES

3 Local fields, J.W.S. CASSELS

4 An introduction to twistor theory, S.A. HUGGETT & K.P. TOD

5 Introduction to general relativity, L.P. HUGHSTON & K.P. TOD

6 Lectures on stochastic analysis: diffusion theory, DANIEL W. STROOCK

7 The theory of evolution and dynamical systems, J. HOFBAUER & K. SIGMUND

8 Summing and nuclear norms in Banach space theory, GJ.O. JAMESON

9 Automorphisms of surfaces after Nielsen and Thurston, A.CASSON & S. BLEILER

10 Nonstandard analysis and its applications, N.CUTLAND (ed)

11 Spacetime and singularities, G. NABER

12 Undergraduate algebraic geometry, MILES REID

13 An introduction to Hankel operators, J.R. PARTINGTON

14 Combinatorial group theory: a topological approach, DANIEL E. COHEN

15 Presentations of groups, D.L. JOHNSON

16 An introduction to noncommutative Noetherian rings, K.R.GOODEARL &

R.B. WARFIELDJR.

17 Aspects of quantum field theory in curved spacetime, S.A. FULLING

18 Braids and coverings: selected topics, VAGN LUNDSGAARD HANSEN

19 Steps in commutative algebra, R.Y. SHARP

21 Representations of finite groups of Lie type, FRANCOIS DIGNE & JEAN MICHEL

22 Designs, codes and graphs and their linkages, P. CAMERON & J. VAN LINT

23 Complex algebraic curves, F. KIR WAN

24 Lectures on elliptic curves, J.W.S. CASSELS

25 Hyperbolic geometry, B. IVERSEN

26 Elementary theory of L-functions and Eisenstein series. H. HIDA

27 Hilbert space: compact operators and the trace theorem, J.R. RETHERFORD

London Mathematical Society Student Texts 26

Elementary Theory ofL-functions andEisenstein Series

Haruzo HidaDepartment of Mathematics,University of California at Los Angeles

H CAMBRIDGEUNIVERSITY PRESS

CAMBRIDGE UNIVERSITY PRESSCambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo

Cambridge University Press

The Edinburgh Building, Cambridge CB2 2RU, UK

Published in the United States of America by Cambridge University Press, New York

www.cambridge.orgInformation on this title: www.cambridge.org/9780521434119

© Cambridge University Press 1993

This publication is in copyright. Subject to statutory exceptionand to the provisions of relevant collective licensing agreements,no reproduction of any part may take place withoutthe written permission of Cambridge University Press.

First published 1993

A catalogue record for this publication is available from the British Library

ISBN-13 978-0-521 -43411 -9 hardbackISBN-10 0-521 -43411 -4 hardback

ISBN-13 978-0-521-43569-7 paperbackISBN-10 0-521-43569-2 paperback

Transferred to digital printing 2006

Contents

Suggestions to the reader xi

Chapter 1. Algebraic number theory 1

§1.1. Linear algebra over rings 1§1.2. Algebraic number fields 5

§1.3. p-adic numbers 17

Chapter 2. Classical L-functions and Eisenstein series 25

§2.1. Euler's method 25§2.2. Analytic continuation and the functional equation 33§2.3. Hurwitz and Dirichlet L-functions 40§2.4. Shintani L-functions 47§2.5. L-functions of real quadratic field 54§2.6. L-functions of imaginary quadratic fields 63§2.7. Hecke L-functions of number fields 66Chapter 3. p-adic Hecke L-functions 73

§3.1. Interpolation series 73§3.2. Interpolation series in p-adic fields 75§3.3. p-adic measures on Zp 78§3.4. The p-adic measure of the Riemann zeta function 80§3.5. p-adic Dirichlet L-functions 82§3.6. Group schemes and formal group schemes 89§3.7. Toroidal formal groups and p-adic measures 96§3.8. p-adic Shintani L-functions of totally real fields 99§3.9. p-adic Hecke L-functions of totally real fields 102

Chapter 4. Homological interpretation 107

§4.1. Cohomology groups on Gm(C) 107§4.2. Cohomological interpretation of Dirichlet L-values 117§4.3. p-adic measures and locally constant functions 118§4.4. Another construction of p-adic Dirichlet L-functions 120

Chapter 5. Elliptic modular forms and their L-functions 125

§5.1. Classical Eisenstein series of GL(2)/Q 125§5.2. Rationality of modular forms 131§5.3. Hecke operators 139§5.4. The Petersson inner product and the Rankin product 150§5.5. Standard L-functions of holomorphic modular forms 157

Chapter 6. Modular forms and cohomology groups 160

§6.1. Cohomology of modular groups 160§6.2. Eichler-Shimura isomorphisms 167§6.3. Hecke operators on cohomology groups 175§6.4. Algebraicity theorem for standard L-functions of GL(2) 186§6.5. Mazur's p-adic Mellin transforms 189

vi Contents

Chapter 7. Ordinary A-adic forms, two variable p-adic Rankin productsand Galois representations 194

§7.1. p-Adic families of Eisenstein series 195§7.2. The projection to the ordinary part 200§7.3. Ordinary A-adic forms 208§7.4. Two variable p-adic Rankin product 221§7.5. Ordinary Galois representations into GL2(Zp[[X]]) 228§7.6. Examples of A-adic forms 234

Chapter 8. Functional equations of Hecke L-functions 239

§8.1. Adelic interpretation of algebraic number theory 239§8.2. Hecke characters as continuous idele characters 245§8.3. Self-duality of local fields 249§8.4. Haar measures and the Poisson summation formula 253§8.5. Adelic Haar measures 257§8.6. Functional equations of Hecke L-functions 261

Chapter 9. Adelic Eisenstein series and Rankin products 272

§9.1. Modular forms on GL?(FA) 272§9.2. Fourier expansion of Eisenstein series 282§9.3. Functional equation of Eisenstein series 292§9.4. Analytic continuation of Rankin products 298§9.5. Functional equations of Rankin products 306

Chapter 10. Three variable p-adic Rankin products 310

§ 10.1. Differential operators of Maass and Shimura 310§ 10.2. The algebraicity theorem of Rankin products 317§10.3. Two variable A-adic Eisenstein series 326§10.4. Three variable p-adic Rankin products 331§10.5. Relation to two variable p-adic Rankin products 339

§10.6. Concluding remarks 343

Appendix: Summary of homology and cohomology theory 345

References 365

Answers to selected exercises 371

Index 383

Preface

Number theory is very rich with surprising interactions of fundamentallydifferent objects. A typical example which springs first to mind is the specialvalues of L-functions, in particular, the Riemann zeta function £(k) (which willbe explained in detail in Chapter 2). For each integer k, we sum up all positiveintegers n, raising to a negative power n"k

C(k)=£ n"k

n=l

Since n"k =—r gets smaller and smaller as n grows, we really get this numbern

(if k > 1) sitting somewhere on the positive real coordinate line. Numbertheorists are supposed to study numbers, and in particular, integers. Thus thiskind of sum of all integers should be interesting. Of course, one hopes to sumpositive integer powers nk. Obviously, even squares n2 get larger and larger asn grows, and there seems to be no way of summing up all squares of integers.Nevertheless, in the mid-18th century, Euler computed the sum of positiveinteger powers. He introduced an auxiliary variable t and looked at

C(-k) = t+2kt2+3kt3+4kt4+---|t=1.For example:

£(0) = t+t2+t3+- • • = y - | t=i (the geometric series).

Knowing that the derivative (with respect to t) of tn = -r- = nt11 ~\ we differ-

entiate the right-hand side of the above formula and get l+2t+3t2+4t3+- • * which

is quite near to t+2t2+3t3+4t4+—. Thus we get

t=l

and similarly, taking the derivatives k times, we get

Still one cannot get the answer, because we cannot replace t by 1 in T—. NOW,

when number theorists get to this point, in a hunt, they start to smell somethinginteresting and will never give up the chance. Let's separate the sum into twoparts, that is, the sum over even integers and the sum over odd integers. Then wesee that

viii Preface

(l-2k+1)C(-k) = (t+2kt*+3kt3+4kt4+--- |t=1)-2(2kt+4kt2+6kt3+8kt4+--- |t=i)

This time we have won, because we can really put 1 in place of t and get anumber, which (after dividing by (l-2k+1)) Euler declared to be the value £(-k).It is then easy to see that £(-k) = 0 for even integers k > 0; thus we mayconcentrate on the odd negative ^-values £(l-2m) for positive integer m. Themost remarkable fact Euler discovered in this context is the following relation(whose proof will be given in §2.2 and in Chapter 8):

The left-hand side is the actual sum of negative powers n"2m over positive in-tegers and the right-hand side is the value of the ratio of suitable polynomials int. This is a remarkable interaction of an infinite sum and derivatives of apolynomial created by Euler.

There is another example of this kind. Let us fix one prime, say 5. Suppose youlive in a country under a very crazy dictator, who decreed that two points are'near' if the distance between them measured by meters is divisible by very highpower of 5; so if you sit 53 = 125 meters distance from a friend, you are 'near' tohim. If you sit 55 = 3125 meters distance away, then you are 'very close' tohim, and so on. This type of topology (called the p-adic topology) was createdby Kummer and his student Hensel in the 19th century. Naturally, the world withthis topology is very different from our own one, but number theorists free fromany worldly restrictions can look into such strange places (see §1.3 for details).Looking into Euler's formula of £(-k), Kummer discovered that if k is close tok' in his (5-adic) sense, £(-k) is again close to £(-k'). In other words, if k-k' isdivisible by 5n4, then £(-k)-£(-k') is divisible by 5n+1. Thus £ is not only afunction of integers but is a function of numbers in this new 5-adic world. Thenewly obtained function is called the 5-adic ^-function. Of course, one can workchoosing a prime p different from 5 and create the world measured by thepower of p: the p-adic topology. Then we again find the p-adic ^-function (thisperspective of viewing results of Kummer as p-adic zeta functions was intro-duced by Kubota and Leopoldt in the 1960's). This is the world in which I havea great interest. The reader will find many p-adic L-functions in this book alongthe way of studying the values of L-functions of modular forms on the algebraicgroups GL(1) and GL(2). The modular p-adic L-functions for GL(2) havemany variables (see Chapter 7 and 10), and the discovery of a new set of naturalvariables for the modular p-adic L-functions may be the only legitimate "raisond'etre" of this book apart from the educational point of view. This means that for

Preface ix

the zeta functions of algebraic groups, the interaction of their values is muchmore intense than in the classical abelian case of GL(1). Moreover, the p-adic orcomplex L-functions and Eisenstein series studied in this book are naturallyassociated to analytic Galois representations into GLn (1 < n < 2). We brieflytouch this point in Chapter 7.

The principal text is an outgrowth of courses given at UCLA (U.S.A.), HokkaidoUniversity (Japan), Universite de Paris-Sud and Universite de Grenoble (France).It is my pleasure to acknowledge encouragement from the small audiences Ialways had in all of these lectures. Many people helped me to write correctMathematics and correct English. Especially, I am grateful to A. Bluher, K.Chandrashekhar, Y. Maeda, J. Tilouine and B. Wilson for reading some of thechapters and giving me many useful suggestions. I should also acknowledge thehelp I got, in writing a readable text in a precise format, from D. Tranah who isthe publishing director at the Cambridge university press. While I was writingthis book, I was partially supported by a grant from the National ScienceFoundation and by a fellowship from John Simon Guggenheim MemorialFoundation.

September 25, 1992 at Los Angeles

Haruzo Hida

Suggestions to the reader

The principal text of this book consists of 10 Chapters. The first chapter sum-marizes results from the theory of Linear algebras, algebraic number fields andp-adic numbers, which is used in later chapters. For the first reading, the readeris suggested to start from Chapter 2 skipping Chapter 1 and, from time to time, toconsult Chapter 1 when results there are quoted in the principal text. AfterChapter 2, all the chapters (and the sections) are ordered from basics to more so-phisticated subjects, although logically several chapters are independent. Withinthe same section, the numbered formula (or statement) is quoted just by its num-ber. If the formula or statement is quoted from another section within the samechapter, the section number precedes the number of the formula or the statement.Namely, the formula (1) in Section 2 is quoted as (2.1), and Theorem 2 in Section3 is quoted as Theorem 3.2. This principle also applies when a numbered state-ment or a formula is quoted from different chapters; for example, Proposition3.2.1 implies Proposition 1 in Section 2 of Chapter 3.

We use the standard symbols in the books. For example, Z, Q, R and C de-note the ring of rational integers, the field of rational numbers, the field of realnumbers and the field of complex numbers, respectively. The symbol N is usedfor the set of non-negative integers including 0. The ring of p-adic integers isdenoted by Zp for each prime p whose field of fractions is written as Qp, thefield of p-adic numbers. We write Q for the field of all numbers algebraic overQ in C. Any subfield of Q which is of finite dimension over Q is called anumber field. Thus Q is already embedded into C. We sometimes but not sooften write this embedding as loo. We fix once and for all an algebraic closureQ p and an embedding ip of Q into Qp. Thus Q is also a subfield of Qp.We denote by Gal(Q/Q) the group of all field automorphisms of Q. Eachelement a e Gal(Q/Q) acts on Q from the right, and the composition givesthe group structure on Gal(Q/Q). We make the group Gal(Q/Q) into aprofinite topological group declaring every subgroup fixing a number field to bean open subgroup (see [N, Chapter 1]).

For each commutative ring A with identity, we denote by Ax for the group ofall invertible elements (i.e. units) in A. We denote by Mn(A) the ring of allnxn square matrices with entries in A. We then define

GL2(A) = {X e Mn(A) I det(A) e Ax} and SLn(A) = {X e Mn(A) | det(X) = 1}.

We write GL(n) and SL(n) for the linear algebraic group which assign to eachcommutative algebra A, the group GLn(A) and SLn(A), respectively. For anymaps f: X —> Y and g : Y —> Z, we write g°f: X —> Z for the map given bycomposition g°f(x) = g(f(x)).

Chapter 1. Algebraic Number Theory

To make this text as self-contained as possible, we give a brief butbasically self-contained sketch of the theory of algebraic number fields in §1.2.We also summarize necessary facts from linear (and homological) algebra in §1.1and from the theory of p-adic numbers in §1.3. For a first reading, if the readerhas basic knowledge of these subjects, he or she may skip this chapter andconsult it from time to time as needed in the principal text of the book. Wesuppose in §1.2 basic knowledge of elementary number theory, concerningrational numbers and algebraic numbers, which is found in any standardundergraduate level text. We shall concentrate on what will be used in the laterchapters. Readers who want to know more about algebraic number theoryshould consult [Bourl,3], [FT], [Wl] and [N].

§1.1. Linear algebra over ringsWe summarize in this section some facts from linear algebra and some fromhomological algebra. We will not give detailed proofs.

Let A be a commutative ring with identity. For two A-modules M and N, wewrite HorriA(M,N) for the A-module of all A-linear maps of M into N. Inparticular, M* = HOIIIA(M,A) is called the A-dual module of M. A sequence ofA-linear maps M — ^ N — £ - » L is called "exact" (at N) if Im(oc) = Ker(p).A sequence > Mi_i -^ Mi —> Mi+i -»••• is called exact if it is exact at Mi

for every i. It is easy to check that if 0 —> M—^—»N——>L —» 0 is an exactsequence of A-modules, then for any A-module E,

(la) 0 -> HomA(L,E) P* >HomA(N,E) —^->HomA(M,E) is exact,

where "*" indicates the natural pull back, i.e. p*<|) = (|)op. Similarly, we canshow

(lb) 0 -> HomA(E,M) a* )HomA(E,N)--^HomA(E,L) is exact,

where a*§ = a°<]). The map P* is known to be surjective if E is a projective

A-module [HiSt, 1.4.7]. For a given A-module M, we consider the exact se-

quence 0 -» R —^—» P —> M —> 0 in which P is a projective A-module. Then

the first extension module ExtA(M,N) for another A-module N is defined to be

ExtA(M,N) = Coker(a* : HomA(P,N) ->HomA(R,N)).

It is well known that ExtA (M,N) is well defined (up to isomorphism) indepen-dently of the choice of the projective module P [HiSt, III.2]. Now, out of thecommutative diagram with exact rows:

1: Algebraic number theory

>La ib ic

we get the following long exact sequence (the snake lemma [HiSt, III, Lemma5.1], [Bourl, 1.1.4]):

Ker(a) —^-> Ker(b) —£-» Ker(c) —±-> Coker(a) —*—> Coker(b) —ÛCoker(c),

where the connecting map d is defined as follows. For any x e Ker(c), we takey e F so that (3(y) = x. Then b(y) falls in Ker(8) because of thecommutativity of the right square of the diagram. Thus we can find Z G M SOthat y(z) = b(y) by the exactness at N. Then d(x) is defined to be z moduloa(E). We apply the above lemma in the following situation. Let0 — > M - ^ N - ^ L - ^ 0 be an exact sequence of A-modules. For a givenA-module E, we take a projective A-module P and an A-module R to get an

exact sequence 0 - » R — ^ P >E —> 0. Then we have the followingcommutative diagram whose rows are exact:

0 -> HomA(E,M) -> HomA(E,N) -> HomA(E,L)

ib M * ^bN* ib L *0 -> HomA(P,M) -> HomA(P,N) -» HomA(P,L) -> 0

iaM* J^N* >UL*

0 -> HomA(R,M) -> HomA(R,N) -> HomA(R,L).

Note that Coker(ax*) = ExtA(E,X) for X = M, N and L. Then the snakelemma shows

Theorem 1. For each exact sequence 0 - > M - > N - > L - > 0 of A-modulesand for each A-module E, we have the following seven term exact sequence:0 -> HomA(E,M) -> HomA(E,N) -> HomA(E,L)

-> ExtA(E,M) -» ExtA(E,N) -> ExtA(E,L).

The above sequence is a part of the long exact sequence of extension groups[HiSt, IV.7.5]. By the above sequence, the group ExtA(E,M) measures thedeviation from being surjective of the map p* in (lb).

When A is a valuation ring with a prime element G3, we can easily

compute ExtA(E,M) for some E and M. By definition, if E is projective,

ExtA(E,M)=0. We now compute ExtA(E,A) when E = A/G3rA. We have the

exact sequence 0 -» A—-—> A -> E —> 0. Then by Theorem 1 and (la), wehave the following exact sequence:

1.1. Linear algebra over rings 3

0 -» HomA(A,A) = A ^ )HomA(A,A) = A -> Ext A (E, A) -> 0,

which shows ExtA(A/G3rA,A) = A/G5rA. Each torsion A-module E of finite

type is a direct sum of finitely many cyclic modules of the form A/G5rA. As is

easily seen from the definition, the functor Ext A satisfies

(2) E x t ^ E e E ' ^ ) = ExtACEJSêExt^CE'JVl) (see [HiSt, III.4.1]).

This shows

Corollary 1. Suppose that A is a valuation ring. Let E be a torsion A-module

of finite type. Then ExtA(E,A) = E (canonically).

Let M and N be two A-modules. We define the tensor product M ® A N with abilinear map i : MxN —> M® AN as a solution of the following universalityproblem. For any given A-bilinear map <]): MxN —> E for any given thirdA-module E, there is a unique A-linear map $* : M ® A N -> E such that

$ = <|)*ot. We write t(a,b) = a®b e MAN. If M®AN exists, the uniqueness of

M<8>AN up to isomorphism is clear. We can construct M(8>AN as follows. LetA[MxN] be the free A-module generated by elements of MxN. We consider theA-submodule X in A[MxN] generated by elements of the form

(x+x',y)-(x,y)-(x',y), (x,y+y')-(x,y)-(x,y') and (ax,y)-(x,ay)

for x € M, y e N and a e A. Then A[MxN]/X satisfies the requireduniversal property of M®AN, where x®y is the image of (x,y) in the quotient.We see easily the following properties:

M<g)AN = N® AM and (M® AN)(g> AE = M® A(N® AE)x®y <-> y®x (x®y)®z <r* x®(y®z),

and

M®AE a 0 i d )N®AE P®id )L®AE->0 is exact

m®e h^ a(m)®e; n®e h-> P(n)®e

for each exact sequence 0 —> M —^—>N —*-—>L —> 0. As a dual version of theextension functor Ext, we can now construct the torsion functor Tor as follows.For each A-module M, we take an exact sequence.

0 -> R —2-» P -> M -» 0 for a projective A-module P.Then we define for another A-module N

Torf (M,N) = Ker(cc®id : R®N

4 1: Algebraic number theory

Similarly to Theorem 1 (see [HiSt, IIL8.3]), we get

Theorem 2. For each exact sequence 0 - * M — > N - » L - » 0 of A-modules,and for each A-module E, we have the following seven term exact sequence:

Torf^MJE) -> Tor^(NJE) -> TorÔUE) -> M®AE -> N®AE ->L®AE ->0.

When B and C are A-algebras, B® AC is naturally an A-algebra via themultiplication (b®c)(b'®c') = bb'®cc'. We now suppose that A is a field K ofcharacteristic 0. For any K-vector spaces V with basis {vi)iei and W withbasis {wj}jGj, it is obvious from the definition that {vi®Wj}(ij)eixj form a basisof V ® K W . Thus if V and W are finite dimensional,

(3) dimK(V<g>KW) = (dimKV)(dimKW).

Let M/K and F/K be field extensions in a fixed algebraically closed field Qcontaining K. Suppose that M/K is finite and F contains all conjugates of Mover K. Then it is well known that the set of all the field embeddingsI = I(M/K) into F is linearly independent over K in HorriK(M,F) (a theorem ofDedekind [Bour3, V.7.5]). Note that a e I induces a®id : M®KF -^ F, whichis a homomorphism of K-algebras. Thus we have a K-algebra homomorphism:

i : M ® K F -^ F1 given by i(m®f) = (fa(m))aei,

where F1 is the product of I copies of F. The morphism i is injective becauseof the theorem of Dedekind. By comparing the dimensions, we know that

(4) M®KF = FI as K-algebras.

This applies when F = C for example and we have M ® Q C = 0M :QI The situa-tion for F = R is a little bit different. We have from (4)

(5a) M®QR = R1 (I = I(M/Q)) if M is totally real.

Here M is called "totally real" if all the field embeddings of M into C in facthave values in R. If M is not totally real, we split I(M/Q) = I(R)III(C) forreal embeddings I(R) and non-real embeddings I(C). Then we can further splitI(C) = £ l l E c for complex conjugation c, and we consider the mapi : M ® Q R -> R W ^ C 1 given by i(m®r) = (a(m)r)a€i(R)H£. Then the range andthe domain of t have the same dimension over R. The injectivity of I followsfrom Dedekind's theorem, and hence we obtain

(5b) M®QR = RKRJxC* for I(MjQ) =

We now note a simple fact about the existence of a basis of Z-modules. If M is

1.2. Algebraic number fields 5

a Z-submodule of finite type in Q, then we can find t e Q such thatM = tZ = {mt | m e Z}. In fact, M can be written as M = tiZ+t2Z+- • -+trZ.Thus M has a unique minimal positive element t provided M * 0. For eachO ^ m e M, by the Euclid division algorithm, we can write m = tq+r with thequotient q e Z and remainder r with 0 < r < t. The remainder has to be 0 bythe minimality of t. Thus M = tZ. If V is a one dimensional Q-vector-space,we can identify V with Q by taking a basis of V. Thus any Z-submodule offinite type in V is generated over Z by one element. Now let V be aQ-vector-space of finite dimension n. Let M be a Z-submodule of finite typewhich contains a basis of V. Then we can find a basis {ti, ..., tn} of V suchthat M = Z^Zt i . This can be shown by induction on n as follows. Taking asubspace W of V of dimension n-1, MflW is of finite type over Z [Bour3,VIII.2]. By induction, MflW has a basis t i , . . . , t n - i , so M/MflW is aZ-submodule of finite type in the one dimensional space V/W and hence has aunique generator over Z. Thus taking any tn e M whose image in M/MflW isa generator, we see that {ti, ..., tn} gives a basis of M.

§1.2. Algebraic number fieldsIn this section, we denote by F a number field and by O the integer ring of F.By a number field, we mean a finite dimensional field extension of the rationalnumber field Q. Then O is the set of all elements in F satisfying a monic inte-gral polynomial equation. Since Z is a principal ideal domain, O is of finitetype as a Z-module ([Bourl, V.1.6]), and we know that

(la) O is a ring and has a basis {©i, ...,©d} over Z for d = dirriQF.

An O-submodule a* {0} of F which is of finite type over O (i.e. finitelygenerated over O) is called a fractional ideal of F. A non-zero O-submodule ais a fractional ideal if and only if O~z>Xa for some X G Fx. By this fact, a is offinite type over Z and a®zQ = F via a®b h-> ab. Taking a generator oci ofQ©ifk whose existence is assured by the above claim, we consider the quotienta/Za\. We see that dirriQ(d/Zai)®Q is one less than that of a®Q. Thus byinduction on the dimension, we have

(lb) Any fractional ideal a has a basis {oci,..., 0Cd} over Z.

Fixing a basis {©1, ..., ©d} of O over Z, which is also a basis of F over Q,

we can express for a e F, a©i = Zj=1aji©j with aij G Q. In parcular, aij G Z if

a G O. We now define p(a) G Md(Q) by

1: Algebraic number theory

(acoi,...,acod) = (coi,...,a)d)p(a) with p(a) =

a n a i 2 ••• a id

a21 a22 ' " a2d

ad2 •"

Md(Q).

Note that p(a) = aid if a e Q, where Id is the dxd identity matrix. We seep(ab) = p(a)p(b) and thus p : F -» Md(Q ) is a Q-algebra homomorphism,which is called the regular representation of F over Q, and a is a root of thecharacteristic polynomial det(Xld-p(a)). We define

Tr(a) = TrF/Q(a) = Tr(p(a)) and tfF/Q(a) = tf(a) = det(p(a)).

We know that the trace Tr is Q-linear and the norm N is a multiplicative map(i.e. N(ab) = N(a)N(b) and N(l) = 1). When K/F is a finite extension and{wi, ...,wr} is a basis of K over F, then obviously {a)iWj}i=i,..Md,j=i,...,r givesa basis of K. The regular representation pK of K with respect to this basissatisfies

pK(a) = p(a)®lr for the rxr identity matrix lr for all a e F .This shows that Afc/Q(a) = NF/Q(a)[K:F].

Let / be the set of all fractional ideals of F. We define the product aB ofa, Be I by aB = {Zj â^bi: finite sum | X{ e O, ai e a and b[S 5}. The setaB is clearly an Osubmodule of F. If OL[ (resp. (5j) generates a (resp. 6), thenociPj generates aB and hence aB is finitely generated; i.e., it is a fractional ideal.We note that

(lc) / is a group with the identity O under the above multiplication.

This follows from the following lemma:

Lemma 1. For a given fractional ideal a, there is another fractional ideal 6

such that aB = aO for a e Fx.

By the lemma, a1 = a'lB and / is a group. We first prove

Sublemma. Let f(X) = IôsaXm-[ and g(X) = l^bpC1^ be polynomials with

coefficients in O faobo * 0). Let 0*Xe O. If all the coefficients of f(X)g(X)

are divisible by X in O,then aibj is divisible by X for all (i,j).

Proof. We claim that if all coefficients of a polynomial P(X) are algebraic inte-gers, then for each root % of P(X) = 0, all the coefficients of the polynomialP(X) P(X)~—r are algebraic integers. In fact, if P(X) = aX+b, then ——r = a as aX-q A-q

1.2. algebraic number fields 7

polynomial and hence the assertion is true in this case. Now we complete theproof by induction on deg(P(X)). Write deg(P(X)) = n and write a for thecoefficient in Xn of P(X). Then P(X)-a(X-^)Xn"1 has degree less than n and

hence by the induction hypothesis, ——r - aXn-1 has (algebraic) integer

P(X)coefficients. Thus ——r itself has (algebraic) integer coefficients. This proves

A-qthe claim. Write £l 9 ..., £n for the roots of P(X), i.e. P(X) =a(X-î)(X-£2) (X-^n); then, by the above claim, aIIiGA(X-£i) has integralcoefficients for any subset A of {1, 2, ..., n}, that is,alliG A(X-Î) I x=o = ialTie A i is an algebraic integer. Returning to the lemma,we apply this fact to the roots of f(X)g(X)A, = 0. Let £i,..., ^m be all the rootsof f(X)=0 and T|i, ..., rjn be all the roots of g(X). Then by the aboveargument, for any subsets A of {1, 2, ..., m} and B of {1, 2, ..., n},

B'ni is an algebraic integer. On the other hand, ai =A/

ab-±S#(A)=iaoIlaeAâ and bj = ±S#(B)=jaor[pGB'np- Thus -j1 is the sum of several

algebraic integers of the form -^niGA^riiGB'ni, anc* n e n c e is a n algebraic

integer, i.e., aibj is divisible by X.

Now we prove the lemma. Let a = aiZ+-"+a<iZ. We may assume that a is in-tegral. Let a ^ (a/1^ = ai) be all the conjugates of ai and put

f(X) = aiXd-1+a2Xd"2+---+ad and g(X) = n J 2 (

Since g(X)=7VF/Q(f(X))/f(X) = If=0biXn-i6 O[X] and f(X)g(X) = j

e Z[X], we can think of the greatest common divisor d of all the coefficients ofg(X)f(X). Let 6= bo0+biO+---+bn0 which is an ideal of 0. Then by the sub-lemma, aibj is divisible by d (i.e. aibj e 6.0) and hence do 3 ab. But by defi-nition, d is the greatest common divisor of Ci, ..., cm and hencedZ = CiZ+c2Z+—K;mZ which is contained in ab. This shows abzDd.0 and soab = do.

Theorem 1. Let <P be the subgroup of I generated by principal ideals aO for

a E Fx. Then the index of £ in I is finite.

The number h = h(F) = (I:(P) is called the class number of F. The quotientgroup CIF = I/(P is called the class group of F. Before proving the theorem, weextend the norm map to ideals. Let a*0 be a fractional ideal. Choosing a basisa = (ai,...,(Xd) of a, we can identify a=Zd. Since a is also a basis of F overQ, we can identify F with Qd via a. Then FR = F®QR can be identified with


the formal real span of a and thus FR = Rd as real vector spaces. That is,FR/# = (R/Z)d, which is compact. For any point x e F, we can find a smallneighborhood U in FR SO that U is isomorphically projected into FR/O. Thus,for any fractional ideal 6 containing a, the image of B in FR/# is discretebecause B has only finitely many generators over Z. Since FR/# is compact,61a is a finite group. We now define for all fractional ideals a

Two ideals a and 6 are said to be relatively prime (or a is prime to F), ifa+6 = O. We then have an exact sequence

0 -> cC\B -> O -> (O/a)®(O/6) -> 0x h-> (x mod a)0(x mod 6).

Thus N(cC\S) = N(a)N(B) if a+£ = 0. Since Oz> a, multiplying by B, we see that6= BO ID aB. Similarly we see that az> aB and hence cf]Bz> aB. If a+B= 0, wecan find a e a and b € £ such that a+b = 1. Then we see that x = xa+xb forany x e flfl£. Thus x e aB. This shows that cC\B = aB if a+6 = O. That is, wehave

(2a) N(a£) = N(a)N(S) if a+£ = a

Let L and L' be free Z-modules of finite rank. Let ( , ) be an inner product onLxL' which is non-degenerate on both sides. Thus rankz(L) = rankz(L'). Let eibe the smallest positive integer in {(x ,y) |xeL and y e L ' } . We take xx andyi such that (xi,yi) = ei. Let Li = {w e L | (w,yi) = 0}. Then L = xiZ©Li.By definition, xiZflLi is orthogonal to everything and hence is reduced to {0}.Let Z E L with (z,yi)^0. By the division algorithm, (z,yi) = eiq+r with0 < r < e i . Then (z-qxi,yi) =r. By the minimality, r has to be 0, that is,z-qxi G Li, or in other words, xiZ+Li = L. We also know that (L,yx) c eiZ.Similarly L' = yiZ©Li' for Li' = { w e Lf | (xi,w) = 0}. Repeating this process,we find two bases xi, ..., xa of L and yi, ..., yd^ L' such that ei = (xi,yj)8ijfor the Kronecker symbol 8ij and ei I ei+i. We can apply this argument to theinner product (x,y) = xAly for x,ye Zd and a matrix A e Md(Z) withdet(A)^0. Then we see I det(A) | = det((xi,yj)) = Ef=1ei = #(Zd/ZdA).Applying this argument to A = p(a) for a e O, we know that

(2b) iV(aO)= I//(a) |.

Thus N: / - ^ Q x coincides with the norm map on P up to absolute value.Thus N is a homomorphism of multiplicative group T. Now take an ideal a


and its basis a = (ai,...,0Cd). We identify a with Zd via this basis. We thendefine the regular representation p f : F -> Md(Q) by

aoc = (accOi = Xj=iaJiaJ = aP'(a) w i t h P'<a) = (ay)-

If we write a = coA for a basis co = (coi,...,O)d) of 0, we see Ap'(a)A"1 = p(a).Thus N(a) = det(p'(a)). Then applying the above argument replacing O by a,we see that(2c) I N(a) I = #(fl/aa) = N(a0).

Finally from the equality #(O/aa) = #(O/a)#(fl/aa), we conclude that

(2d) N(?La)=N(aO)N(a).

Combining (2a-d), we see that

(3a) N: / -> Q+ = {a e Q | a > 0} is a homomorphism of multiplicativegroups.

Indeed, for any integral ideals a and 6, we can find a e 0 such that aa+£ = 0.Then we see that

and hence (3a). Since A^K/Q : /—» Q+ for any finite extension K/F is a multi-plicative map satisfying NK/QÔK) =A^F/Q(aO)[K:F-5 for a e Fx, we see easilythat(3b) for fractional ideals a of F,

To prove Theorem 1, we first prove the following lemma:

Lemma 2. There exists a constant C depending only on F such that any ideala in O contains a non-zero element a e a with \N(a) I < CN(d).

Proof. Let O- Zcoi+---+ZcOd. Let g be the greatest integer [^JN(a)] not ex-

ceeding ^]N(a). Consider the set of numbers

S = {niCQi+—+ndcode o\ 0 < n i <

Then #(S) = (g+l)d > N(a) = #(O/a). Thus there are at least two distinct elementsP, y G S such that p = y mod a. That is 0 * a = p-y = mi©i+- • •+mdC0d withI mi | < %lN(a). Let M = maxij( | coi

(j) |) . Then

|N(a) | = Yl*=l | micoi(j)+.• •+mdcod(j) I < ddMdA^(a).

We can take ddMd as C.

We now give the proof of Theorem 1 (which is due to Hurwitz). Let cbe a class in II(P. Take an integral ideal a in c"1 and choose a as in the


lemma. Since a => ocO, we can find another integral ideal 6 such that aB = ocO.Thus Be c. On the other hand, N(a)N(B)= \N(a)\ <CN(a). Thus N(£)<C.Thus every class c contains an ideal of norm less than C. The number ofintegral ideals of norm less than C is finite because the number of submodulesof Zd with index less than C is finite. Hence h is finite.

By the above proof, the constant C gives a bound of integral ideals withminimum norm in each ideal class and hence is very important. For example, ifone can show that all integral ideals with norm less than C are principal, thenh = 1. The constant ddMd given in the proof of the lemma is not the bestpossible bound. The following estimates of C are well known:

(4) Minkowski's estimate: C < UD^\ t d* (see below);

- I | Dp I when F is totally real [Si];

here Dp = det(Tr(o>iCOj)) for a basis {coi,..., C0d} of O is the discriminant of

F/Q, t is the number of complex places of F and r = - or - +1 according

as d = 1 mod 6 or not. (Here [a] is the greatest integer not exceeding a.)

Exercise 1. Let&+= {ao\ oce F, oca>0 for all field embeddings a of F into R},

I(m) = {5 = - I n and d are integral and prime to m)

for a given ideal m, and H?+(m) = &-f){aO \ a e Fx, a = 1 modxm}, wherea = 1 modx7rc means that aO e I(m) and there exists |3 e fP+ such thatpOe /(/n), (3G O, a p e O and a p ^ p m o d w . Then show that &+(m) is asubgroup of finite index in I(m).

The finite group CLp(/n) = I(m)/^P+(m) is called the strict ray class group modulonu

Exercise 2. (a) Using (4), show that the class number h(F) = 1 for the follow-

ing fields: Q(Q, CKC+C'1) for C7 = 1 but £ * 1, and Q(3V2).

(b) Show that h(F) > 2 for F = Q(V:5).

Exercise 3 (Minkowski). Using the estimate (4), show that | Dp | > 1 for any

number field F & Q.

An ideal p of O is called a prime ideal, if the residue ring O/p is an integraldomain (i.e. having no zero divisor). When a prime ideal p is non-zero, O/p is


a finite integral domain. Thus for every 0 * a e O/p, the sequence of elementsa, a2, a3,..., a11,... in O/p has to overlap. Thus for some i > j , aâ^. Soai-j _ i Thus we can always find a positive integer h such that ah= 1.Therefore a"1 = ah-1 G O/p. Thus every non-zero element of O/p has an inverseand hence O/p is a finite field. Since ideals of a field are either {0} or the totalring, there are no proper ideals in O containing p. Thus p is maximal. Bydefinition, the ring O is normal (i.e. any element in F integral over O belongsto O). A normal integral domain whose non-zero prime ideals are maximal iscalled a Dedekind domain (for further study of such rings, see [Bourl, VII.2]).For any ideal a, by Zorn's lemma, there exists a maximal ideal p containing a.

Then p'la is again an ideal of O. In fact, multiplying pz>a by p"1, we getO = pp~l 3 ap~l. Thus a=pB with an ideal B of O, and we have N($)<N(a).

Therefore, inductively, we can decompose a = IL • / ^ for finitely manymaximal ideals p. We now claim as follows:

Theorem 2 (Dedekind). Each fractional ideal a can be decomposed uniquely asa product a = Ylp primepe^ of finitely many prime ideals (allowing negativepowers). The ideal a is integral (i.e. is contained in O) if and only if e(p) > 0for all p.

Proof. The ring O/p has no zero-divisors, for if a,b G O and ab G p, then eithera G p or b G p. We now show for ideals a and B of O, if a prime ideal pcontains aS, then either pz> a or p z> B. We may suppose that a is not con-tained in p. Thus we have a G a with a e p. Since for each b G £,pi) aS & ab, we see b G p and hence pz> & We now suppose that an ideal a inO has two prime decompositions a = p-^p2- • *pm = qx #2"' #n lowing repetition ofprimes. Since either p a o r p ^ if p 3 a£, by renumbering the prime ideal^ ' s , we may assume that p1 3 qv Since q^ is a prime ideal, it is a maximalideal, and hence px = qv Dividing both sides by pv we get a new identityVi 'Tm = #2*"&i* Repeating the above process, we finally know that m = n andpi = ^ for i = 1, ..., n after renumbering the ^ 's . This shows the uniqueness ofthe prime decomposition. By definition, each fractional ideal / can be written as

af- - for integral ideals a and S. By the uniqueness of prime factorization for

ba and fB and 6, the ideal / has a prime factorization which is unique.

If p and q are distinct maximal ideals of O, then p+q = O. Taking p G p andq G q such that p+q = 1, for any given positive integer n, we can find a large Nsuch that 1 = (p+q)N e pn+^1. Thus pn+<fl = O. Similar reasoning shows thata+6= O if and only if there are no common prime factors in the prime factoriza-tions of two ideals a and 6. We say that two fractional ideals a and B are rela-tively prime if in the prime factorizations of a and B there are no common


prime factors in both the denominator and the numerator. We write (a,S) = 1when a and S have no common prime factors.

Let ja(F) be the group of all roots of unity in O. Let Oi,..., a r : F -» R denoteall distinct real embeddings and choose complex embeddings Gr+i,...,Gr+2t: F -> C so that {or+i, ..., Gr+t,ar+t+i = cor + b ..., ar+2t = car+t} makes theset of all complex embeddings of F into C, where c denotes complex conju-gation. We simply write a® for a?1 for a e F. For each fractional ideal a ofF, taking a basis a = (ai , . . . , ad) of a over Z, we find the following formula:

(5) I det(ccj(i)) | = for D = DF.

To see this, we embed F into FR = RrxCl by i : a n (aw)i=i,...>r+t. Then we de-

fine a measure dji on FR by ®ir=1dxi®t

=11 dxjOdxj | , where for a variable

z = x+iy on C (x, y e R), |dz®dz| = 2dxdy. Then for d(=r+2t) linearly

independent vectors VJ = (xij)i==i>...>r+t e FR, it is an easy computation to see that

the volume with respect to d|i of the parallelotope V = V(vi, ..., Vd) spanned

by vi, ...,Vd is given by

1,1 x l ,2

xr+t,l xr+t,2

T+1,1 xr+l,2

•r+t,l xr+t,2

xl,d

xr+t,d

xr+l,d

xr+t,d

Thus writing

A = A(a) = A(a) =

u ia(2)

!

, a l

a21 }

^(2)U 2

a2

- a^n(2)^ d

fy(d)

ad j

we see that | det(A) | is the volume of V(i(ai),...,i((Xd)), which is the funda-

mental domain for F R / # . Thus we see N(a) = [CKa] = r. Since

AlA = (Tr(aiaj)), we get (5).

|det(A(O))|

Now we want to determine the structure of the unit group (?. Let |Li(F) be the

group of all roots of unity in O. We simply write oc^ for a*1 for a e R First

we prove


Lemma 3 (Kronecker). For an integer a <= 0, if \ athen a is a root of unity.

= 1 for all i = 1,..., r+t,

Proof. We can identify F ® Q R with RrxCl so that the projection of F to thei-th factor R or C is given by Oi (see (1.5b)). Since a basis of O over Z is abasis of F over Q and hence is a basis of F ® Q R over R, O is a (closed) dis-crete Z-submodule of RrxCl (because identifying RrxCl with Rd via the basisof O over Z, we know that the image of O in R is Z ). On the other hand,S = {(xi)e RrxCl | |x i | = l } = {±l}rxSit (with the circle Si of radius 1 in C)is a compact set. Thus CHS is a discrete and compact set and hence is finite.Let |1 = {XG O\ | x ^ | =1}. Then \i is a subgroup of O* and is contained inSflO and hence is a finite group. Thus ja is made of roots of unity.

Lemma 4 (Minkowski). Forgiven n-real linear forms Li(x) = S=1aiqxq on

Rn, suppose that D = det(apq) ^ 0. Then for any positive numbers Ki,..., Kn

such that II^Ki > ID |, there exist m = (mi, ..., mn) e Zn with not all mi

zero such that \ Li(m) | < Ki for all i.

Proof. Consider the parallelotope PQ = (x e Rn | |Li(x)| <y for all i}.

Similarly we consider the translation

Pm = {x G Rn | | Li(x-m) | <j for all i}

whose center is an integer vector m. We can give the following illustration ofthe situation when n = 2:

If Pm and Pm. form^m 1 have non-trivialintersection, then for x GPJiPm1, we have

m)| < y and

|Li(x-m')| <-y for all i.

f T S o w e s e eLx(x) = -1 Lj(x) = 0 I Li(m-m') I < Ki. Thus

what we need to prove is that if IIf=1Ki > ID |, then Pm and Pm- intersect for

some m ^ m1. We shall show that if Pri/lPm' = 0 for all m ^ m1, then

Ilf=1Ki < | D | . In fact then the volume J of the Pm's in a square

S ( L ) = { X G Rn | |xj | <L} for any given L is less than 2nLn. Let

c = Maxxep0{ | xi |,..., | xn | }. Then for m e Zn, | rrii | < L if and only if


S(L+c)z>Pm. There are (2L+l)n such m's for each integer L. This shows that

(2L+l)nvol(P0) < 2n(L+c)n. By making L large, we know that vol(Po) ^ 12n(L+c)n

since lim — — = 1. On the other hand, we see thatLôo(2L+l)n

vol(Po) = J- '-Jp dx1dx2---dxn

^ Kn.

This shows that Ki Kn < | D | . By our assumption, PmnPm' = 0 for allm * m\ Po does not contain any non-zero integer point. Since Po is a compactset, and Zn-{0} is a closed discrete set, 8 = Min (| xi-yi I,. . . , I xn-yn |)

xePo.yeZMO}

exists and is a positive number. Thus if 0 < e < 8, the system of inequalities|Li(x)| <Ki+e has no integer solutions. Thus

Ki Kn < (Ki+e) (Kn+e) < ID |,which shows the desired assertion.

Corollary 1. Let a be an ideal of O. There exists a constant C depending only

on F such that for any given positive numbers Ki, ..., Kd such that

IIi=1Ki > CN(a) and Ki = Ki+t for r < i < r+t, we can find a non-zero element

as a such that | ocw | < Ki for all i.

Proof. When all the field embeddings O[ (i=l, ..., d) of F into C actually

take F into R (i.e. F is totally real), then we apply the above lemma to linear

forms Li(x) = Sj=10Cj Xj for a basis cci,..., 0Cd of a over Z. Then (by (5)),

I det((Xj(l)) | = N(d) | VD | for the discriminant D = DF of F and hence is

non-zero. Let C = £+1 VD | for some e > 0. Then for any set of positive num-

bers Ki, ..., Kd with Ki Kd ^ CN(a), we can find slightly smaller K'i< Ki

such that Ki Kd > CN(ri) > K'I Kfd > N(a)|VD|. Applying the lemma to K'i,

we find O ^ m e Zd such that |Li(m)| < K'i < Ki. Then for 0 * a =mi0Ci+***+md0Cd s a, I a^ I <Ki. When some embedding of F into C is not

real, we apply the lemma to the constant K'i = Ki for 1 r . Then

the desired assertion is true again for C = £+|VD1 for any e > 0.

Exercise 4. (a) Write down all the steps of the proof of the above corollarywhen F is not totally real.(b) By using the argument of the proof of Lemma 4 and Corollary 1, show that

we can take |VlD | as the constant C in Lemma 2.


Now we prove the Dirichlet unit theorem:

Theorem 3 (Dirichlet). There exists a free Z-submodule E of rank r+t-1 in0* such that C? = ji(F)xE, where we consider the multiplicative group E asan additive Z-module.

Proof. First we shall show that the multiplicative group cf contains r+t-1 mul-

tiplicatively independent elements (i.e., if one writes the group cf additively, we

shall show that in b*, there are r+t-1 linearly independent elements). Fix a ba-

sis C0i,...,C0d of O over Z. By Lemma 4, we have a positive constant C so

that we can find an integer vector m e Zd such that | S^mjCGi^ | < iq for all i

if Ki Kd > C; thus, we can find a non-trivial algebraic integer O ^ a e O suchthat I a ^ I < Ki for all i. In particular, applying this to Ki = *Vc for all i, wecan find 0 * a e 0 such that |N(a) | < C. Then we redefine Ki = | oc(l) | fori > 1 and Ki = CI a(1) | /1 N(a) | and applying Lemma 4, we can find 0 * cci e

0 such that | a i ( 1 ) | < C -i r and | ai ( l ) | < | a(l) | for all i > 1. In the same|iV(a)|

manner, we can find inductively 0 ^ an G O such that

locn(1)| <c|(Xn-i(1)l/l^(ocn-i)l and |ocn(i)| < |an.!( i )l f o r a l l i > l .

Then

I N(a } I - n d I nL.^ I < c ' a n A ! ' n d I rv , a) I - r\ i y / \ S ^ n ) \ — AJi_i I tX-n I I Ar/ |11^_2 I ^ n - 1 I — v—

Since the number of ideals in O of norm less than C is finite, there are integers0 1. Similarly, we can find a unit £j (j = 1, ..., r+t-1) such thatI £j(l) | < 1 for all i * j and 1 1. Now we want to showthat £i,...,er+t_i are multiplicatively independent. That is, we want to show thatthe relation(6) Ceini es118 = 1 for C e |H(F) and s = r+t-1

holds only when C, = 1 and ni = 112 = ••• = ns = 0. For a e F, we write/i(a) = log(I a ( i ) I ) for l < i < r and h(a) = 2log(\a(i)\) for r< i<r+t . Thenthe multiplicative independence of £j is equivalent to

(7) de t (R)^0 for R =

/s(£2) ... /S(£S)J


Since | E J ( I ) | < 1 for all i * j and l 1, we know that

Zi(ej) < 0 if i * j and /j(ej) > 0 and

Generally det(aij) ^ 0 under the following three conditions:

(i) aij < 0 for i ^ j , (ii) an > 0 for all i and (iii) ^ S ,aji > 0 for all i.

In fact, if xA = 0 for A = (aij) and a non-trivial (xi, ..., xs) e Rs, then we see

that aipXi+---+aPpXp+---+aSpXs = 0 . However, from 5L1ajp>0, we know that

a p p > -(aip+---+ap.ip+ap+ip+-"+aSp) and from app = | app | and | aip I =-aip for

i & p, we know that

I appxp | > -(aip+- • •+ap.ip+ap+ip+- • -+asp) I xp | if xp * 0.

Then taking the index p so that I xp | = max( | X[ | ) ,

I appxp | > | aip+- • •+ap-ip+ap+ip+- • -+asp I I xp |

^ I aipxp | +• • •+ | ap_ipxp | + | ap +ipxp | +• • •+1 aspxp | (because aip = -1 aip | )

> | aipXi | +• • •+ | ap_ipXp_i I +1 ap+ipxp+i | +• • •+ | aspxs |

| - • •+ap.ipxp.i+ap+ipxp+i+- • -+aspxs | ,

which is a contradiction. Thus A is invertible. We conclude that the only

possible solution of (7) is ni = n2 = ••• = ns = 0 and hence £ = 1.

Next we shall show t h a t H = {ein i . ---esn s I n = (ni, ..., ns) e Zs} is of finite

index in cf, which finishes the proof of the theorem. We consider e = (/i(e), ...,

/s(e)) e Rs for e e 0*. Since R is invertible, we can find a solution x e Rs

such that Rx = e; that is

18( i ) | = | (ei(i))Xl (es

(i))Xs I for all i < s-1.

However, this is true even for i = s because | N(e) \ = |iV(ei)| = 1. Letni = txi] be the maximal integer not exceeding Xi and define r\ = eini.---*es

ns.

Then | (en"1)^ I < Sup ((ei(i))Xl (es(i))Xs) = M, which is independent of the

0<xi<l

choice of e. The number of integers with | a( l ) | < M for all i is finite and

hence (O*:H) < +oo.

Exercise 5. Show that for any proper subset S ^ 0 of I = {1, 2, ..., r+t}, there

exists a unit e e (? such that I £(l) I < 1 for i e S and I £(l) I > 1 for any

i e I-S.

1.3. p-adic numbers 17

§1.3. p-adic numbersWe briefly recall the construction and properties of p-adic numbers in this sec-tion. We refer to [Kb] for more details. Recall the construction of real numbersout of rational numbers using Cauchy sequences. A sequence {an}nEN of ratio-nal numbers is called a Cauchy sequence if for any given small positive e, thereexists a large integer M such that if m,n > M, then | am-an | < e. Then definingan equivalence relation on the totality C of Cauchy sequences by {an} ~ {bn} ifI an-bn I -> 0 as n -> °°, we define the set of real numbers R to be Cl~. Byassigning the constant sequence {a} to each rational number a, the set ofrational numbers Q is embedded into R. The multiplication (resp. addition) oftwo Cauchy sequences {an} and {bn} is defined to be the Cauchy sequence{anbn} (resp. {an+bn}). This well defined ring structure induces a field structureon R which makes R an extension field of Q. By construction, any Cauchysequence of R has its limit inside R.

Obviously we can perform the above process of completing Q for any absolutevalue II || which is a function on Q having values in the non-negative real line(and may be different from the usual absolute value | I) satisfying

||a|| =0 <=>a = 0 and ||a+b|| < ||a| + | b | .

We can extend the absolute value to the completion by b = lim bn for the

number b represented by the Cauchy sequence {bn}. We list some examples ofdifferent absolute values in a somewhat more general setting. Let F be anumber field and O be its integer ring. We fix a prime ideal p * (0) of 0. Forany O ^ x e F, we decompose the ideal xO into a product of (non-zero) primeideals in O as guaranteed in Theorem 1.2.2. Let v(x) = vp(x) be the exponent ofp in xO. When the prime ideal p does not show up in the prime decompositionof xO (i.e. xO is prime to p), we simply put v(x) = 0. We also put v(0)=oo(i.e. (0) is divisible by p infinitely many times). Then we put | x | p = N(p)'v^x\Then x h-> | x | p is a function having values in the non-negative real line,| x | p = 0<=>x = 0 and | x | p = 1 <=> xO is prime to p. It is easy to check the

following fact which implies the triangle inequality || a+b || < | a || +1| b ||:

(1) | x+y I p < max( | x | p, | y | p), and the equality holds when | x | p * \y\F

Let us now compute I n! | p for a given prime p and n e N. Let m be aunique integer such that pm < n < p m + 1 . We define a sequence of integers0<ni<pm + 1"1 and 0 < a i < p for i = 0, I, ..., m inductively as follows. Firstwe put no = n. We define ao e N as the quotient of no divided by pm and nias the remainder: no = aopm+ni. Then we repeat this process replacing no by ni


and define ai and n2 by ni = aipm"1+n2. Thus after having defined ai andni+i, we define ai+i and nj+2 by n^â^jp111"1"1+ni+2. Then we have

n = aopâip"1"^—Ham with 0 < &{ < p.

Let s = ao+ai+---+am. Then we claim that vp(n!) =—r. Writing kj for the

number of multiples of p1 in the integers from 1 to n, we see that lq is the

largest integer not exceeding —, i.e.,P

and thus

Thus we have

(2) | n ! | p = p(s"n)/(p-1) for n e N.

Here is another example. Let a : F —»C be a field embedding. Then we define| x | a = | x ° | using the usual absolute value | | on C. Then obviously | | a

is an absolute value. We denote by Z the set of all absolute values on F. Twoabsolute values are said to be equivalent if they give an equivalent topology onF. Here the topology associated to a norm | | is given by the metricp(x,y) = I x-y |. Each equivalence class of absolute values of F is called a place.Thus we have attached a place both to each maximal ideal p and to eachcomplex embedding a. It is known that the set of places of F corresponds bi-jectively to the disjoint union of the set of all (non-zero) prime ideals of O andthe set of all complex embeddings of F modulo right multiplication by complexconjugation (see [Bourl, VI.6.4] or [Wl, I]). Anyway, we hereafter identify theset of all places of F with the disjoint union of the set of all (non-zero) primeideals of O and the set of all complex embeddings of F modulo rightmultiplication by complex conjugation c.

Exercise 1. Show that I I a and | | T are different as places of F if the twoembeddings a and x into R are different.

Let I I v be a place of F. Then v is either a complex embedding or a primeideal of O. We perform the completion process (sketched for Q for the usualabsolute value) for this absolute value | | v- The resulting field will be denotedby Fv. Thus Fv can be identified with the equivalence classes of Cauchy se-quences under | | v , and there is a natural embedding of F into Fv. Whenv = p, Fp is called the /?-adic completion of F. When F = Q, any (non-zero)prime ideal of Z is spanned by a unique positive prime number p. In this case,


we write Qp for Q(p). The closure of Z in Qp is called the p-adic integer ring,denoted by Zp.

Let n be an integer. Then we can find a unique integer [n]p such thatn = [n]p mod p and 0 < [n]p < p. Let ao = [n]p, and applying this process to

—^k, we define ai as n p . We iterate this process and define, after* L J p

having defined am, am+1 = r ^J p

{bm = ao+aip+a2p2+---+ampm}mez

is a Cauchy sequence converging to n under | | p, and we can writen = Sm=oampm • We can inductively solve the formal equation

{Z°° rtampm}{Z°° xmpm} = 1 if ao is prime to p (i.e. n is prime to p).m=0 m=0

By definition, every p-adic integer z in Zp actually has such an expansion

z = Z°° an(z)pn for unique integers an(z) with 0 < an(z) < p. In particular, z isn=0

invertible in Zp if and only if ao is prime to p. Thus all the ideals of Zp areexhausted by pnZp and they are all distinct. We can writepnZp = {x e Qp | | x | p < p"n}. Thus Zp is a valuation ring and hence is aprincipal ideal domain. In particular, all finitely generated torsion-free modulesover Zp have a basis (i.e. are in fact free).Let Op be the closure of O in F^ Taking an element G5 in O with v (G5) = 1and fixing a representative set R in O for O/p, we can expand in the samemanner as in the case of Zp any ze Op into a unique expansion

z= lZ=o^z)mn with an(z)G R-Thus again all the ideals of Op are of the form U5nOp for some n (allowing

n = oo), and therefore Op is a valuation ring. The above process of expanding

ze Op into a power series of G5 can be performed in the finite ring O///11 and of

course in this case the series is a finite sum: zm = 5^~oan(zm)G3n for

Zm = z mod y11 with an(zm) = an(z) mod pm. Then the sequence {zm e O/f^}m

satisfies zm = Zn mod pn whenever m > n. Thus z naturally determines an

elementz=limzm in lim(O/pm).

m m

On the other hand, for any given element z. =limzm, we can recoverm

z = ^n=oan(z)n3n in Op because {an(zm)} uniquely determine {an(z)}. Thuswe have an expression


(3a) Op = \im(0/pm) and Zp = lim(Z/pmZ).m m

It is easy to check that the topologies of both sides coincide (see [Bour2,1.4.4,III.3.3, III.7], [Bourl, IH.2.6]). In particular, this shows that

(3b) O^pmOp = 01 f1 and Zp/pmZp = Z/pmZ.

For any z e Op, we simply put z™ = 2^="oan(z)G3n e O. Thus we can always findan infinite sequence in O converging to a given z in Op; i.e. O is dense in Op.When O=Z, an(z) is always non-negative by our choice, i.e., z^ e N for allm. Thus

(3c) the set of natural numbers N is dense in Zp and O is dense in Op.

Exercise 2. Let q = #(O/p) for a (non-zero) prime p of O. Show that for an

integer x in O, (i) {xqn}nGN is a Cauchy sequence in C ,, (ii) if x is prime to p,

C,= limxqn is a (q-l)-th root of unity, (iii) if x is prime to p,

| £_xqn4(q-i) | p < p-n a n ( i ^ a l l t j i e (q. i ) .^ r o o t s of u n i t v ^ e obtained in this

way.

Naturally Op contains Zp for a unique prime p in Z such that pf]Z = pZ.Thus Op is the integral closure of Zp. The natural homomorphism of 0®zZp

to Op is surjective, because 0®zZp is compact and its image contains the densesubring O of Op. Thus Op is finitely generated over Zp, and we can find abasis {wi, ..., Wf} (f<r) of Op over Zp. In particular, for any a e Op

defining a matrix p(a) by (wia,...,Wf(x) = (wi,...,Wf)p(a), we know that asatisfies the characteristic polynomial of p(oc), which is a monic polynomial withcoefficients in Zp. Thus Op is integral over Zp. We now show the converse. Itis well known that any valuation ring is normal (i.e., if x e ¥p satisfies a monicequation with coefficients in Op it belongs to Op). In fact, writing x = uG3"m

with I u I j , = 1 and m > 0, if xn = X i^x11"1 with aj e Op we have

ton+-+anG5m(n-1)

whose right-hand side belongs to Op. Hence m has to be 0. This shows thatx G Op and shows the normality of Op. In particular, Op is the integral closureof Zp. For this reason, Op is called the p-adic integer ring of ¥p.

Let K/F be a finite extension. Let OK be the integer ring of K. For a non-zeroprime ideal p of O, we decompose pOK = Tl^^ for prime ideals T of OK-Thus OK/fCk = n2>(OK/ê(5>)). Since as an 0-module, OK/2>= (O/p)m forapositive integer f(#), by (2.3b),


(4a) [K:F]

Then the ^P-adic completion R=OK,<P is free of rank e(£)fC?) because R/pR isof dimension e(fP)f(#) over O/p by (3b) and (4a). Similarly as in (1.5b), weknow that

(4b) K®FFP=YIT\Pi € R for sufficiently large integer a. Thus FR = E. This implies that, if

wi, ..., w r e R are O^-linearly independent, then r < d . This shows that R is

free of rank [E:Fp] over Op. Since R = Op, R is p-adically complete. If

p R = <Pi&l &<** for non-zero prime ideals ^ of R, then

R = lim(R//R) = fl lim(R/2fR),

because R/paR = n L ( R / ^ a e i R ) by the Chinese remainder theorem. Since Ris an integral domain, s has to be equal to 1 and R is a valuation ring. By def-inition, if we write pR = & and f = dimo/p(R/(P), then

(5) [E:¥p] = ef and | a \T= \ a | f:F] for a e F.

Thus we can extend | | p to E by putting | x | p = \ x | " -1 for x e E. The

norm | | p is the unique norm extending | | p on F and defines the same

topology as that given by | | #.

We now introduce the p-adic exponential function exp and the p-adic logarithmfunction log for later use. To define these functions, we use the followingpower series expansions:

_.n / 1 \n+l__n

(*) exp(x) = X~=Q fj- and log(l+x) = £~=1 H)_JL .

By the strong triangle inequality, any power series f(x) = Z°loanxn is convergent

at xe Op if and only if lim | anxn I p = 0. Here I I p is the normalized p-adic

norm: | x | p = | x | p' p . Thus the radius of convergence is given by

R = (lim sup U n l J V

(i.e. R is the largest real number which is a accumulation point of the sequence

{Unlp1/n}). We know that | n ! |p/n = p-(1"(s/n))/(p-1) by (2) for the integer s = sn


= Z°° am(n) for the finite p-adic expansion n = L°° am(n)pm. Note thatm=0 m=0

I s | < (p-l)(l+logp I n I) for the complex logarithm logp with base p. Thus

lim I — I = 0 and we haven-»«> n

(6a) The radius of convergence of exp with respect to I | p is p"1^'1^

As for the logarithm function, since I n | p < p"^logp'n' \

(6b) The radius of convergence of log(l+x) with respect to | | p is 1.

Writing D(x,r) = {y e Fp \ | y-x | p < r}, we define

exp : D(0,p"1/(p"1}) -> Opx and log : D(1,1) -» Op

by the p-adically convergent power series (*). Let p be 4 if p = 2 and p = p

if p > 2. Then for x e Z p with I x-1 | p < p \ log(x) e D(0,p"1/(p"1}), and wedefine xs = exp(s log(x)) for s e Zp. Using the formal identities in the powerseries ring, we can verify the following properties:

(7a) exp(x+y) = exp(x)exp(y) and log(xy) = log(x)+log(y),(7b) log(xs) = slog(x) for s e Zp,(7c) exp(log(x)) = x and log(exp(x)) = x,

Exercise 3. Give a proof of the above identities.

If f(s) = 2T_ an(s-a)n is a power series converging around a e Op, then its for-

mal derivative j-(s) = Z°°_1ann(s-a)n"1 also converges at a. This is clear from

the inequality lim sup | an | ^ lim sup I (n+l)an+i | , since | n+11 p < 1.

Writing f n) for the formal n-th derivative of f, we have

(8a) an = L^L for all n e N.

In particular, we see

(8b) ^ =

Let Q be the p-adic completion of the algebraic closure Q p of Q p under | | p

which is the unique extension of I l p on Q p . Let A = { x e Q | x | p < l } .

For any x e Qx, we see that x | x | p e Ax. Since Q p is dense in Q., for any

positive n and any x e Ax, we can find a finite extension Kn/Qp and xn e Kn

such that I x-xn |p < p"n. As seen in Exercise 2, co(xn) = lim xnp m ' is well


defined in Kn. Note that co(xn) is independent of n, for which we write co(x).Then we have

where q = p[K«:Qp]. Thus 1 co(x)-xqn(q-1} | p < p"n. Then for x e Ax, co(x) =

lim x1*1" is a unique root of unity with | x-co(x) | D < 1. We thus see that

where |i«, = {£ <= Qx | £N = 1 for some integer N prime to p},

p Q = { p r | r e Q} and r .o={xe Ax | | x - l | p < l } .

The projection to Too written as X H ( X ) is given by ©(x lx lp^x lx lp) . On

Too, log is well defined. We then extend log to Qx by

(9a) log(x) = log«x» (XG Qx).By definition, we still have(9b) logtxy"1) = log(x)-log(y).

We conclude this section by giving a brief explanation of the Frobenius element.Let F be a p-adic field with p-adic integer ring O. We fix an algebraic closureF of F. Let O be the integral closure of O in F. We write p and p for themaximal ideals of O and O, respectively. We also write F = O/p andF = O/p, which is an algebraic closure of F. We note here that F is not p-adically complete [BGR, 3.4.3]. By definition, every automorphism of F leavesO and p stable and induces an automorphism of F over F. Thus we have anatural homomorphism of groups:

p : Gal(F/F) -> Gal(F/F).

The kernel of p is called the inertia subgroup, which we write I. Note that allnon-zero elements of F are roots of unity. Then, it is easy to check that p issurjective by considering the extension F(Q for various roots of unity £([Bourl, VI.8.5]). There is a canonical generator n in Gal(F/F): rc(x) = xq forq = #(F). The element % is called the Frobenius element of Gal(F/F). If K/Fis a (finite or infinite) Galois extension of F (see [N, Chapter I] for the Galoistheory of infinite extensions) and if K is fixed by I, K is called unramifiedover F. If this is the case, on the residue field F' of OK, p induces a canonicalisomorphism Gal(K/F) = Gal(F'/F). Thus in this case, we have a naturally spec-ified element p4(7c) = Frob in Gal(K/F), which we call the Frobenius elementof Gal(K/F).

Now we suppose F to be a number field in a fixed algebraic closure Q of Q.We pick a maximal ideal p of the integer ring O of F. Let K/F be a (finite orinfinite) Galois extension of F inside Q. We pick a maximal ideal T of PKover p, where OK is the integer ring of K. Then a e Gal(K/F) naturally acts


on maximal ideals of OK- We denote by D = D((P/p) the stabilizer of <2 inGal(K/F). This group D(<P/p) is called the decomposition group for Tip. Bydefinition(10a) D((Pc/p) = cD(V/p)G-1 for each a e Gal(K/F).

Since a e D preserves P, a is continuous with respect to the #-adic topologyon K and hence induces an element of Gal(K<p/Fp). It is known that

(10b) D(T/p) = Gal(Kp/Fp ([N, IV.l]).

We write l(T/p) for the inertia subgroup in D((P/p). If l(T/p) = {1}, we say fP isunramified over p. If all the maximal ideals over p in 0^ are unramified overp, we call p unramified in K/F. If Tip is unramified, we can consider itsFrobenius element Frob( P/p) in D(T/p). By (10a), we know that

(10c) Frob(2*7p) = aFrob(^P/p)a'1 if Tip is unramified.

Thus the conjugacy class of Frob(T/p) is well determined by p, which we writeFrobr Sometimes, we identify Frob^ with Frob(^) , which is actually a rep-resentative of the Frobenius conjugacy classes. We quote the following wellknown density theorem:

Theorem 1 (Chebotarev). Suppose that only finitely many prime ideals in F areramified in K/F. Then under the Krull topology on Gal(K/F) (see [N, LI]), theset of Frobenius elements

Z = {Frob(ffl^) e Gal(K/F) | <Efp unramified in K/F}is dense in Gal(K/F).

We do not prove this theorem here, because we need a large amount of eitheranalytic number theory or class field theory to give a proof. We only refer forthis to [N, V.6].

Chapter 2. Classical L-functions and Eisenstein series

In this chapter, we will deal with basic analytic properties and some alge-braic properties of abelian L-functions in a classical setting. To give an illustrationof what we will do, let us start with a typical example of such results.

§2.1. Euler's method of computing ^-valuesIn the following, a continuous C-valued function f defined in an open subset Uof C is called holomorphic (or analytic) on U if f is differentiable on U and

— = 0 for the variable z on U. By Cauchy's theorem, any holomorphicdzfunction on U can be expanded into a power series of z-u convergent absolutelyon an open neighborhood of each u e U. A function defined on a dense opensubset V of U is called meromorphic if there exist two holomorphic functions gand h * 0 such that f = f on V.

The simplest example of an L-function is the Riemann ^-function given by the

infinite sum

C(s) = X I - i n S for complex numbers s € C.

This infinite sum converges absolutely if Re(s) > 1 and gives an analytic func-tion in this region. In fact, it is clear that

I C(s) I < C(Re(s)) < l + J~x"Re(s)dx

and the integral converges absolutely if Re(s) > 1. Moreover we shall find in

§2.2 an analytic function f(s) defined on all s e C such that f(s) = (s-l)^(s)

if Re(s) > 1. This function f(s) is necessarily unique by the Cauchy integral

formula. The uniqueness can be shown as follows. If $ is an analytic function

on the domain Q.T = ( z e C | | z | < r } , then by Cauchy's integral formula (see

[Hor, 1.2]),

4>(s) = (2TI V z T)- 1 J | z | = t ^dz if | s | < t < r.

Moreover we see that

36 = ^ 4>(s+AsH)(s)3 S As->0 As

26 2: L-functions and Eisenstein series

Here the interchange of the integral and the limit is possible because

r f(z-s-As)-1-(z-s)"1]l im <

As—»0 I ^ g

converges uniformly on the circle (ZG C I |z | =t}. In particular, we have

—(0) = (2ft V—l) . , <t)(z)z~ dz.3s J|zl=t

Repeating the above process, we have

Exercise 1. Give a detailed proof of the above formula.

Since | s | < | z | , we see that — = z"x «- and | z-1s | < 1. Thusz"s l-z^s

^-= z"1—Vz~s l-z^s+z"n-1sn+

Of course, this series is absolutely and uniformly convergent on any compactsubset in Qr. Thus we have

= (2K V=T)"1J|z|=t<l)(z)(z-1+z-2s+z-3s2+- • •+z"n-1sn+- • -)dz

Here we can again interchange the integral and the summation because the powerseries expansion of (z-s)"1 converges uniformly on the compact subset{z| | z | = t} . If $ is analytic on C, changing variables by Z H Z+W (thuss H» s+w) in the above argument, we see that

This implies that if $ = 0 on a neighborhood of w, we see (|) n)(w) = 0 andhence (|> = 0 identically. Thus if f(s) = (s-l)£(s) = g(s) on {s G C I Re(s) > 1}for two analytic functions f and g, we know that f = g on C. Now we write

for the meromorphic function f(s)/(s-l). We will show here and in §2.2 thathas a singularity at s = 1, Ress=i£(s) = 1; and also that £(s) has the

following functional equation:

(2TE)SC(1-S)

" 2F(S)COS(TIS/2) '

Here F(s) is the Gamma function defined by J e 'V^dt if Re(s) > 0 and

satisfies F(n) = (n-1)! for positive integers n. Therefore, essentially the valueof £(s) can be defined by the infinite series when either Re(s) > 1 or

2.1. Euler's method of computing £-values 27

Re(s) < 0. Now there is an interesting way to compute the value of £(l-n) forpositive integers n, which was invented by Euler in 1749. We consider instead of

the alternating sum:

We want to compute (l-2m+1)£(-m) for m > 0. Euler's idea is to introduce anauxiliary variable t and consider

Then Euler pretended that the above series were convergent at s = -m andconcluded by replacing t by 1:

(1) (l-2m+1)^(-m) = f ^ T Y — 1 | t=1 for every integer m>0.

Here the right-hand side is the value of a rational function at t = 1 and hence arational number. Thus if the above argument is correct, we have

(2) £(-n) E Q for n> 0.

Of course the above argument needs justification, but the result and the formula (1)are actually true.

Exercise 2. (a) Show that the rational function t—I f J does not have

(t-1) as a factor in its denominator.(b) Explain why the argument of Euler is a little problematic.

Of course, Euler was fully aware of the shakiness of his argument. Here is how

he justified it. First he replaced t by ex. By the chain rule, we see that

t-jrf(t) I = T~f(ex) I _Q. Thus if you believe the formula (1), we have

(3) (l-2m+1)C(-m) = QL)m( Tf-r | | _ n for each integer m > 0.

Instead of x, we put 27cV~lz and write e(z) = e (/ = V-I) . We then

consider the function F(z) = , / y By (believing) (3), the Taylor expansion of

F at z = 0 is given by

(4) F(z) = X (F(n)(0)zn/n!) = X ((l-2n+1)C(-n)(27CV^zf/n!).n=0 n=0


By another formula of Euler, e*e = cos0 + V-IsinG, we know thatiz , -iz iz -iz

cos(z) = —~—, sin(z) =

From this fact,cot(z) = V ^ l —

Therefore we havee(z)+l

On the other hand, the function cot has the following partial fraction expansion:

(5)

For the moment, let us believe this expansion (which is absolutely convergent ifz ^ Z). By the expansion of geometric series, we know, if | z | < 1 andz * 0, that

Then we see that

(6) ei(z) = ^ + f ; £n=lr=0

ln=l,k=l J k=l

Here only the terms for odd r survive and then we have written r+1 = 2k.

Exercise 3. (a) Suppose that z g Z. Show that absolute convergence of

E {—— + —:} and also show that Z —— is not absolutely convergentn=i z+n z-n z + n

(b) In (6), justify the interchange of the summations with respect to k and n (i.e.show rigorously the equality marked by ? in (6)).

From (6), we know that

! ; 2(l-22k)C(2k)z2k"1 (=} (VzT7i)-1(ei(z)-2ei(2z))k=l

? e(2z)+l e(z)2-2e(z)+l (e(z)-l)2

z(e(z)-l)(e(z)+l) " ' e(2z)-l " " (e(_ 1 _ e(z) e(-z) e(z) v( ,

(= - £ 2(l-22k)C(l-2k)(27cV::Tz)2k-1/(2k-l)!.k=l


This shows that C(2k) =

By specializing the functional equation £(s) = at s = 2k, this2r(s)cos(ics/2)

equality is in fact true, because cos(kTt) = (-l)k and F(2k) = (2k-l)!. At thetime of Euler, the functional equation was not known and in this way, Euler pre-dicted its form.

To make sure of our logic, we summarize our argument. Introducing an auxiliary(d y^f ex ^ I

variable t, Euler related the value I —I - — r | _0 with £(-m); so we write

(l-2m+1)am for this value, which is not yet proven to be equal to (l-2m+1)£(-m).Then by definition, the formula (4) read

F ( z ) = -n=0

On the other hand, by using the partial fraction expansion of the cotangent func-tion, we computed the power series expansion in (6):

^ e (z )+ l_ l__

+i ( T + " i z k=i

Since F(-z)-F(z) = fl , - 2 n \ i > equating the power series coefficients of

the two sides using the above two formulas, we obtain

Thus we know

Proposition 1. £(2k) e 7C2kQ for all 0 < k e Z.

On the other hand, by specializing the functional equation (which we have notproved yet) at s = 2k, we have

C(2k)=

Then we conclude, assuming the functional equation, that

C(l-2k) = a*.! = (l-22Vfxf "TTT> I I x=0-Here are some remarks, (a) By the formula (1), one can compute C(2k) or£(l-2k). Here are some examples:

C ( 2 ) = 1 + 2 - 2 + 3 - 2 + - = \ , ....


For more examples, see the table given in [Wa, p.352].(b) The primes appearing in the numerator of £(2k), for example 691, are calledirregular primes and have arithmetic significance. In fact, the class number of thecyclotomic extension Q(CP) for a root of unity £p with £p

p =1 but £p * 1 isdivisible by p if and only if the prime p appears in the numerator of C(l-2k) forsome k with 2k<p- l . This is the famous theorem of Kummer proved in themid 19th century and immediately implies the impossibility of a non-zero integersolution to the Fermat equation xp + yp = zp if p is regular (i.e. not irregular).We refer to [Wa] for more details of this direction of research and to [Ri] for theapproach using modular forms.(c) The nature of the value £(2k+l) for a positive odd integer is quite differentfrom the even values £(2k) and they are supposed not to equal a rational numbertimes a power of n.

We insert here a sketch of the proof of (5) due to Eisenstein and Weil [W2, II].We shall show that the right-hand side of (5) satisfies the differential equationy' = -y2-7i2. The solution of this equation which goes to °° at z = 0, as is eas-ily shown by a standard argument, is unique and equals Ttcot(Ttz). We put

l v - » ~ r 1 1 ^ >*

z+ X 1 fe + z^ lf r =

if r > 1.de

These series are absolutely convergent. Here note that - p = -rer+i. Taking two

independent variables p and q and putting r = p+q, we get — = — + —.pq pr qr

a aDifferentiating once by — and —, we get (keeping the fact that r = p+q in

dp dqmind)

_1 1_ J _ _2_ _2_ _1_ J_ _1 2_ _2_p V = PV + qV + pr3 + qr3 Or p V " PV " qV = pr3 + qr3 '

In this equality, we put p = z+n, q = w+m-n with integers m and n. Thenr = z+w+m and summing up with respect to n keeping m constant, we have

r-3Xn{jU 1-} = 2(z+w+m)-3{e1(z)+e1(w)}.Now we sum with respect to m:

(7) e2(z)e2(w) - 82(z)e2(z+w) - e2(w)e2(z+w) =

= 2{ei(z)+ei(w)}]Tm(z+w+m)-3 = 2e3(z+w){ei(z)+£i(w)}.

Differentiating (6) with respect to z (noting the fact that —- = -e2), we getQZ

e2(w) = w"2 + Y~=i (2r-l)2^(2r)w2r-2 = w"2+ 2£(2) + 6i;(4)w2 + —


Now expanding £2(z+w) into a power series in w at w = 0 regarding z as a

constant, we have, by the formula —L = -rer +i,

e2(z+w) = X ~ o (e2(r)(z)wr/r!) = £ ~ 0 (-l)r(r+l)er+2(z)wr,

£3(z+w) = Xr°l0 (-Dr(r+l)(r+2)er+.3(z)wV2.

Then the constant term of the left-hand side of (7) in the power series expansionwith respect to w is given by

2C(2)e2(z) - £2(z)2 - 3£4(z) - 2£(2)£2(z) = - £2(z)2 - 3e4(z).The constant term of the right-hand side is (by using (6))

2£3(z)£1(z)-6£4(z).Thus we have

(8a) 3e4(z) = e2(z)2 + 2e3(z)ei(z).

Similarly by expanding both sides of (7) into a power series in x = z+w atx = 0 regarding z as a constant and equating the constant term, we have

(8b) £2(z)2 = £4(z) + 4C(2)£2(z).

Eliminating £4 using (8a,b), we know that

(8c) £i£3 = £22 - 6£(2)£2.

Differentiating this formula, we know from —L = -r£r+i thatQZ

(8d) £2£3 - 4^(2)£3 = £i£4.

Multiplying (8b) by £i and eliminating £i£4 using (8d), we haveei£22-4^(2)£i£2 = £2£3-4C(2)£3 or equivalently £2(£3-£i£2) = 4£(2)(£3-£i£2).

Since £2 is not the constant 4^(2), we know that £3 = £i£2. In (8c), we replace£3 by £i£2 and then divide by £2 to obtain

£ l2 = £2 - 6C(2).

Then by the facts - p = -62 and 6£(2) = n2, we know that £1 satisfies the

differential equation y1 = -y2-7C2. The fact 6£(2) = 7c2 will be shown inde-pendently of this argument.

Now we give a generalization of the formula (1):

(l-2m+1)C(-m) = ( t ^ J * ^ ) I t=i for each integer m>0,(^J^) t=iwhich was found by Katz [Kl]. Instead of 2, we fix an integer a > 2. Wedefine a function \ : Z —> Z by

1 if n gfe 0 mod a,11-a if n = 0 mod a.


We note that Y^l^{b) = £j=15(b) = 0. We consider, instead of j ^ , the

rational function O(t) = — 7 ~ a i — • Then we see thatt — l t — 1

<D(e(z)) = (^ny\ncot(nz)-mcot(anz)) = -(V-LTC)"1 ]T~= I 2(l-a2k)C(2k)zz2k-1

In the special case of a = 2, we have2t+l t 2

+ l (t+l)2-2(t2+l)2?T £i

By computation, we see that

O ( t ) = (t+Dd+t+tV-'+t^-ata-a

l-a+2(t+t2+---+ta"1)-

ta-l

ta-l2+-+tb-1)2la_1^(b)(l+t+t2+-+tb-1)

S L 1 ^ ( b ) ( t t t ) I ^(b)tNow we put T(t) = ^ , pj -= b - l S

a . The lastl+t+r+-+ta"1 l-ta

equality follows from Z^=1^(b) = 0. When a = 2, we have ^ ( t ) = J^T. In

general, we have -2vP(t) = 3>(t)-(a-l). Then by the formula which we havealready looked at,

*(e(z)) = -(i7i)-1X~=12(l-a2k)C(2k)z2k-\

we now know that

To relate this formula with the values of C, at negative integers, we consider

Putting g(t) = Z ^(n)tn, we getn=l

( d^m

and hence formally

(•5On the other hand, we have

2.2. Analytic continuation and the functional equation 33

d^m

= [ t ^ j g(t) t n a >. I t = l

z=0

This formal computation can be justified by applying a functional equation (to beproved in the following section) to the right-hand side of

(l-a2k)Cd-2k) = •

Thus we obtain

Theorem 1. Let a > l be an integer. Then for each positive integer m,we

( A \m . Zu_iS(b)tb

have (l-am+1)C(-m) = t^- Y(t) I t , w/z r ^F(t) = b - 1 . .\ oij t=i 1-t

Exercise 4. We have only proved the theorem when m is an odd positiveinteger. Give a proof of the theorem for even positive integers m. (First show

d W^ f e— -—5f n = 0 for 0 < m £ 2Z by making the substitution X H - Xdxy ^l + e j • x=oand then show £(-m) = 0 by using the functional equation.)

Exercise 5 (the Lipschitz-Sylvester theorem). By using Theorem 1, show thatam+1(l-am+1)£(-m) e Z for every integer a> 1, where m is a non-negativeinteger.

Exercise 6. (a) For each prime p, show that (Z/pZ)x is a cyclic group,(b) For a positive even integer k, show that if the denominator of £(l-k) is di-visible by a prime p, then k is divisible by p-1.

§2.2. Analytic continuation and the functional equationAs seen through Euler's argument, we now know the importance of the function

F(z) = i , / \ in the theory of the Riemann zeta function £(s). Here we modify

this function a little to deal with £(s) directly (instead of (l-2"s)£(s)). Pute"z

G(z) = —— for z G C. First we give an integral expression for £(s). Fori~e

that, we need several properties of the F-function. Let us recall some of them. We

may take the following integral as a definition of the F-function:

(1) F(s) = Jo~ e-y-My if Re(s) > 0.


This integral is convergent only when Re(s) > 0. To show this, we remark that

for a fixed real number e > 0 and for any real number a, there exists M > 0

such that if y > e, then My"2 > e^y0"1. Thus writing a = Re(s), we see that

I r(s) I = I Jo~ e-V-xdy I * J~ I e-V11 dy = J~ e V ' d y .

In particular, we know that

I f e-VMy I <: f e V ' d y < M [~ y-2dy = M[-y"T = Me"1.JE Je. JE £

Thus the integral J e"yys"1dy converges for all s and gives an analytic function

of s. The problem of divergence lies in the other integral:

I Jo£ e^y-My I < J* e'V^dy < J* y^dy < [ya/a]* = ea/a if a > 0.

To find the analytic continuation of F(s), we consider eyys~l as a function of thecomplex variable y. Here note that the function y i—> ys is not well definedbecause ys = QslogW and log is multivalued. To fix a branch, we write

y = | y | e/G with 0 < 9 < 2TC and define log(y) = log \ y \ +/0 andys = e

s /^(y). When 0 = 0 or 2K (i.e. y is on the positive real axis), wewrite ys when 0 = 0 and y.s when 0 = 2n. Note that y11 = y.n for integersn. We fix a positive real number e and denote by 3D(e) the integral path whichis the circle of radius e with center 0 starting from e = I e I el0 and withcounterclockwise orientation, by P+(e) the path on the real line from +°o to eand by P_(£) the path on the real line from e to +<*>. We consider that 0 = 0on P+(e) and 0 = 2% on P_(e). We write the total path asP(e) = P+(e)U3D(e)UP-(e). We similarly write 9D(e,ef) for the boundary of theannulus cut along the real axis:

{P+(ef)-P+(e)}UaD(e)UaD.(eI)U{P-(e')-P-(e)} for e > e' > 0,

where 3D.(e') = 3D(e') as a path but has the clockwise orientation:

Then jP +(e)e"yys"1dy and

Jp (£)e~yy-s~1(ty converge for alls and give analytic functions onC. Note that


+(e)e"yyS"ldy = - Je~ e-V'dy and / p ^ e ' V ^ y = e2ltis J£" e ^ d y .

Changing variables by y = eel9 (dy = *ee'ed9), we have, for a sufficiently small e,

I JaD(E)e"yyS"ldy I ^ Mea J^dG = 27iMea if a > 0,

where M is a constant independent of e. Thus if a = Re(s) > 0, we know that

^ o W ' V ^ d y = (e27t's-l)r(s).

The function e y**"1 has no singularity on 3D(e,e') and hence, by Cauchy'sintegral formula (see [Hor, 1.2]),

f y y Zz:poles ta v^^-yf-1 = o.Then we have

Thus we know that the holomorphic function Jp(8)e"yys"1dy of s is independent

of £ and gives (e27ns-l)r(s) if a > 0. Thus we have the meromorphic con-tinuation of F(s) given by

(2) r(s) = (e2ms-l)-1Jp(E)e-yys"1dy for all s e C.

Since (e271^-!)"1 has singularities only at integers s e Z, which are simplepoles, F(s) has at most simple poles at non-positive integers. Through integrationby parts, we have the well known functional equation

r(s+l) = Jo~ e'Vdy = [-e"yys]Q+s Jo~ e ' V ^ y = sr(s).

The analytic continuation of F(s) can be also proven by using this functional

equation, which shows that F(s) has in fact simple poles at integers m < 0.

Exercise 1. Show that for each positive integer n

Now we go into the integral expression of £(s) using G(z). Expanding G into ageometric series, we know that

(3) G ( y ) = - ^ = £ e"ny.X " e n=l

This series is convergent when | e'y | < 1 (<=> Re(y) > 0). We first formallyintegrate G on R+:


n=l n=l

n=loo

X /.OO

and I made at the equality markedn=l

"V

by "?". For that, we look at the poles of G(y) =——. Since l-e'y has zerosx-e

only at 2%4-irL = {27cV-Tn|nG Z}, G(y) can have a pole only at 0 on R.

Exercise 2. Show that | yG(y) I is bounded (independently of y) in the unitdisk of radius one with center 0 (hint: show lim yG(y) = 1 and deduce the

result from this).

On the other hand, since e"y -» 0 as y -> +©°, there is a constant M > 0 such

that I yG(y) I < Me"y/2 by the above exercise. This shows first of all, for

a = Re(s),

I J~ G(y)ys"1dy I < Jo°° I G(y)y I y°-2dy < M Jo°° e"y/2ya'2dy = 2°- iMr(a- l ) .

Thus I G(y)y dy is convergent if a > 1. Since the domain of integration

R+ is not compact, the uniform convergence of (1) is not sufficient to get the in-(•OO

terchange of Z°° and . In order to assure the interchange, we shall use then—1 JQ

following dominated convergence theorem in the integration theory (due toLebesgue): If a sequence of continuous (actually integrable) functions fn(x) (onan interval [a,b] in R; a and b can be ±<») is dominated by a continuous andintegrable function and f(x) = lim fn(x) at every point x, then

rb rblim fn(x)dx = lim fn(x)dx.

We apply this result to Gn(y)ys"1 = E ^ e ' ^ y 8 " 1 which is dominated by the

integrable function I G(y) | y0"1 (if a > l ) and converges to E e'^y8"1. Inn=l

fact, we see that

I GnCyV11 = | I ^ ^ ^ V 1 1 * S;=1h"myys"i = I G(y) I y0"1 if y > 0.Therefore we have

j ; -"y y -My= lim j;Gn(y)dy= j ; UmGn(y)dy= f fn=l n=l


This justifies the interchange and we have

Proposition 1. F(s)C(s) = J~ G(y)ys"1dy if Re(s) > 1.

We cannot extend (naively) this integral expression to the left half plane{s |Re(s)<l} because £(s) has a simple pole at s = l.

Since G(y) has singularities only at y = 0 in a small neighborhood of R+, weknow, in the same manner as in the case of the analytic continuation of F(s), that if0 < e < In and Re(s) > 1,

r(s)C(s) = Jo~ G(y)ys"1dy = (e27l/s-l)-1JP(e)G(y)ys-1dy.

The integral of the right-hand side converges for all s and gives an analyticfunction of s. Thus we have as a corollary of the proposition.

Corollary 1. (e27ns-l)F(s)£(s) can be continued to a holomorphic function onC and has an integral expression:

(e2™-l)F(s)C(s) = JP(e)G(y)ys-1dy for 0 < e < 2TC.

Exercise 3. (a) Prove Corollary 1 rigorously along the lines of the proof of theanalytic continuation of F(s). (b) Compute Ress=iC(s) by using the aboveintegral expression.

We now want to compute the value of Jp(e)G(y)ys"1dy at s = 1-n for positive

integers n. As in Exercise 1, we know that

We define the Bernoulli numbers Bn by

We see that Bn = (— J{xG(x)} I x=Q. Since Resy=0G(y)y"n = f , we know that

"n^v = (27cV—T)—p. Then by Exercises 1 and 4(b) below, we know

pTheorem 1. For each positive integer n, we have £(l-n) = — - if n is even,

nG(0) = - i and C(l-n) = 0 if n > 1 is odd.

This agrees with Euler's computation via Exercise 1.4.


Numerical Example. We list here several Bernoulli numbers:

Bo = 7o 23' " 4 2 "2.3-7' * 1 0 ~ 66 "2.3-11'

691

30

691"2730 ""2.3-5.7.13'

(See [Wa, p.374] for more Bernoulli numbers).

Exercise 4. (a) Show that B2n is positive if n is odd and B2n is negative ifn is even (Use the functional equation of £(s)).(b) Show that Bn = 0 if n is odd and n> 1.(c) Show that the denominator of Bn consists of primes p such that n isdivisible by p-1 (use Exercise 6(b) in the previous section).

Now we prove the functional equation. The idea is to relate the function of s1 xgiven by Jp(e)G(y)ys"1dy with the integration of G(y)y'

tegral path for each integer m > 0:(2m+l)<r 2

s"x on the following in-

-2m-l

We denote this inte-gral path by A(m),where the orientationis clockwise on theoutside rectangle.Since the orientationof A(m) is clockwiseif you look from apoint inside of A(m),by Cauchy's integralformula, we knowthat

A(m)

where Resy=w(|)(y) is the coefficient of (y-w)"1 in the Laurent expansion

<Ky) = ^n>-ooc(n)(y-w)n- S i n c e l*~y = 0 if and only if y e 2% VÎZ, the

function G(y)ys"1 =_y

^ " 1 has simple poles only at 27cV-ln for integers nx~e

withO< |n I < m inside A(m). The residue at y = 27cV-Tn can be easilycomputed and is given by

/ ^ / x s u I - J - i \ 2 M i \ s - \ s % i l 2 i f n > 0 ,(4) Res^GWy-)!


Exercise 5. Write down every detail of how one obtains the formula (4).

Thus we have

n=l

Therefore, if Re(s) < 0, we already know that lim L^n55"1 converges to

£(l-s) and hence we obtain

(5) inmJA(m)G(y)ys-1dy = (27c)sems(e7lis/2-e-7Cis/2)C(l-s) if Re(s) < 0.

On the other hand, if we denote by Q(m) the square of side length 4m+2 cen-tered at 0 with clockwise orientation and by P(e,m) the complement of Q(m) inA(m), then we have

mHmJA(m)G(y)ys-1dy = ta

y)ys-1dy +

JimJQ(m)G(y)ys-1dy

We now show that lim Jo(m)G(y)ys"1dy = 0. Then after a minor modification,

we will have the functional equation. We write Q±(m) for the upper and loweredge of Q(m) and Qr(m) (resp. Q/(m)) for the right (resp. left) edge of Q(m).Here we only prove that

The computation for the other edges will be left to the reader as an exercise. Firstnote that, writing y = X+TCV-I (2m+l) on Q+(m) for x = Re(y), we have

e-y = e-7Ue"2m7Vx = -e"x, I l-e"y I = l+e"x > e"x

and|G((2m+l)7cVz:l+x)((2m+l)7cV=:r+x)s-11 < M | (2m+l)7cV::l+x|a-1

for a constant M depending only on s and for a = Re(s) < 0. Hence, we

have I G(y) | = -^—- < 1 for all y e Q+(m). This shows that1+e

|G((2m+l)7tV::r+x)((2m+l)jcV::T+x)s-1|dx—2m—1

f—2m-l


If a < 1, then o-l < 0 and we see that

((2m+l)V+x2)( a-1 ) / 2

Then we have

f2m+1

*—2m—1•—2m—1

which goes to 0 as m ^ +°o if a < 0. This shows the assertion.

Exercise 6. Show that lim Jgr(m)G(y)ys"1dy = 0. (First show that on Qr(m),

I G(y) I < 2 and proceed in the same manner as above.)

By the above estimate and Exercise 6, we obtain

(e27n's-l)r(s)C(s) = (27i)sem's(e7C/s/2-e-^/2)!;(l-s)

or multiplying by e~ms, we have

= (27i)s(e7Cfs/2-e-m"s/2)C(l-s).e7Us/2_e-ms/2 %$

By the fact that : :— = {2COS(-TT)} , we deduceJ JKIS _ -7WS l V 2 ' '

Theorem 2 (Riemann). We have £(s) = (27C) ^ 1 " s )2r(s)cos(7is/2)

§2.3. Hurwitz and Dirichlet L-functionsNow we extend our argument to Dirichlet L-functions. Let % be a character of the

multiplicative group (Z/NZ)X. Thus % : (Z/NZ)X -> Cx is a homomorphism

of the finite multiplicative group (Z/NZ)X into Cx. We extend % to a function

on Z by putting %(n) = %(n mod N) if n mod N is in (Z/NZ)X and

%(n) = 0 if n mod N is outside (Z/NZ)X and define the Dirichlet L-function of

X by

Any x e (Z/NZ)X satisfies xr = 1 and hence %(x)r = 1, where r = cp(N) is

the order of (Z/NZ)X. Thus %(n) is a root of unity and in particular satisfies

I %(n) I < 1. It is an algebraic integer of a cyclotomic field. Then we have

\L(s,x)\ * XI=i 15C(n)ns | < J^=1 n"° = ^ ) for a = Re(s).

This shows that L(s,%) is absolutely convergent if Re(s) > 1 (this convergenceis uniform on every compact subset in the region with Re(s) > 1 and henceL(s,%) is an analytic function of s in this region).

2.3. Hurwitz and Dirichlet L-functions 41

Exercise 1. Show the following Euler product expansion of L(s,%) and its

convergence if Re(s) > 1: L(s,%) = U p ( l

More generally we think of a function c|>: Z/NZ —> C and consider

To continue this function to the whole complex plane, we rewrite it as

oo N N

L(s,§) = ^<|)(n)n~s = ^(()(a)^(Nn + a)~s = ^(|)(a)N~s^(n-l-—)~s.n=l a=l n=0 a=l n=0

Thus the problem of analytic continuation of L(s,<|)) is reduced to the same prob-

lem for X (n-hrj)'s. More generally we consider the following zeta function forn=0 -N

all 0 < x < 1:

£(s,x) = £~ = 0 (n+x)-s,

which is convergent if Re(s) > 1. This zeta function is called the HurwitzL-function (and was introduced by Hurwitz in the 188O's). We then have

In the same manner as in the case of the Riemann zeta function, we have the fol-

lowing integral expression of £(s,x):

(2) r(sK(s,x) = J~ G(y,x)ys"1dy if Re(s) > 1,

where G(y,x) = ^y-^ - = - ^ y = £~ = Q e-(n+x)y.

Exercise 2. (a) Show that £(s,x) is absolutely convergent if Re(s)>l .(b) Show the formula (2) along the lines of the proof of Proposition 2.1.

Note that the zeros of l-e'y are situated at 27iin for n € Z and are all simple.In particular, yG(y,x) is bounded in any small neighborhood of 0. Thus forsufficiently small e > 0 (actually, any e with 0 < e < 2n does the job), theintegral Jp/8xG(y,x)ys"1dy is convergent for all s and gives an analytic functionof s. The same computation as in the proof of Corollary 2.1 gives

Proposition 1. (e27liS-l)F(s)^(s,x) can be continued to a holomorphic functionon C and has an integral expression:

(e27U's-l)r(s)C(s,x) = JP(e)G(y,x)ys-1dy for 0 < e < 2TC.


This combined with the formula (1) yields

Corollary 1. The function (e27US-l)r(s)L(s,(|)) of s can be continued to aholomorphic function on C and has an integral expression:

-idy for 0< e < 27C/N.

As a byproduct of the analytic continuation, we can compute the value of £(l-n,x)using the formula

{(e27C/s-l)r(s) | s=1.n}Cd-n,x) = JaD(£)G(y,x)y-ndy = (2TC VzT)Resy=oG(y,x)y-n.

By Exercise 2.1, we know that

(e2™ D H s ) I ('1)n"1(27cV=II)

(e i;i WiThus we compute Resy=oG(y,x)y"n. Write F(y,x) = yG(y,l-x) and expandF(y,x) into a power series in y (regarding x as a constant):

We can interpret this argument in a manner similar to §1 as follows. Writingtx

t = ey, we have F(y,x) = yfy. Thus pr = - + Z°° / t n t y11- This showsi~ x i~ x y nÔ \n~T" x) *

^ - a-f^ = £^(B n + 1 (x ) -a^B n + i (x ) ) y n for a n y i n t e g e r a > 1? a n d thus

(1"a } n+i - I'dTj \t-rV-ij I '=!•

Then Bn(x) is a polynomial in x of degree n with rational coefficients. Theseare called "Bernoulli polynomials". We can make this more explicit as follows:

Therefore, equating the coefficients in y11, we see thatBn(x) = y n Bj xn-j

n! ^ j = o j ! (n - j ) !In other words, we have the formula

(3b) Bn(x) = ]T*=o (^Bjx1 1^ G Q[x].

Thus we know thatResy=0G(y,x)y-n = Resy=0F(y,l-x)y-n-1

= the coefficient of y11 in F(y,l-x) = B n ( 1 " x ) .n!

This shows that


( ) ; j j

Thus we conclude that £(l-n,x) = (-1)11"1 n^ . Here we note that(l-x)y e-xy

This equality yields

(4) Bn(l-x) = (-l)nBn(x).

Thus we obtain

Theorem 1. We have, for any integers a > 1 and 0 ) = E^=1(|)(a)N"sC(s,^), combined with Theorem 1,

implies

Corollary 2. Let Q(%) denote the cyclotomic field generated by the values of

the character %. Then we have

Moreover, suppose %(-l) ^ (-l)n. ^/zen we have L(l-n,%) = 0 /or n> 0 if% is non-trivial, and £Q-n) = 0 if n> 1. In general, for ty : Z/NZ --» C,we have, if SL is prime to N,

L(-m,Hm+1W = fffffyWt" a f ^ f ] I t , /or a// m > 0,^ dtJ l&T^ ST^^J 't=1

The vanishing of L(l-n,%) follows from (4) if %(-l) * (-l)n. The number

Bn>% = Z asiXteJN^Bnfe) is called the generalized Bernoulli number. Using

the notation Bn>5c, the above formula takes the following form:

Examples of Bernoulli polynomials:

B0(x) = 1, Bi(x) = x-i , B2(x) = x2 - x + \ , B3(x) =

Using the above formula, we can get

X ^ ^ i f X is non-trivial.


Since % : (Z/NZ)X -» C x is a group character and (-1)2 = 1, %(-l)2 = 1.Thus %(- l )=±L

Now we prove the functional equation of L(s,%). We proceed in the same way asin the case of the Riemann zeta function. We consider the integral on the pathA(m) for each positive integer m given in §2. By the Cauchy integral formula,

The function G(y,x) has a simple pole at 2ft V—In for integers n and theresidues at 27iV—In can be computed as

, „ , N s K U « > W I 2 n n | ^ i f n > 0 ,R e s y = 2 7 t ( n ( G ( y ) x ) y ) = j e . 2 7 t i n x + 3 ( s . 1 ) 7 l i / 2 , ^ , s , .{n<Q

because as already computed, Resy=2mn z=l and the value of &'xyysA isl-e~y

c-2ninx+(s-l)ni/2 | 2n% | s-1 Q r e-27imx+3(s-l)7rf/2 | 2n% | s-1 a c c o r d i n g a S n > 0 Or

n < 0. Therefore

This shows, if Re(s) < 0,

JmJA(m)G(y,x)ys-1dy =

In exactly the same manner as in the case of the Riemann zeta function, the integralon the outer square Q(m) converges to 0 as m —> +°°. Thus we have

(e2*'s-l)r(s)C(s;x) = JP(e)G(y,x)ys-1dy = |KmJA(m)G(y,x)ys-1dy

= (27t)s{e3nis/2Y~ e2xmxns-l_ems/2y~ -2xmxns-lj i

Then by the formula L(s,<|>) = I^=1<))(a)N"s^(s,^), we see that

(e27t's-l)r(s)L(S)%)

where e(z) = e27liz.

Now we need to compute Xa=i %(a)e(na/N). To treat this sum, we insert here adefinition. For each proper factor D of N, we have a natural homomorphismp D : (Z/NZ)X -> (Z/DZ)X which satisfies pD(n mod N) = n mod D. If forsome D, there exists a character %o • ( Z / D Z ) X -> C x such that%(n) = %O(PD(H)) for all n e (Z/NZ)X, we say % is imprimitive. A character% is called primitive or primitive modulo N if % is not imprimitive. Now wesuppose


(5) % is a primitive character modulo N.

We do not lose much generality by this assumption:

Exercise 3. Let % = %O°PD for a proper divisor D of N. Then show

Ms,X) = {np|N(l-%o(p)p"s)}^(s,Xo),where p runs over all prime factors of N.

First we prove the following lemma (the orthogonality relation) in group theory:

Lemma 1. Let G be a finite abelian group and % : G -> Cx be a character.

If % is not identically equal to 1, then Xg^GX(g) = 0. Similarly if g ^ 1,

X%%(S) = 0» where % runs over all characters of G.

Proof. Since % has values in the group of roots of unity, which is cyclic, we may

assume that G is cyclic and % is injective by replacing G by G/Ker(%). Let N

be the order of G. Then % induces an isomorphism of G onto the group |IN of

N-th roots of unity. Thus X*(g) = X s = 0 because nê^N(X-C) = X N -1.geG ;ey.N

Let G* be the set of all characters of G. Defining the multiplication on G* by%V(g) = %(g)V(g)» G* is a group. If G is cyclic of order N, then the characteris determined by its value at a generator go. Thus G* s % h-> %(go) e (IN ={£ e C x I £ = 1} defines an injection. For any given £ e [i^, defining%(gom) = £m> % is a character having the value £ at go- Thus G* = (IN- Thenassigning g the character of G* which sends % to %(g), we have a homomor-phism: G -> G**. Since %(g) takes all the N-th roots of unity as its values atsome %, this map is surjective. Then by counting the order of both sides, weconclude that G = G**. In general, decomposing G into the product of cyclicgroups, G* will be decomposed into the product of that character groups of eachcyclic component. Thus G = G** in general. Then replacing G by G* andapplying the first assertion of the lemma, we get the second.

Lemma 2. Define the Gauss sum G(%) = Z^=1%(a)e(a/N). Suppose that % is

primitive modulo N. Then ^ %(a)e(na/N) = %~l(n)G(%) for all integers n.

Note here that %~l(n) = %(n)"1 is again a character of (Z/NZ)X, which is primi-

tive. In particular, this implies Za=1%(a)e(na/N) = 0 if n g (Z/NZ)X because

%"1(n)=0 by our way of extending %A outside (Z/NZ)X.


Proof. Define \|/ : Z/NZ -> Cx by \j/(t) = e(t/N). Then, \|/(t) does not de-

pend on the choice of t. In fact, if t = s mod N, then t = s+Nn for an integer

n and thus e(t/N) = e((s/N)+n) = e(s/N)e27l/n = e(s/N). Thus \|/ gives a ho-

momorphism of the additive group Z/NZ into the multiplicative group Cx. Thisfact is obvious because

\j/(t+s) = e(t+s) = e27lf(t+s) = e27lfte27C/s = \|f(t)\|f(s).

Then S^=1%(a)e(na/N) = Sae(Z/NZ)XX(a)\|/(na). We first treat the case where

n mod N e (Z/NZ)X. Then the multiplication of n induces a bijection x h-> nx

on (Z/NZ)X. Thus we can make the variable change in the above summation; so,

rewriting na as a, we have

£ %(a)\|/(na) = ^ ( n ^ a ^ a ) = %l(n) £x(a)\|f(a) = X'l(n)G(x)-ae(Z/NZ)x a€(Z/NZ)x ae(Z/NZ)x

Now assume that n £ (Z/NZ)X. In this case %"!(n) = 0 by our way of extend-

ing x"1 outside (Z/NZ)X. Thus we need to prove that

Let p be a prime which is a common divisor of N and n. Write N = pD andn = pnf. Then we have

£x(a)\j/(na) = ]T%(a)e(n'a/D) = I ^ n 'ae(Z/NZ)x ae(Z/NZ)x ae(Z/DZ)x

because e(n'a/D) only depends on the class of n'a modulo D (but not N). Weshall show that S t e a m o d D%(b) = 0. We see that

X E 1 X D%(b)«Xb=a mod D%(b) = Ebâ mod D%( a b a ) = X(a)Xb.l mod

Let H = ( x 6 (Z/NZ)X | x = 1 mod D). Since H = Ker(pD), it is a sub-

group of (Z/NZ)X. If % is trivial on H, then we define %0 : (Z/DZ)X -> Cx

by %O(PD(C)) = %(c). Xo is well defined because if PD(C) = PD(C'), then

c = c'h for h G H = Ker(pD). Thus %(c) = %(c'h) = %(c')%(h) = %(c')

because of the triviality of % on H. Then % = %O°PD> which contradicts the

primitivity of %. Therefore we can conclude % is non-trivial on H. Thus the

orthogonality relation of characters (Lemma 1) shows that

2.4. Shintani L-functions 47

Now by using this lemma, we finish the computation:

(e27tis-l)r(s)L(s,x)

We see easily that

c2nis-l |2COS(TCS/2) if x ( - l ) = 1,e37t;s/2_x-i(_1)e™s/2 {2^pLsin(ns/2) if %(-l) = - 1 .

This shows

Theorem 2. Suppose that % is primitive modulo N. Then we have

'G(%)(2TC/N)SL(1-S,%-1) .,

^>3" ^2r(s)cos(7is/2)

if

G(x)(27r/N)sL(l-s,%-1) ^. 2Vzir(s)sin(7Cs/2)

Exercise 4. Using the above functional equation, show that L(s,%) is a holo-morphic function on the whole complex plane C if % is a primitive charactermodulo N > 1. (The main point is to show that L(s,%) is holomorphic ats = 1; use also Corollary 2.)

Exercise 5. Suppose that % is primitive modulo N. By using the functional

equation (and also the power series expansion of L(s,%) at s = -j), show

G(x)G(%"1) = %(-l)N for general primitive %, and supposing that L(w,%) * 0,

show G(%) = ^3C(-1)N if % has values in {±1}. (The fact that G(x) =

has values in {±1} is true without the

shing of

non-vanishing assumption.)

assumption of the non-vanishing of L(s,x) at ^- Try to prove it without the

§2.4. Shintani L-functionsIn this section, we introduce the contour integral of several variables and ShintaniL-functions [Stl-6] and later, we will relate them with Dedekind and HeckeL-functions of number fields. We now take another branch of log different fromthe one in the previous section; namely, for z e C, we write it as z = | z | tlQ


with -K < 0 < n using the polar coordinate and define log(z) = log I z | +/0.Accordingly, we define the complex power zs = es/o^(z^ by this logarithmfunction. We put

H ' = { z e C | Re(z) > 0}: the right half complex plane,

R ± = j x e R | i x > 0 ) : the right or left real line,

R± = R±U{0}, N =

By our choice of log, we have the luxury of

(ab)s = a V , as = a? and (a1)8 = a"s

for any two a,b G H' and s e C.

To define the Shintani L-function, we need the following data: (i) a complex rxmmatrix A = (ay); (ii) % = (%i, ..., %r) e Cr with |%il £ 1 for all i; and(iii) x = (xi, ..., xr) G Rr such that 0 < Xi < 1 for all i but not all Xi are0. We define linear forms Li on Cm and Lj* on Cr by

m r

Li(z) = X aikzk> Lj*(w) = X akjwk (z = (zi,...,zm), w = (wi,...,wr)).k=l k=l

We suppose throughout this section that

(1) Re(aij) > 0 for all i and j .

This assumption guarantees that Li(z) and Lj*(w) for z e R+m-{0} and

w G R+r-{0} stay in H', because H ' 3 R+H1 and H ' D H ' + H ' . Then weformally define the Shintani L-function by

(2) C(s,A,x,x) = Xn G NrXnL*(n+x)"s for s = (si sm) G Cm,

where we write L*(n+x) = (Li*(n+x), ..., Lm*(n+x)) G C m and for

w = (wi, ..., wm) G H'm, we write ws = I I ^ w / J . When A is the scalar 1

and % = 1, then £(s,A,x,%) = ^(s,x) = Z°° (n+x)"s and thus the Shintanin=0

L-function is a direct generalization of the Hurwitz L-function. We leave the proofof the following lemma to the reader as an exercise:

Lemma 1. £(s,A,x,%) converges absolutely and uniformly on any compact sub-

set in the region Re(si) > — for all i.


Exercise 1. (a) Prove the above lemma. (Reduce the problem to the case where

all entries of A, %, and x are 1 and use the fact that

#{k e Z+ r | ki + —+kr = n} < Cn*"1 for a constant C > 0. Actually,

£(s,A,x,%) converges if Re(si+---+sm) > r and Re(sj) > 0 for all j.)

(b) When all the entries of % are equal to 1 and A is a real matrix, show that

£(s,A,x,%) diverges at s = —(1,...,1) (actually it diverges if si+---+sm = r).

Now we give another exercise which generalizes the fact that

T(s) = P e ' V ^ d y if Re(s) > 0:Jo

Exercise 2. If a e Hf and s e H', then f e ' ^y^dy = a"T(s), where asJo

Jot - 1 ft __ .7Calready explained, a's = | a r s e writing a = | a | e with — < a < — . First

r°° 1 1

interpret the integral J e~ayy dy as a"s times the integral of e"yy on the line

in H' from 0 to +<*> with argument a. Then relate this integral with the T in-

tegral J e^y^dy by using the following integral path:and show that the integral on the innercircle of radius e (resp. the outercircle of radius N) goes to 0 ase -^ 0 (resp. as N-> +°o).

c N Now we want an integral expression of£(s,A,x,%) converging in the domain

with sufficiently large real part. We consider the following functionG(y) = G(y,A,x,%) with variable y in R+m given by

(3) G(y) = X n e Nr Xnexp(- L*j(n+x)Vj).

The convergence of this series can be shown as follows. First of all, we see that

This shows that

G(y) =


Since Li(y) e H' as already remarked, | %iexp(-Li(y)) | < 1 and the geometricseries in the inside summation converges absolutely. We then have

(4)

Now writing ys = n™= 1y;s i for s e C m , 1 = (1 ,1 , . . . , 1 ) e C r a ,

dy = dyidy2"-dym and Fm(s) = n^jIXsi), we have, by Exercise 2,

(5) J J - J ~ G(y,A,x,z)ys-1dy = JJ...J" In€Nr%nexp(-I™1L*j(n+x)yj)ys-1dy

G(y,A,x,X) = f t rV('XffVr\ 'U 1-Xiexp(-Li(y))

-s = rm(s)C(s,A,x,x).

As in the case of the Riemann zeta function (Proposition 2.1), we can justify the

interchange of the integral [)•••[ and the summation £ r marked by "?" if

Re(si) > — for all i, thus (5) is valid (if Re(sO > — for all i). In fact, if A is

a real matrix and % = 1, the convergence of E Nrexp(-Z^1L*j(n+x)yj) to

G(y,A,x,l) is monotone on R+r and hence we can interchange the integral andthe summation at the equality marked by "?". We also know from this that

G(y,A,x,l)yRe(s)"1 is an integrable function if Re(sO > — for all i. In general,

we know thatI G(y,A,x,%)ys"11 < G(y,Re(A),x,l)yRe(s)"1

and using the dominated convergence theorem of Lebesgue,

If a sequence of continuous (actually integrable) functions fn(x) (on an interval[a,b] in R; a and b can be ±oo) is dominated by a continuous and integrablefunction and f(x) = lim fn(x) at every point x, then

n—>o<>rb rb

lim fn(x)dx = lim fn(x)dx,Ja. n—>°° n—>ooJa

we can justify the interchange.

Things have worked in exactly the same way so far, but we encounter a seriousdifficulty in converting the above integral into the contour integral convergent forall s G Cm. Probably, many people before Shintani considered the zeta functionof type (2) and tried to get its analytic continuation, but because of this difficulty,we had to wait until 1976 [Stl] to get the analytic continuation of £(s,A,x,%).


First, we explain why the naive conversion to the contour integral does not work.For simplicity, we only treat the case where % = 1 and x = 1. As already

e-zseen, — has a simple pole at z = 0 whose residue is equal to 1; in other

1-e

words,1 - e

—£ is holomorphic at z = 0. Thus each factor of

has a simple pole at the hyperplane S[= {ye C m | Lj(y) = 0} if m > 2 (ifm = 1, then Si = {0}). On the other hand, if we denoteD(e) = {ye Cl | y | < e } , then D(e)m gives a neighborhood of 0 in Cm andthus D(e)mriSi •*• 0 if m > 2. This implies that we cannot avert the singularityby taking the path 3D(e)m.

Shintani's idea is to convert the integral (5) into a contour integral by means of aningenious variable change. We divide R+

m into the following m regions:

R+m = U£ = 1 D k , Dk = {y = (yi, ..., ym) | yk > yi for all i * k } .

This decomposition can be illustrated in the case of m = 2 as follows:Y2 On each Dk, we shall make the following variable

change:

(6)(0 < t i < \

) = M(f;,f2,...,rm)k) and tk = 1).

Yi Thus yi = u t i , y2 = u t 2 , ..., yk = u, ...,ym = utm. Since the computation is all the same for

any of the Dk, we assume k = 1 and compute the jacobian matrix:

du du

Thus we know that dy = um"xdudt andTr(s) = S1+S2+—f-sm. On the other hand, we see that

a0

0

i 0

hu

0

0

0

u

0

(7) a n dk-l

= uTl(s)'mts-1 for

Thus we have


mJJ-J G(y,A,x,x)ys-1dy = £

k=l

= X O o - J o Gk(u)t,A,x)%)uT^-1ts-1dtdu)k lk=l

where Gk(u,t,A,x,%) = U],—^—•*—^- . This integral expression can be' " ' l X e x p C u L W )

converted into a contour integral convergent everywhere. First let us examine the

singularities, i.e., the zeros of l-%iexp(-uLi(t)). Write %. = l % J e ' i with

0 < 6i < 2K. Then writing z = uLi(t), we see that the zeros of l-%iexp(-z) are

located as follows (note that /og lx j ^ 0 because |%.| < 1):

e ^ The small circles in the figure at left indicate zeros.

Therefore we can take 8 > 0 so that

if 0 < | uLi(t) I < 5 for all i. We fix such a 8.

Then we can find 0 < 8' so that if | u | < 8' and

T 7 ^ - 2 ^ | ti | < 1 for all i, then | uLj(t) | < 8 . We also

fix such a 8'. On the other hand, we know from

(7) that l imRe(Li( t)) = Re(a i k ) > 0. Put a = Min{Re(aik) | i = 1,. . . , r } .

Thus we can find 1 > 8" > 0 so that if I til < 8" for all i * k, then

Re(Li(t)) > y- Thus the only possible zeros of U\=l{ l-%iexp(-uLi(t))} in the

neighborhood

U = {(u,t i , . . . , tk_i, tk +i , ..., tm) | | u | < 8', | til < 8M for i * k}

are at Ufl{u = 0} . On the other hand, if R s u > 5 ' and | ti | < 8 " ,

Re(uLi(t)) > a8'/2 (i.e. | %iexp(-uLj(t)) | < 1), and hence no poles are expected

in this remaining case. Thus on the following integral path, we do not have any

singularity of Gk(u,t,A,x,%) for 0 < e < min(8',8M) independent of t and u:

P(e): I ^ +oo for u

P(£,l): U , - 1 for ti,

where the circle is of radius e centered at 0. Note that if t is on the real line,

Re(Li(t)) > a always because ay e H' for all i and j . Thus I Gk(u,t,A,x,%) |

decreases exponentially as u goes to infinity when t e P(e,l)m"1. Thus the in-

tegral of u on the real line from e to -h» always converges. On the other hand

P(e,l)m"1 is compact and therefore the integral on this path also converges always.

Thus


gives an analytic function of m variables on the whole complex space Cm. Thuswe have the following result:

Theorem 1. £(s,A,x,%) can be continued to the whole space Cm as a mero-

morphic function and has the following integral expression valid for all s e Cm:

' " X ) "

Here we insert a general formula. We assume that %i * 1 for all i. Then

G(y,A,x,%) has no singularity on a neighborhood {y | lyil <£ for all i} for

sufficiently small e and therefore has a power series expansion in y around 0.We look at the expansion

(8a) G(y,A,x,x) - IT exPexP(-xiLi(y))

for G(y,A,x,x) in (4), where (n+1)! = n ^ i i i + l ) ! and y" = II™ 1yi"i. We

also write the coefficient of u ^ ' ^ I W i ' " 1 in Gk(u,t,A,x,%) as B / ^ f o r

/ G N. If %i 9t 1 for all i, then G(y,A,x,%) is holomorphic at y = 0 andGk(u,t,A,x,%) is holomorphic at u = 0 and t = 0. Thus we can compute theexpansion of Gk(u,t,A,x,%) using that of G(y,A,x,%), simply replacing yi byuti (i ^ k) and yt by u. Then, we have

(8b) Bnl(x) = m - ^ B ^ C x ) for 1 = (1 1).

Thus we conclude, if Xi * 1 f°r all U that

(8c) C((l-n)l,A,x,x) = (_i)m(n-i)BniM for a l l 0 < n e Z.

This formula will be used later to get a p-adic interpolation of this type ofL-function and can be proven without using the variable change y h-> (u,t).

fa bÊxercise 3. When A is a 2x2 matrix and % = (1,1), get the explicit

formula of £((l-n,l-n),A,x,%) for positive integers n in terms of Bernoulli poly-nomials given in (3.3b).


§2.5. L-functions of real quadratic fields and Eisenstein seriesIn this section, we interpret the Shintani zeta function in terms of Dedekind zetafunctions and Hecke L-functions of quadratic fields and later introduce Eisensteinseries in this context. Although we are ready to do the same thing for generalfields, the detailed exposition in the case of quadratic fields helps demonstrate whatis going on. First we treat the quadratic field F = Q(Vd) for a square-freeinteger d. We take the standard basis {0)1,0)2} of the integer ring O of F([N-Z, 9.5]); i.e.,

[Vd if d = 2 or 3 mod 4,0)1 = 1 and 0)9 = i /—

I ( l+Vd)/2 if d SE 1 mod 4.

Let / be the ideal group of F and I(m) the subgroup of / consisting of idealsprime to a given integral ideal m. Similarly, let ?P+(m) be the subgroup of /made of principal ideals aO with a = 1 mod m and a ° > 0 for all real em-beddings a of F into R. Then as seen in Theorem 1.2.1 (see also Exercise1.2.1), the (strict) ray class group Cl(m) = I(m)/@+(m) is finite. We consider acharacter % : Cl(m) —> Cx. Such a character is called a (finite order) Heckecharacter modulo m. Then the Hecke L-function of % is defined by

£(s,%) = £ « e I(m), oz>n X(n)N(nys (s e C).

Exercise 1. (a) Show that L(s,%) is absolutely convergent if Re(s)>l .(b) Show that L(s,%) has the following Euler product expansion:

£(s,x) = nPa-x(p)N(Pysy\where p runs through all prime ideals prime to m.

Now suppose d > 0 . Then F = Q(Vd) is a real quadratic field and there aretwo real places. Let a be the non-trivial field automorphism of F, so that

(Vd)a = -Vd. Now O* = {±1 }x{eon I n e Z} for a fundamental unit eo by

Dirichlet's theorem (Theoreml.2.3). Let E = {5 e Ox \ 8 > 0, 5 a > 0},

which is a subgroup of finite index of Ox (in fact E D (OX) and hence

(O*:E)|4). Then we can find a generator e of E and E = {en|n e Z}. We

may assume that e < 1 < ea. The following fact is very important to express the

Hecke L-function as a sum of Shintani L-functions.

Lemma 1. Let F+ = {a e F | a > 0 and oca > 0}. Then each a e F+

c a n b e u n i q u e l y w r i t t e n a s a = e n ( r + s e ) f o r 0 < r e Q , 0 < s e Q and

n E Z .

2.5. L-functions of real quadratic fields and Eisenstein series 55

Proof. We embed F into R2 by X H (x,xa) e R2. It is clear from the figure

that FR = F<g>QR = R-span of {l,e} = R2, where e = (e,ea) and 1 = (1,1). Thecurve in the figure is defined by xy = 1.Multiplication by e (as a map of FR into itself) takesthe line passing through 0 and 1 to that passingthrough 0 and e. We denote by V the cone{s«e+r«l I r > 0, s > 0}. Then we see easily fromthe figure that (R+)2 = UneZenV which is a disjointunion. Thus any a e F+ falls in a unique enV,namely a = en(s«e+r-l) for unique r > 0 and

s > 0. Since 1 and e form a basis of F over Q, we can find r' and s1 in Qsuch that e~noc = s'e+r' in F. This in particular means thate"noc = s'«e+r'«l = s«e+r-l. Since 1 and e form a basis of R2, we now knowthat s = s1 e Q and r = r' e Q. This finishes the proof of the lemma.

Now let x : C\(m) -» C x be a Hecke character and consider the HeckeL-function

£(S,X) = £»<= /(m), oz>« %(n)N(ny\

Let h = #Cl(O) and let us fix a representative set [a\9...,%} for C1(O) con-sisting of integral ideals (i.e. ideals of O). Then for any fractional ideal 5, we canfind a unique a\ such that 6 = a q for a e F+. Applying this to 6m l

9 wecan find (3 e F+ such that 5m~l = paj, namely, 6 = $a}m. Thus{aim,...,%/n} forms another representative set.

Exercise 2. Show that we can take the representative set {#!,...,%} so thateach O{ is prime to the given ideal m.

Hereafter we extend % to the whole ideal group / so that %(6) = 0 for6e I-I(m). Then we can write

Therefore to express the Hecke L-function as a finite sum of Shintani L-functions,it is sufficient to do so for the partial L-function

LetRi = {a e a(lml | a = xi + x2e, 0< xi < 1 and 0 < x2 < 1}.


This is a finite set. In fact, W = {[0,l]+[0,l]e} is a compact subset of R . Onthe other hand, a(xml - aZ+bZfora Z-basis {a,b} since a\Xml

is a fractional ideal. Since {a,b}forms a basis of R 2 via theembedding F —» R2 which takesa h-> (a ,a a ) , a(lml is a discretesubmodule of R2. Thus the subsetWf)ailml is discrete and compactand hence is finite. Then Ri isfinite since W z> Rj. T h eshadowed area is W and the dotsindicate points in Ri. Now for any

Diagram of W and Rt: a £ ^-i ^ i ^ b y L e m m a ^ w e

can uniquely write a = en(r+se) with 0 < r e Q and 0 < s e Q. Let [s]denote the largest integer not exceeding s and (s) = s-[s]. Thus 0 < (s) < 1.Similarly we define

[<r> if 0 < (r) < 1,

U if (r) = 0,and {r} = r-(r)e Z.

Then r+se = (r)+(s)e + {r}+[s]e and (r)+(s)e Since Oz> o[m, we

have a\ m => O. This shows especially that {r}+[s]e e a( m and hence(r)+(s)e G (0,l]+[0,l)erifli"1w"1 = Ri. Thus, we see that

+ = UkeZekIIXi+X2eGRi{xi+x2e+m+ne | (m,n) e N2}.

This shows that

where N(xi+x2e+m+n£) = (xi+x2£+m+ne)(xi+x2ea+m+nea). Now we shallshow that

Exercise 3. Show that there exist y e O and an integral ideal 6 prime to m

such that yO = mB.

We can write (xi+x2e+m+ne)aim = y(xi+X2e+m+ne)y"1fli^« Since yO=mS,yla\m = OiB'1, which is prime to m, and hence %{y~la\m)* 0. On the otherhand, y(xi+X2£+m+ne) = y(xi+x2£)+Y(m+ne) = y(x!+x2£) mod m because


y(m+ne) e yO = mb. Thus %(y(xi+X2E)) = %(y(xi+x2e+m+ne)). (Actually weshould remark the fact that y(xi+X2e+m+ne)/y(xi+X2e) € F+ to assure thisequality). This shows that

Thus finally we know that

This combined with the analytic continuation of the Shintani zeta function yields

Theorem 1. Let ¥ be a real quadratic field and % : Cl(m) —> Cx be a Heckecharacter. Then the Hecke L-function L(s,%) can be continued to a meromorphicfunction on the whole s-plane. Moreover, it has the following expression in termsof Shintani zeta functions:

eOJ,(Xl,X2),l),

where 1 = (1,1) and e is a totally positive fundamental unit of F.

The analytic continuation of L(s,%) was first shown by Hecke in 1917. Actually,one can show, by Hecke's method, that L(s,%) is entire if % is non-trivial andonly has a simple pole at s = 1 even when % is trivial. Here the word "entire"means that the function is analytic everywhere on the s-plane. We will come backto this question later in Chapter 8.

Corollary l(Siegel-Klingen). For a positive integer n, L(l-n,%) e Q(%).

Proof. We here give a proof due to Shintani. We will come back later to thisproblem and give a proof due to Siegel and another proof due to Shimura (seeCorollary 5.2.2 and Theorem 5.2.2). What we need to prove is that for a positiveinteger n,

£((l-n,l-n), o ,(xi,X2),l) e Q.

By the study of the Shintani L-function, we know that

(e4*<M)(e2™-l)r(s)2|s=1.n}


w h e r e 0 ( U f t ) = exK-x.ud + t)) x exP(-x2u(e + e«t))1 - exp(-u(l +1)) 1 - exp(-u(e + eat))

G , t) = expC-x^Ct + l)) x exp(-x2u(et + £a))a^ ' 1 - exp(-u(t + 1)) 1 -exp(-u(et + ea)) '

Writing (n!)"2B'n(xi,X2) for the coefficient of u^11'1^11"1 in the power series ex-pansion of G(u,t), we see that B'n(xi,X2) is a polynomial with coefficients in F.Moreover if we denote by B'n

a(xi,X2) the polynomial obtained by applying a to

all coefficients of B'n(xi,X2), then (n!)~2B'na(xi,X2) gives the coefficient of

U2(n-i)tn-i i n Ga(u,t). Thus by the Cauchy integral formula and the fact that

(e4*'s-l)(e2*/s-l)r(s)21 s=1 _ = 2 ? ^ ? } , we have, noting the fact that(n-1)!

X i e Q ,

^ ^ x ^ U ) = r1n-2TrF/Q(Bln(xi>x2)) e Q.

We now want to introduce Eisenstein series which will be studied in detail inChapters 5 and 9 and which is one of the simplest examples of modular forms.First let us give a brief definition of modular forms. We writeH = {z € C I Im(z) > 0} for the upper half complex plane. Then the group

Gl4(R) = {oce M 2 ( R ) | d e t ( a ) > 0} acts on H via zh-> cc(z) = ^ ± 1cz + d

for a = , (cz+d * 0 because z is not real). We see easily that

(a(3)(z) = a(p(z)). To see a(z) stays in H, we use the following identity: forz1 = oc(z),

a bYz w\ fa(z) a(w)Ycz + d 0c dj[l l j = [ 1 1 A 0 cw + d

Then replacing w by the complex conjugate of z and taking the determinant, we

see that det(oc)Im(z) = Im(a(z)) |cz+d| 2 and hence if det(oc) > 0 and Z G H ,

then a(z) e H. For any discrete subgroup T of GL^R), a modular form on T

of weight (s,k) (s e C and k e Z) is a function f: H -> C such that

f(Y(z))=j'(y,z)klj(y,z)|2sf(z) for all ye T,

where j(y,z) = det(y)"1/2(cz+d) if y = , . We see easily that

j(y8,z) = j(y,8(z))j(8,z) (a cocycle relation).Then we see thatf(yS(z)) = j(y5,z)k | j(y5,z) 12sf(z) = j(y,8(z))k | j(y,8(z)) 12sj'(8,z)k | j(8,z) 12sf(z)

= J(T,5(z))k | j(y,8(z)) 12sf(8(z)) = f(y(8(z)))


and hence the definition is at least consistent. When f is holomorphic (resp.C°°-class), then f is called a holomorphic (resp. C°°-class) modular form. Theimportance of modular forms lies in the fact that it is a non-abelian replacement ofDirichlet and Hecke characters (see Chapter 9 for more details about this fact).This point will be clarified later in Chapters 8 and 9. We will study modular formsin detail in later chapters: Chapters 5 to 10. Here we introduce an example ofmodular forms: the Eisenstein series. Take a positive integer N, integers0 < a < N, 0 0. Put as a function ofz e H and s e C

E'k,N(z,S;a,b) = £( m,n ) G Z2.{ 0 }, (m,n).(a,b) mod N I mZ+n IOne can easily verify that this series is absolutely and locally uniformly (with re-

spect to s and z) convergent if Re(s) > l-y- We can write

I -2smz+nE'k,N(z,S;a,b) = 2,(m,n)eZ2-{0}, (m,n)E(a,b)modN(mZ+

= X'(m,n)Ez2((a+Nm)z+(b+Nn))-k I (a+Nm)z+(b+Nn) -2s

-2s

where Z ' means the summation over all (m,n) e Z for which

(Tr+m)z+(TT+n) * 0. To relate this series with Shintani L-functions, we consider,

for 0 k(z,s;u,v) = Xf(mn)eZ2(u z + v + m z + n)~k ' uz+v+mz+n | "2s.

Then we have-k_ /_ _ a bx(1) E'k,N(z,s;a,b) =

We now split the summation of 9k into four pieces:

Here we write u* = 1-u (0 < u* < 1) and v* = 1-v (0 < v* < 1). Weconsider the function summed on a cone:

£(CDI,G)2;S;U,V) = X(m,n)€N2(ucoi+vco2+mcoi+no)2)"k I ucoi+vco2+mcoi+nco21-2s


for a>i,G)2e C and for u, v not both 0- Then it is easy to see from the abovefigure that, except when (u,v) = (0,1),

q>k(z,s;u,v) = 5(zJ;s;u,v)+^

When (u,v) = (0,1), we have

9k(z,s;0,l) = §(z,l;s;(U) + $(-z,l;s;l,0) + £(z,-l;s;l,0) + £(-z,-l;s;0,l).

We can choose a,p G C X SO that cc,ocz e H' and P,Pz G H' for any givenz E C -R. The choice of p = a works well, but we keep the freedom ofchoosing P differently. Then we see that uocz+va+naz+ma G Hf andupz+vp+npz+mp e H' for any (m,n) e N2. Recall that (xy)s = xsys if

x,y e H', where we define xs by I x | seI'8s for x = | x | e1"9 with - - < 0 < - .

Thus we have(uccz+va+maz+na)'s(uP z +vp+mP z +n P )"s

= {(uaz+va+maz+na)(uP z +vP+mp z +np)} "s

= {aP | uz+v+mz+n 12}"s = (ap)"s | uz+v+mz+n | ~2s.

From this, we conclude that

2(uaz+va+maz+na)"k"s(up z +vp+mp z +np)'faz

)^

In the above argument, replacing (u,v) by (u*,v*), (z,l) by (-z,-l) and (a,p)by (-a,~P), we obtain

faz Qz\(-a)-k(ap)-s^(-z,-l;s;u*,v*) = C((k+s,s),â

Fp J,(u*,v*),l).

Similarly choosing a',p' e Cx so that a'(-z),a' e H' and p'(-z),pf G H \we have

and

T k ( o c ' P T s ^ l * ) C ( ( k + ) ^ Z " Z )(-aTk(oc'PTs^z,-l;s;u,v*) = C((k+s,s),^,Z p(

Z)J,(u,v*),l).

Thus we have a theorem of Shintani ([St6, §4]):


Theorem 2. The Eisenstein series E'k,N(z,s;a,b) can be continued to a mero-morphic function of s to the whole plane and is real analytic with respect to z.Moreover writing u = (-^, ^ ) and supposing u * (0,N) ifN > 1 and u = (0,1)if N = 1, we have the following expression in terms of the Shintani L-function:

E'k,N(z,s;a)b) = N-2s-k{(ap)sakC((k+)H j

J,u*,l)

(a'p')salkC((k+s,s),r'

faz

a,|3,a',p' are m/:en ^ r/zaf az , a e Hf, Pz,P e H1, a'(-z), a ' € H' andp ' ( - z ) , p ' e H \

The analytic continuation of Eisenstein series was first proven by Hecke in the1920's.

Exercise 4. (a) Show that the following subset of SL2(Z) is a subgroup of

finite index: T(N) = {a e SL2(Z) | a = L A mod NM2(Z)}.

(b) Show that E'k,N(z,s;a,b) is a modular form of weight (k,s) on the subgroupr(N).

Now we compute the residue of E'o,N(z»s;a,b) at s = 1, which is a key to prov-ing the class number formula for imaginary quadratic fields. We first deal with the

(a. bresidue at s = 1 of the Shintani zeta function. For each matrix A =

{c d

with a,b,c,d € Hf, we have seen the following integral expression:

(e47lfs-l)(e27lIS-l)r(s)2C((s,s),A,x,(l,l))

= Jp(e)Jp(e,i)Gi(u't)u2s-1ts-1dtdu + Jp(£)Jp(el)G2(u,t)u2s-1ts-1dtdu,

where_ fexp(-xiu(a+bt))l Jexp(-x2u(c+dt))]

U l l u ' t j " [l-exp(-u(a+bt))j x {l-exp(-u(c+dt))f_ fexp(-xiu(at+b))l Jexp(-x2u(ct+d))]- j j x {l-exp(-u(ct+d))J*

Even if t moves around the interval [0,1], Gj(u,t)u is finite provided that

(Gj(u,t) has a pole of order 2 at u = 0). Thus if Re(s) > 0, we do not have to

make the contour integral with respect to t; that is, we have if Re(s) > 0


(e47l/s-l)r(s)2C((s,s),A,x,(l,l))

G2(u,t)u2s-1ts-1dtdu.

Note that the expansion of Gi(u,t) at u = t = 0 is, for the Bernoulli polynomialBj(x) introduced in §3,

i v(u(c+dt))k-1

2/j=o v-lrBjCxi) j , x2 / k=o ^ B k ( X 2 ) k!In particular, the coefficient of u"2 is (a+bfT^c+dt)"1. Now we compute thevalue

0 ( a + b t )1

( c +dt)d t

A.tl{A)A{-log(2i+b)+log{c+d)+log{&)-log{c)} if det(A) * 0,

^ . f d . . ( A , - 0 .

We can compute similarly the integral: J p(£) J G2(u,t)u2s~1ts~1dtdu and have

Jp(e)Jo G2(u,t)u2s-1ts-1dtdu

f 1 d ) - Z o ^ ( d ) } if det(A) / 0,

. f 4 . . ( A , - 0 .

Exercise 5. Explain how one can compute J . . y ^ ^ d t .

Now we start the computation of the residue of E'ojvjCz.sja.b). We have

(<xz

i(a'p')sC((s,s)^ a, p J.Cx.x1),!)

= -2-i(z-zy1{log(-a'z)-log(-pz)+log(p)-hg(a')}.

If ze IT, then z € H', and a and (3 can be taken in H1. Hencelog(az) = log(a)+log(z) and log($z) = log($)+log(z).

On the other hand, if we write a1 = I a' I em and z = | z | e10, then —<0<—2 2

and — < 0 —7i + a<— and—<a<—. This shows2 2 2 2

log(-a'z) = log{ | oc'z | )+(a+0-7c)/ =

Similarly, if p1 = | pf | c^, then - | < %-Q+b < \ and - | < b < \ This

shows that

2.6. L-functions of imaginary quadratic fields 63

(-$ z) = log( I p'z |

Thus we haveccz

Thus Ress==iElo N(z,s;a,b) = —9 . One can easily verify the same formulaNlm(z)

even if Re(z) < 0 and we can conclude with

Corollary 2. Ress=iE'0,N(z,s;a,b) = 2

Exercise 6. Prove the above formula when Re(z) < 0.

§2.6. L-functions of imaginary quadratic fieldsIn this section, we interpret special values of Eisenstein series as the values ofL-functions of imaginary quadratic fields and later prove the class number formula.Let F = Q(V-D) be the imaginary quadratic field with discriminant -D and mbe an integral ideal of O. We consider a quasi-character % : I(m) —> Cx suchthat %((a)) = a k a c m for a = 1 mod m, where c denotes complexconjugation, occm = (ac)m, (a) = aO is the principal ideal generated by a and(k,m) is a pair of integers. Such a character is called a Hecke character (of ©o-type(k,m)). When (k,m) = (0,0), % gives a character of the ideal class groupCl(m) = I(m)/2(m). The Hecke L-function is then defined by

^X) = yLn.I(m),o^X(n)N(nr (s E C).

Note that %(ot) = akoccm = a k - m N ( a ) m = a c ( m " k ) N(a) k . Thus writing

N : 1(0) —» Cx for the Hecke character given by N(a) = NF/QU) for all ideals

a, L(s,%) =L(s-m,Xi) = L(s-k,%2) for %i = %N'm and %2 = XN'k. Note

that %i is of type (k-m,0) and yji is of type (0,m-k). Thus without loss of

generality, we may assume that % is of type (k,0) or (0,k) for k > 0. Since

the argument is the same in either case, hereafter we assume that % is of type

(0,k) with k > 0.

Exercise 1. Assume F = Q (V d) for a square-free integer d > 0. Let

% : I(m) -> C x be a quasi-character such that %((cc)) = cckaam whenever

a = 1 mod m for a pair of integers (k,m) and the unique non-trivial field auto-

morphism a of F. Show that k = m. (Use the existence of non-trivial units.)


By Exercise 1, we do not lose generality by assuming k = m = 0, i.e. % is offinite order, when F is real quadratic.

As in the case of real quadratic fields F, we take a representative set {a\,...,%}of Cl(m) consisting of integral ideals. Then we see that

where [i(m) = {£ E ( f l ^ s 1 mod /rc}, which is a finite group and trivial ifm is sufficiently small. To express SaGa.-i a H l moda-imackA^(a)"s as a sum of

special values of Eisenstein series, we pick a basis (0)1,0)2) = (0)14,0)2,1) of of1

and a positive integer N such that N e m. We may assume that a* is prime to

N and Z{ = 0)1/0)2. By changing 0)2 to -0)2 if necessary, we may assume that

Zi = 0)1/0)2 e H. Let

R i = {(a,b) e Z 2 | 0 < a < N, 0 2-k I ^ I -2s+2kE'k,N(0)i/0)2,s-k;a,b).( a f b ) e R i G>2

Thus we have

/(m), oz>nX(n)N(nys

I ^ ( w ) I ' 1^(a,b)ERic°2,i"k i co2,i I •

2 s + 2 k E lk f N (z i , s -k ;a > b) .

Theorem 1. L(s,%) ca« Z?e continued to a meromorphic function on the wholes-plane.

In particular, when k = 0, and m= O, and % is trivial, then

CF(S) = w-1X"1^(«i)"Ilo)2.il"2lEIo.i(zi.s;O,l),

where H is the class number of F and w = \(f\. Note that

Idetf"1 "2

2.6. L-functions of imaginary quadratic fields 65

On the other hand, we see thatfCGj ©2^ _ _ _ 9 ,

det _ _ = O i CG2-CO2CO1 = 0)2 0)2(zi-Zi) = l o ) 2 r 2 V - H m ( z i ) .

By the residue formula of E'o,i(zi,s;O,l) (Corollary5.2):

we have Dirichlet's residue formula:

Theorem 2. Let H be the class number of the imaginary quadratic field ofdiscriminant -D. Then we have

— T = .wVD

Let R = Z [V^D] 3 Z. Thus R is a subring of the integer ring O andR = Z[X]/(X2+D). Let p be an odd prime in Z prime to D. Then we seeeasily that X +D mod p is reducible if and only if -D is a quadratic residue

mod p. In fact, if — = 1, then we can find a e Z such that

a 2 = -D mod p and R/pR = F[X]/(X-cc)(X+a) = F 0 F for F = Z/pZ.Thus pR = p\C\p2 m R for two distinct prime ideals p\ and p2> IfD = 0 mod 2, then O = R and thus

(1) pO = p\p2 for p\*p2 if and only if — = 1.

If D is odd, then for co = , the minimal polynomial of

co: X2-X+N(co) = 0 is reducible over F if and only if there is a e Z suchthat a 2 =-D mod p, because 2 is invertible in F. Thus (1) is still true. For

example, if D is a prime, then D = 3 mod 4 (i.e. (=£\ = (-l){D'1)/2 = -l),

and we know from the quadratic reciprocity law that

D

Thus the map p H-> — is a Dirichlet character modulo D. More generally, we

have the following fact:

Exercise 2. Using the quadratic reciprocity law, show that the

map: p n — is induced by a Dirichlet character %D : (Z/DZ)X -> {±1}.

Thus we see that pO = p\p2 if and only if %D(P) = 1- Now we look at theEuler factor of the Dedekind zeta function of F. We see


XD(P) = !

XD(P) = - 1

XD(P) = 0

Euler factor at prime ideals dividing p

(l-iV(Pl)-s)(l-N(^)-s) = (l-p"s)(l-XD(p)p-s)

l-N(pYs = (l+p-s)(l-p-s) = (l-p-s)d-%D(p)p-s)

l-iV(p)-s = (l-p-s)(l-xD(p)p-s)

l = C(s)L(s,%D).

Thus we haveCF(S) = Up

Thus we know that

L(l ,xD) = Ress=iC(s)L(s,%D) = Ress=iCF(s) =

On the other hand, by the functional equation, we can relate L(1,%D) with the

value L(0,%D) = -D"1I^:i1xD(a)a (see §3). Thus we have

Theorem 3 (Dirichlet's class number formula). H = -—Xa=Ti %D(a)a-

Exercise 3. Using the above class number formula, show(i) For a prime p > 3 with p = 3 mod 4, the number a of quadratic residues

in [0, S] exceeds the number b of quadratic non-residues in the same interval;

(ii) If p > 3 and p = 3 mod 8, then a-b = 0 mod 3.

§2.7. Hecke L-functions of number fieldsLet F be a general number field and let I be the set of all embeddings of F intoC. Let I(R) be the subset of I consisting of real embeddings and putI(C) = I-I(R). Then the number of real places of F is given by r = #I(R) andthe number of complex places is given by t = #I(C)/2. We start with the study ofthe fundamental domain of F+/E, where

E = ( e e O x | £ ° > 0 for all o e I (R)},

F+ = {a e F x I a a > 0 for all a e I(R)}.

Thus if I(R) = 0 , then we simply put F+ = Fx. The result we want to provefirst is

Theorem 1 (Shintani [Stl, St5]). Let E' be a subgroup of finite index in E.Then there are finitely many open simplicial cones Q = C(vii,...,Vimi) with

vij G F+ such that C = UjCi and F+ = U£GE-eC are both disjoint unions. We

can take the Q's so that there exists ua>i e Cx for each a e I and i such

that Re(ua)iVija) > 0 for all j = 1, ..., mi.

2.7. Hecke L-functions of number fields 67

Here an open simplicial cone C(vi, . . . ,vm) in an R-vector space or Q-vectorspace V with generators vi e V is by definition

C(vi, ..., vm) = {xivi+---+xmvm I Xi > 0 for all i},

where the m vectors v are supposed to be linearly independent. We divideI(C) = X(C)U£(C)c into a disjoint union of two subsets X(C) and its complexconjugate Z(C)c for complex conjugation c and write £ for I(R)LE(C). Inthe theorem, we regard Q as an open simplicial cone in the real vector spaceV = F ^ = F<8)QR = R I ( R ) x C I ( C ) . We then embed F into V bya h-> (oc°)ae £ G V. Then F is a Q-vector-subspace of V which is dense in V.We put

V+ = R+I<R>x(Cx)L(C),

where R + = ( x e R | x > 0 ) . Then F+ = V+flF.

Proof of Theorem 1. Since the proof is the same for any E', we simply treat onlythe case E' = E. Consider the hypersurface X in V+ defined by

X = {(xa)aG s | N(x) = ITa G i

| 2 =

Then, for S = {x e C \ | x | = 1}, we have p : X = S^R1"""1. In fact theprojection to Sl can be given by xh ) (xa/1 xa | )aeZ(C) G Sl and the projectionto R1**"1 is given by xi—> /(x) = (/a(xa))z-{T}» where we exclude oneembedding x e Z and /a(xa) = log{ I xa |) or 2log( I xa |) according as a isreal or complex. By definition E acts on X by componentwise multiplication.The image of E in R1"1"*'1 is a lattice by Dirichlet's theorem (Theorem 1.2.3) andhence X/E is a compact set. Thus we can find a compact subset K of X suchthat X = UeGEeK. We can project V+ to X via x H> N(x)~1/dx for

d = [F:Q], which will be denoted by n. This is obviously continuous and

surjective and hence takes the dense subset F+ to a dense subset of X. We can

find a small neighborhood U of 1 in the multiplicative group X such that

eUflU = 0 if 8 * 1 in E. We may assume that U = CoflX for an open

simplicial cone Co in V+ with generators in F+. Thus UXGKfto(F )x^° - ^*

Since K is compact, we can choose finitely many xi e 7i(F+)flK such that

UjLjXiCo ^ K. We write F+flxiCo = C0,i. Note that eCo,iflCo,i = 0 if

6 * 1 because eUflU = 0 if 8 * 1 in E and Co is the R+-span of U.

Moreover Co,i is a cone with generators in F+. In fact, taking yi e F+ such

that 7t(yi) = xi, then Co.i = XiCo = yiCo. Since Co = C(vi, ..., Vd) with

Vi G F+, we see that Co,i = C(yjVi, ..., yiv^) is a cone generated by vectors in

F+. Now admitting the following lemma, we finish the proof of the theorem:


Lemma 1. Let C and C be two polyhedral cones whose generators are in F(where a polyhedral cone means a disjoint union of finitely many open simplicialcones). Then CflC, CUC and C-C are all polyhedral cones whose generatorsare in F.

We have F+ = U"=1UeeEeC0,i and Co/leCo,! = 0 if e * l . If n = l , C0,i

is the desired cone. When n> 1, we define C i j = Co,i and for i> 2,

Ci,i = Co,i-UeeEeCo,i. Since Co,i ( i > l ) intersects with eCo,i for only

finitely many e, Cij is a polyhedral cone by the lemma. We now have

C i , i n { U £ G E e C i , i } = 0 ( i > 2 ) and F+ = U i = 1 U £ G E eCi , i and

Ci,iDeCifi = 0 if e * 1. We now construct inductively (on j) polyhedral

cones Cj,i with generators in F+ for each 0 < j < n by Cy = Cj.î if i < j

and Cjj = Cj_i,i-UeeEeCj_ij (Cy = CJ.IJ) for j < i. Then we see that

Cj,inU£GEeCj,k = 0 for i > j > k, F+ = Ur=1UeEEeCj,i,

and Cj/leCj.i = 0 if e * 1.

Then Cn,i is a disjoint union of finitely many simplicial cones which give thedesired simplicial cones. This proves the first assertion of the theorem. We cansubdivide the cones Cj so that the last assertion follows.

Exercise 1. Give a detailed proof of the last assertion of Theorem 1.

Proof of Lemma 1. We may assume that C and C are open simplicialcones. Write C = C(vi, ..., vm), where the vi's are linearly independent. Byadding vm+i, ..., v<i, we have a basis vi, ..., va of V. Let V* be the dualvector space of V and Vi* be the dual basis of the vi's, i.e., Vi*(vj) = Sij. Inparticular, for v = xiVi+-"+XdVd, xi = vi*(v). Thus, we see that

C = ( v e V | Vi*(v) > 0 for 1 m}.

Since the vi are in F, vi* induces Q-linear forms on F. (We call such a form aQ-linear form.) Similarly, there are d linearly independent Q-linear forms Wi*on F and an integer n between 1 and d such that

C = {v e V | wi*(v) > 0 for l n } .

Thus C-C and CflC are disjoint unions of sets of the following form for finitelymany Q-linear forms Li on a vector subspace W of V (W may not be a propersubspace but W is an R-span of a Q-vector subspace of F):

X = {w G W | Li(w) > 0 for i = 1, ..., / }.


We may assume that {Li} is a minimum set of linear forms to define X. Forexample, when X = CDC, then we take W to be

( x e v | Vi*(x) = 0 and Wj*(x) = 0 if i > m and j > n } .

Since CUC = (CnC^LKC-C'jUCC-C) is a disjoint union, it is sufficient toshow that X is a polyhedral cone with generators in F. When dim(W) = 1 or2, the assertion is obvious. Thus we shall prove the assertion by induction ondim(W). Let

Xj = {w G W | Li(w) > 0 for i * j and Lj(w) = 0}-{0},X = {w e W | Li(w) > 0 for i = 1, ..., /}-{0}.

Since Xj is contained in Ker(Lj) which has dimension less than dim(W), by theinduction hypothesis, Xj is a disjoint union of open simplicial cone. Moreover,Xj-Ui jXi is a disjoint union of open simplicial cones, and hence it is easy to seethat X-X = UjXj is also a disjoint union of open simplicial cones. Write thesecones as UjXj = UkC(vki,...,Vkik). Let u be a point in XflF, which existsbecause X is open in W and Ff|W is dense as already remarked. Sincevki>---»vkik a r e in a proper subspace of W and u is not in the subspace spannedby Vki,...,Vkik, u,Vki,...,Vkik are linearly independent. Write Ck(u) forC(u,Vki,..-,Vkik). Then we claim that

X = UkCk(u)UR+u (disjoint).

By definition, Li(x) > 0 for all i if x e X. Thus in particular, if x = Xu forX e R , then Li(x) = A,Li(u) > 0, Li(x) > 0 and Li(u) > 0. Thus,RuflX = R+u. Suppose that x e X is not a scalar multiple of u. Let s be theminimum of Lj(x)/Li(u) for i = 1,..., /. Then s > 0. There is an index i(maybe several) such that s = Li(x)/Li(u). Then Lj(x-su) = Lj(x)-sLj(u) > 0and Li(x-su) = 0. Thus x-su e X-X. Therefore x-su G Ck for a unique k(because X-X is a disjoint union of the Ck's) and thus x G Ck(u). Thisshows the desired assertion.

Exercise 2. Write down the proof of X-X = UjXj explaining every detail.

Let Z[I] be the free module generated by embeddings of F into C. We can

think of each element t, e Z[I] as a quasi-character of Fx which takes a e Fx

to a^ = I I a G l aa ^ a G Cx. A quasi-character X : I(m) -» Cx for an ideal m

of O is called an arithmetic Hecke character if there exists ^ G Z[I] such that

X((a)) = oc for all a G P(W), where

P(m) = {a G F+ I (3(a-l) G m for some (3 G O prime to m}.


Theorem 2 (Hecke). If X is an arithmetic Hecke character modulo m, then

L(s,X) = £ne /(m), ODn X(n)N(n)~s can be continued to a meromorphic function on

the whole complex s-plane C.

Proof. Let {a\9 . . . ,%} be a representative set of ideal classes for //fP+ (seeExercise 1.2.1). We may assume that a\ are all prime to nu Sincestill gives a representative set, we can write

Now we take the fundamental domain C = U^Cj for F+/E with disjoint open

simplicial cones Cj. Here note that for a positive rational number u, Cj = uCjby definition. In particular, NCj = Cj for any positive integer N. Thus we mayassume that Cj is generated by totally positive integers in 0. Fix a set ofgenerators {vi,..., vb} of Cj in O and consider the Shintani L-function

where va = (v i a , ..., vba) e C b , x = x1v1 + «"+xbvb with Xi e (0,1],

x a = viaxi+---+VbaXb and n«va = viani+--+Vbanb. By Theorem 1, we also

have ua e Cx such that all the entries of uava have positive real parts. Then

where we have to choose the branch of /<?g(ua) suitably and A = (t(uava))aGi is

a bxd matrix with coefficients in H'. Thus by Theorem 2.4.1, £j(s,x) can becontinued to an analytic function of s. Now we try to express L(sX) as a sum of£j(s,x). Let

Rij = ai"1w'1n{xiv1 + - '-+xbvb I 0 < xk < 1 for k = l , . . . , b } ,

Cj(Z) = {nivi+---+nbvb | 0 < nk e Z for all k}.

Then Rjj is a finite set and any oce a{1m1np+ can be written as a = e(x+p)for unique e e E , x e Rij for a unique j and P G CJ(Z). Thus we have

L(S,X) = £ . = 1 XaE(aimnF+)/E

X t J X x ^ C j ( ( s - § a ) a € I , x ) for SG C.

In fact, choosing yeO such that yO = mB and 5 is prime to m, we can write

(x+p)ai/rt = Y(x+P)y'1Oim. Since yO = mB, yla\m = a\b"x, which is prime to


m and hence X(yA 0{m) ^ 0. On the other hand, y(x+p) = yx+yP = yx modx m

because yP e yO = mB. Thus ^(y(x+P)/yx) = (y(x+P)/yx)^. (Actually we should

remark the fact that y(x+(3)/yx e F+ to assure this equality.) This shows that

which finishes the proof of Theorem 2.

Exercise 3. Write down every detail of the proof of the equalityA,((x+p)aiwi) = (x+p)^x"^(xfljm), because in the proof we implicitly assumedthat xo{m is prime to nu

Corollary 1. / / n e Z with n < £a for all a, then L(n,X) e Q(k).

This has meaning only when F is totally real, because we will see in Chapter 8that L(n,X) is zero otherwise.

Now we briefly discuss Hilbert modular Eisenstein series. We will take up thistopic in adelic language later in Chapter 9 in more detail. Here we suppose that Fis totally real, i.e. I = I(R). Consider SL2(0) and its congruence subgroup foreach integral ideal m

i fl 0>lT(m) = {a e SL2(O) | a = I Q x I mod mM2(O)}.

We consider the product H1 of d copies of the upper half complex plane Hindexed by the set I of embeddings of F into R. Then we let SL2(0) acton H1

fa bY faCT bCT>|so that cc((zo)) = (or(za)), where ^ dJ = | o dCTjE SL2(R) acts on each

component H via the linear fractional transformation. To define modular forms ina weak sense, we consider not only weights in Z[I] but also those which areformal linear combinations of symbols a and ac with o e l , where c denotescomplex conjugation. We denote by Z[lUlc] for Ic = { a c | a e l } the moduleof weights. A function f i r f - ^ C is called a Hilbert modular form on T(m)of weight (k,s) if

f(y(z))=j(y,z)k|j(y,z)|2sf(z) for all y e T(m\

fa bwhere k = S a e n j i c k a a e Z[lLJlc], s e C and for y =

c d i

k = noeit^zo+d^^zo+d0)1^} I j(yz) 12s = naGi I (cazo-fda) |2sj(y,z)k =

Then, for (a,b) e (O/m)2, the Eisenstein series of weight (k,s) is defined by


E'k>m(z,s;a,b)

where the summation runs over{(m,n)e O2-(0,0) | (m,n) E (a,b) mod

and

Here E(m) = ( e e E | e = 1 mod m], which is a subgroup of E of finiteindex, and E(m) acts on O2 by (m,n)e = (me,ne). Then E'k,m(z,s;a,b) isconvergent if s has sufficiently large real part. We now fix finitely manysimplicial cones Cj so that C = UiCi gives a representative set of Fx/E(m).

Exercise 4. Show that (Fx)2 = Uee E(m)e(CxFx) is a disjoint union.

By dividing Fx as a disjoint union of finitely many simplicial cones C j , we

know from Exercise 5 that (Fx)2 = UeGE(m)e{UiJCixClj}. Note that Q x C j

is again an open simplicial cone C({(vi,0),(0,Wj)}) for Vi e Q and Wj e Cj.

We define, for x = xivi+---+xava with xi e (0,1] and y = yiwi+---+ypwp

with y ie (0,1],

Cij(s,x,y) = X(m,n)ENaxNbnaeI(x°za+ya+m.v°z0+n.wa)"Sc.

Then in exactly the same manner as in the proof of Theorem 2, we can express

E'k,m(z,s;a,b) as a finite sum of Cij((s+ka)aeujic,x,y) for s e C. Since

£ij(s,x,y) can be continued to a meromorphic function on the whole s-plane and

gives a real analytic function of z for a fixed s, we have

Theorem 3 (Hecke). The Eisenstein series has a meromorphic continuation tothe whole s-plane. When E'k,m(z,s;a,b) is finite at se C,then E'k,m(z,s;a,b)gives a real analytic Hilbert modular form ofweight (k,s) on T(m).

We will return to the study of Eisenstein series in more detail later in Chapters 5and 9.

Chapter 3. p-adic Hecke L-functions

In this chapter, we first construct p-adic Dirichlet L-functions for Q usingEuler's method of computing the values L(n,%). We follow Katz's article [K2] inthis part. Then we generalize the result for Hecke L-functions of totally real fieldsusing the method in [K6]. Roughly speaking, a p-adic L-function is a p-adic ana-lytic function whose values coincide with those of its complex counterpart atenough integer points to guarantee its uniqueness. Here we mean by a p-adicanalytic function on an open set U of Zp a function on U having values in afinite extension F of Qp which can be expanded into a power series in (z-u)(with coefficients in F) for each u e U convergent on an open neighborhood ofu in U. A p-adic meromorphic function on U is a quotient of two p-adic analyticfunctions on U. The p-adic L-function we discuss was first constructed for Q byKubota and Leopoldt [KL] as a continuous function and later shown to be p-adicanalytic by Iwasawa [Iwl]. The result was generalized to totally real fieldsindependently by Deligne and Ribet [DR], Cassou-Nogues [CN] and Barsky [Ba].Katz's method in [K6] is an interpretation of the methods of [CN] and [Ba] interms of formal groups.

§3.1. Interpolation seriesWe recall the definition of the binomial polynomial: as a rational polynomial withvariable x, for a non-negative integer n,

©• n ! lx " *• x '

1 if n = 0.Then it is obvious that this polynomial has integer values on the set N of

non-negative integers. For negative integers -m (me N), we have

("™) = (- l ) n(m +n

1 1 ' 1 ) G Z. Thus in fact, ( * ) has integer value on Z. Such

a polynomial is called numerical By the binomial theorem, we have a formalidentity in the formal power series ring

n=0

When XG N (={ne Z | n > 0}), this power series is actually a polynomialof degree x and the identity holds for any value of X. Thus, for x e N,

By comparing the coefficients of Xm of (1+X)x(l+X)y = (l+X)x+y, we have

74 3: p-adic Hecke L-functions

i cocan=0

Let K be a field extension of Q and A be a subring of K. Let f: N -> A beany function. We define the n-th coefficient an(f) of f by

(2) an = an(f) = X(-l)k(jj)f(n-k) = X(-l)n-k(j)f(k) e A.k=0 k=0

Using these coefficients, we define a new function f* : N -> A byf*(x) = XI=o a*( f )(n) for XG N«

This is actually a finite sum X an(f) ( n ) because ( _ ) = 0 when n > x. Wen=0 V n / V n /n=0

now compute the value of f*. By definition, we have

( j ) { tn=0 n=0 [k=0

The last equality follows from (la). This shows that

(3) f(x) = XI= 0 an®(n) on N.

Exercise 1. Let (ai,. . . , ar) be an r-tuple of non-negative distinct integers and(bi,.. . , br) be an r-tuple of real numbers. Show the existence of a polynomial Pwith coefficients in R such that P(aO = bi for all i = 1, ...,r.

We now prove the uniqueness of the expansion (3). We suppose the existence ofcoefficients {a'neK}nGN such that

Then by (3), putting bn = an-a'n, we see that

Supposing the existence of coefficients bn, not all zero, we take the smallest nwith bn * 0. By the minimality of n, bm = 0 for all 0 < m < n. Thus

O » G ) - 0 and in particular X ^ b m ^ - O . Since (£)=0 if

n < m, we conclude from this that bn = ( n ) b n = 0, which contradicts our

3.2. Interpolation series in p-adic fields 75

choice of n. Thus bn = 0 for all n and the expansion as in (3) is unique.Summarizing this, we have

Proposition 1. For a given function f: N ~> A, there exists a unique se-

quence of elements {an(f)}neN in A such that f(x) = ^ ~ = 0 &n(n) for all

x e N. Moreover { ( n )} n G N 8^ves a basis of the ring of numerical polyno-

mials.

§3.2. Interpolation series in p-adic fieldsWe are now able to describe, using interpolation series, the space C(Zp;A) ofcontinuous functions on Zp having values in any closed subring A of the p-adiccompletion ¥p of a number field F. A continuous function f: Zp -» Op is infact determined by its restriction to the natural numbers N by the density of N inZp (see (1.3.3)). Thus f is uniquely determined by the interpolation series of itsrestriction to N:

(1) f(x) = X^ = o an(f)(*) on N.

We are going to show that the right-hand side of the above identity converges inOp uniformly for any x e Zp. After showing this, we will know that the right-hand side gives a continuous function which coincides with f on N and hence isequal to f on all Zo. What we need to show is lim an(f) = 0. Then the conver-

gence of (1) and the uniformity of the convergence follow from the strong triangle

inequality (1.3.1) and the fact (see (2) below) that | ( * ) | p < 1 for all x € Zp.

Exercise 1. Show the convergence of (1) and the uniformity of the convergenceassuming lim an(f) = 0.

n-»oo

Let us now show that Urn an(f) = 0. For any given polynomial P(X)

(= ao+a]X+- • *+adXd) with coefficients in Qp, we see that

I P(x)-P(y) I p < max (| an | p | x-y | p | x^+x^V**'+yn~11 p)0<n<d

< max( |an | p ) |x-y | p on Zp.

Thus x h-» P(x) is continuous on Zp. In particular, the function x h-» ( nj is

continuous on Zp. Since this function has values in N on the dense subset Nof Zp, its continuity shows that it takes Zp into Zp. Thus

(2) I C D ' P - 1 fora11 XG Z P and nG N*


Since Zp is compact (because it is a projective limit of compact (even finite) setsZ/pnZ [Bour2,1.9.6]), any continuous function f: Zp ~> Op is uniformly con-tinuous [Bour2, II.4.1]; thus, for any given positive e, we can find a small posi-tive 8 such that | x-y I p < 8 => | f(x)-f(y) I p < e. Now assuming x,y e N,we take e to be p"s and write 8 = p~l (t = t(s)) for this e. We may of courseassume that t > 0. Then | f(x+pl)-f(x) | p < p's. If we let fy(x) = f(x+y), thenby definition, we have

Using the identity (Lib), (x+y) = S t o C X n L ) ' w e h a v e

m=0 n=0

Now we are going to interchange the summation with respect to m and n in theabove formula. Since x,y e N, m (resp. n) actually runs from 1 (resp. 0)to x+y (resp. m), the summation is in fact finite and thus the interchange of thetwo sums is legitimate. Thus we have

n=0 m=n n=0 k=0

In particular, we have, by the definition in § 1 of an(fy),

(3) 2>n+k(f)(k) = an(fy) = XM)n"k(J)f(k+y).k=0 k=0

Now replacing y by p l (= 8"1), we see that

(4) an+pt(f) = - X . W f ) + XC"1) (k) f ( k +P l )" an(f)j=l V J J k=0

(3.1-2) P^YP1! V^= - X an+j(f) + X(-!)r

j=i v J y k=o

Since | f(k+pf)-f(k) I p < p"s for all k, | P | p < p"1 for j = 1 p l -1

and I an(f) I p < 1, the strong triangle inequality shows that

(5) I an+Pt(f) IP < max(p-11 an+i(f) I p,...,?"11 an+pt(s)_i(f) I p,p"s).

3.2. Interpolation series in p-adic fields 77

In particular, | an(f) I p < p"1 for n > p 1 ^ . Using this inequality for n > p 1 ^ ,we have | an+pt(2)(f) | p < p~2 if n > pt(1). In other words, | an(f) | p < p"2 ifn > p ^ + p 1 ^ . Repeating this process m times, we have

| a n ( f ) | p <p" m if n>pt(1)+pt(2)4---+pt(m)-

This shows the desired assertion: | lim an(f) | p = lim | an(f) I n = 0. We haven—>oo ? n_>oo ^

proven the following theorem due to Mahler [Ma] for A = Op:

Theorem 1. Let A be a closed subring of F r Then we have

(i) For each function f : Zp —> A, an(f) e A;

(ii) A function f: Zp —> A is continuous if and only if lim an(f) = 0. In this

case, the interpolation series E°°_ an(f)(n) converges to f(x) for all x e Zp.

Proof. We have already proved the assertion (i) and the sufficiency in the assertion(ii) for A = Op. The necessity for (ii) is obvious because the interpolation seriesconverges uniformly on Zp and coincides with f on the dense subset N in Zp.The sufficiency for arbitrary A can be shown as follows. Since Zp is compactand f is continuous, f is bounded. Thus we can find a sufficiently large integera > 0 such that paf has values in Op on Zp. Then we apply the result alreadyproven for A = Op to paf and recover the result for general A.

We write O(Zp; A) for the space of continuous functions on Zp having values inA. Since Zp is compact, for f e C(Zp;A), f(Zp) is a compact subset of A[Bour2, 1.9.4] and hence is bounded. Thus the uniform norm | f | p

= Supx6zp I f(x) I p is a well defined real number. This norm satisfies the strongtriangle inequality. Since the uniform limit of continuous functions is again con-tinuous, the A-module C(Zp;A) is complete under this norm (i.e. it is a BanachA-module).

Corollary 1. We have | f | p = Supn | an(f) \v for f e C(Zp;A).

Proof. By definition of an(f), we have

I an(f) | p = I ^ L o H ^ G ^ a O ^ Iflp.On the other hand, for each x e Zp

m m

lf(*)lp= l iHmoXan(f)(J)|p=iBmoIXan(f)(ylp < Supn | an(f) I p.n=0 n=0

This shows that I f I p < Supn | an(f) | p and the result.


§3.3. p-adic measures on Zp

In this section, we introduce the theory of p-adic measures supported on Zp. Inlater sections, we generalize this notion to measures on arbitrary profinite groups.The theory of p-adic measures was initiated by B. Mazur in the 70's in order toconstruct p-adic standard L-functions for the algebraic group GL(2) ([Mzl] and[MzS]). Increasingly the usefulness of the theory is recognized by everymathematician working with p-adic L-functions because of its conciseness and itsrelevance to the Iwasawa theory.

Let A be a closed subring of F^. An A-linear map (p : O(Zp; A) —> A is calleda bounded p-adic measure if there exists a constant B > 0 such thatI q>(<|>) | p e C(Zp;A). Let Meas(Zv\A) be the space of allbounded p-adic measures on Zp having values in A. Then we can define theuniform norm on Meas(Zp;A) by

I 9 I P = S u p | ^ | p = 1 1 (p((|)) |p = S u p ^ 0 ( | cp((|)) |p/| (|) | p ) .

Obviously, Meas(Zp;A) is a p-adic Banach A-module under this norm. Inparticular, if tym converges uniformly to 0 as m —> °°, we see that

I q>(4>m)-<P(<l>) I p ^ I nT<l> I p

a n d thus l im (p(<t>m) = 9(<t>)- W e s o m e t i m e s w r i t e (p(<|)) as J(t>d(p = Jzp<|>d(p.

Since each continuous function <|> has a unique expansion into an interpolation se-

ries (|)(x) = X ° ° - o a n ^ \ n ) ky Mahler's theorem (Theorem 2.1) with

an(<|>) e A satisfying Hmâ^) = 0 and since (J)m = £™=oan(<|>)(n) con-

verges to $ uniformly, we know that

Thus the measure 9 is determined by the sequence of numbers

{J(*)dcp |ne N} c A .

This sequence of numbers is bounded because

I p .

Conversely if a sequence of bounded p-adic numbers bn in A is given, the infi-

nite sum Eoo_0bnan(cl)) converges absolutely because of the strong triangle in-

equality (1.3.1) and lim an((|)) = 0. Thus we can define a measure 9 by

3.3. p-adic measures on Zp 79

Note that

i j^dcp i p = i x r = 0b n a n ( ( i ) ) i p

< maxn I bnan(<t)) | p < (Supn I bn IP) I <|> I p by Corollary 2.1.

Thus cp is a bounded measure having values in A and

| (p | p < Supn | bn I p.

Since (nj = 1, | f n ) | p = 1, and thus we conclude from J ( n)d(p = bn

that | (p | p > I bn IP for all n. This shows that I (p I p = Supn I bn | p. We haveproven the following fact noted by Katz:

Theorem 1. Given a bounded sequence {bn} of numbers in A, we can define

uniquely a bounded p-adic measure (p satisfying J ( n Jd(p = bn for all n by

All bounded measures on Zp are obtained in this way. Moreover we have

I 9 I p = S u p n I b n I p.

Since (n j is a polynomial of x with coefficients in Q, the value J ( n Jd(p is

uniquely determined by the values of Jxmd(p for all m > 0. Thus we record

Corollary 1. Each bounded p-adic measure having values in A is uniquely de-termined by its values at all monomials xm (m = 0,1,2,- ••)•

We should remark that the measure corresponding to an arbitrarily assignedbounded sequence of bn is not necessarily bounded: for example, if we assignthe value of a measure 9 by

f mj f 1 if m = 1,Jxmdcp = ( 0 otherwise>

c(x\ (-l)Pthen we see easily that n d(p = ——— and thus, (p is no longer bounded.JvP ) p

Exercise 1. Give a detailed proof of the above fact.

If we assign the value am to Jxmd(p for a e Z p , then naturally

](n)d(p = ( n J e A, whose absolute value is bounded by 1. Thus in this case,

the desired bounded measure exists and is called the Dirac measure at a (i.e. theevaluation of functions at a).

80 3: p-adic Hecke L-f unctions

Exercise 2. Suppose that Jf(x+y)d(p(x) = Jf(x)d(p(x) for all y G Zp and

f G C(Zp;A). Show that (p = 0 if 9 is a bounded measure.

§3.4. The p-adic measure of the Riemann zeta functionLet £(s) (s G C) be the Riemann zeta function defined in §2.1. We already knowthat £(-m) e Q for m e N. In this section, for each positive integer a> 2prime to p, we show the existence of a p-adic bounded measure £a on Zp hav-ing values in Zp such that

JxmdCa = (l-am+1)C(-m) for all m G N.

If such a measure exists, it is unique by Corollary 3.1.

As in §2.1, for the function £, : Z --» Z given by

1 if n ^ O mod a,W = 1

we consider the function[l-a if n = 0 mod a,

(la)

Then by Theorem 2.1.1, we have

(lb) (l-am+1)C(-

In order to show the existence of the measure £a> writing ( nj = Z£=ocn!

with cn,k G Q, we need to prove that, for all n G N,

I J Q d C a I P = I JZ=o cn,m(l-am+1)C(-m) I p = I 3nY(t) 11=1 I p < 1,

where we write, as a differential operator,

To show this boundedness, we prepare with two lemmas:

tndn

Lemma 1. We have an = ~T—r-n! dln

3.4. The p-adic measure of the Riemann zeta functions 81

Proof. The assertion is clear for n= 0 and 1. We prove the assertion in generalby induction on n. Thus, assuming the above formula is true for n, we compute3n+i. An easy computation using the definition of the binomial polynomial showsthat

/ x \ (x-n) . / x \U l j = oST)!n!W-

Then we see that

| 1

which proves the assertion.

By Leibniz's formula, we see thatdn(fg) = y n / n \ drf dn"rg

dtn Z, r = 0 W dtr dtn-r •

Multiplying both sides of the above formula by tn/n!, we get

(2) 3n(fg) = X^= o Orf)On-rg).

Lemma 2. Let Rf = { - ^ | P(t), Q(t) e Zp[t] and | Q ( l ) | p = l} . Then

R1 is a ring and is stable under 3n for all n.

P P'Proof. Taking Q and QT from R', we have

P P ' PQ'±P'Q P P 1 PP'Q Q 7 " QQ' G R a n d Q x Q7 = 7 6 R

which shows that R1 is a ring. We now show R' 3 9nR' by induction on n.When n = 1, we see that

which shows the result. Now assuming the result is true for dm for m < n-1,

we shall show R' 3 dnR\ Applying (2) for f = Q and g = Q-1 with

P/Q G R\ we have X"=o^r®^n-r(5 ^ = °* S i n c e 9 ° ^ = ^ ' w e h a v e

By the induction hypothesis, 3n-r(Q"1) e R' for r > 1. Since R' is a ringcontaining Zp[t] s 3rQ, the above formula shows that 3n(Q"1) ^ R'- Thenagain by (2), we have


Since 9n(Q"1) e R', again by induction assumption, we see that dn(-Q) e R \

We are now ready to prove

Theorem 1. Let a e N with a > 2 and (a,p) = 1. Then there exists a

unique bounded p-adic measure £a on %p having values in Zp such that

JxmdCa=(l-am+1)C(-rn) far all me N.

Proof. As already remarked, we only need to prove

U(n>Calp= kBy definition, we see that

and hence *F e R'. Thus by Lemma 2, we see that d^ e R'. This impliesP i i

3n¥ = Q for P, Q e Zp with | Q(l) I p = 1. Thus we see that P(l) e Zp

and

§3.5. p-adic Dirichlet L-functionsSince the absolute value | | p is an ultra metric (i.e. a metric satisfying the strongtriangle inequality (1.3.1)), Zp is totally disconnected (i.e. Zp is a disjoint unionof arbitrarily small open sets). In fact, if we write

D(x,p"r) = { y e Z p | | y - x | p < p - r } = {y€ Z p | I y-x I p < p" r + 1} ,

this is an open set and Zp = Up D(x,p~r) (disjoint union), because

D(x,p-r) = { y e Z p | y = x mod p r Z p } .

By this, there exist locally constant but non-constant continuous functions. Wecompute in this section the integral of such functions with respect to d£a-

It is clear from the argument in §3.4 that to each F(t) e R1, we can associate a

p-adic measure |ip on Zp with values in Zp in the same way:

3.5. p-adic Dirichlet L-functions 83

(la) / © ^ F = (9nF)(l) = 5 f ^ ( D for all n e N.

Now expanding F into its Taylor expansion around t = 1, we see that

)T" (for T = t-1).

In this way, we can embed R' into Zp[[T]]. Conversely if a p-adic measure 9on Zp having values in Zp is given, then we can define the corresponding powerseries <E>cp(t) by

<M0 = £1=0 (J(n)d<P(x))Tn (T = t-1).

By Theorem 4.1, the measure 9 is determined by the power series Oq>. Thus themap 9 H» O(p induces an isomorphism: Meas{Zp\Zv) = Zp[[T]]. Choosing abasis {wi,...,wr} of Op over Zp, for each measure 9 in Meas(Zp,Op) and for<|> e (XZv;Op), we can write 9(<t>) = 9i((|>)wi+---+9r((|))wr. Then it is plain thatq>i e Meas(Zp;Zp). Thus we know that

Meas(Zp;Op) = p

Thus we know that(lb) the map cp h-> Ocp induces an isomorphism: Meas{Zp\O^) = Op[[T]].

Exercise 1. Show Jxmdcp = (t^)mOcp |T=o (T = t-1) for all m e N.

When I y I p < 1 (ye Op), the following infinite sum is convergent:

(1+y)X=ir=o G V for xeZp.Thus we define the p-adic power yx (x e Zp), when | y-11 p < 1, by

(2a) yx = X n = 0 W ( y ' 1 ) n f o r X G ZP*

This definition coincides with the definition given in §1.3 because the two defini-tions coincide with each other if x is a positive integer. It is easy to check that theusual exponent rules hold: yx+z = yxyz, yxzx = (yz)x and y° = 1 because ofthe density of N in Zp. Using this trick, we know that

oo

f . f/'xN i i

( 2 b ) J y x d 9 ( x ) = £ J ( J 1 J d 9 ( y - l ) n = *q>(y) for y e Op if I y - 1 \ p < 1.n=0

Exercise 2. For two measures 9 and \\f in Meas(Zv;Op), define the convolu-tion product 9*\|/e Meas(Zv;Op) of 9 and \|/ by


Jfd(q>*\|0 = JJf(x+y)d(p(x)d\|/(y).Show O(p*y = O ( p O v in Op[[T]].

The space of measures 5Vfe&?(Zp;0p) is naturally a module over the ring O(Zp;0p)

in the following way: for f e C(Zp;Op) and 9 e Meas(Zp;Op),

We want to determine Ofq> for some special f supposing the knowledge of <Xy

First let us deal with the function f(x) = zx for z e Op with | z-1 | p < 1. By

(2b), we see that

ttfip(y) = Jyxdfcp(x) = Jyxzxdcp(x) = J(yz)xd(p(x) = *<p(yz).

Thus

(3) For ze Op with \z-l\ p< I, we have OzX#q)(t) = ©<p(zt) in Op[[t-l]].

Let jipii be the group of pn-th roots of unity in an algebraic closure of Fp. LetK = Fpfjipn] be the field extension of F , obtained by adding all elements in jipn.Let R be the integral closure of Op in K. Then, as seen in §1.3, there is aunique norm | | p on K extending that on F. Since there are no p power rootsof unity except 1 in characteristic p, £ mod & = 1 for the maximal ideal (P.Thus I £-1 I p < 1 for any £ e |j,pn and hence we can think of £x forx e Zp. If x E k m o d p 1 1 for k e N, we have £x = £k because Cp" = 1.If § : Zp/pnZp = Z/pnZ -> K, then

(4) 4>(x) = p " n I ^ ^J?

This follows from the orthogonality relation

V rx"b = Jp n if x = b mod p n Z p ,Zrf?€p.pn^ [0 otherwise.

Thus, applying (3) for z = £ and writing

b for

we see that, for a locally constant function $ e CCZpjO ,) factoring through

Z / p n Z (i.e. (() = (()'op for the projection p : Z p —» Z / p n Z with

(|)' : Z / p n Z -> Op),

(5) <E>4>q> = M>]<l>9 for (pe fêo5(Zp;Op) and

3.5. p-adicDirichletL-functions 85

Here note that [^JO^ actually belongs to Op[[t-1]] because of (5), although thisfact is not a priori clear from the definition of [(t)]O<p. Note that

(6a) [c|)](tm) = ^m)tm and ^ |

(6b) if O(£t) = O(t) for all £ e îpn, then [<|>](O0) = ([<|>]0)O,

where O, 0 e Op[[t-1]]. In fact,

where we have again used the orthogonality relation. Since tetm = mtm, the

second formula of (6a) is obvious from the first. As for (6b), we compute

= *(t)[«e(t).

Exercise 3. Let R = { ] | | | P(t), Q(t) e Zp[t] and | Q ( l ) | p = l } .

Then show that R 3 [<|)]R for a locally constant function <|>: Zp —> Zp.

We know from the definition (4.1a) that

Using the geometric series, we see that —- = l+ta+t2a+---+tna+---. Therefore

apn

m=l b=l m=0 l - t a p

We put 0(t) = ££fi£(b)tb and O = l/(l-tapn). Then we have W = <D0 and

By (6b)

m=l m=l m=l

Writing (J)a(m) = <|)(am) and supposing (|) actually has values in FnO^ (and

hence has values in C 3 F), we can consider the complex L-function


oo

L(s,$-ak+1(t)a) = X (<Kn)-ak+1$a(n))n-s.n=l

As seen in §§2.1-3, this sum is convergent in C if Re(s) > 1 and has a mero-morphic continuation to the whole complex plane, and

If % is a character of (Z/pnZ)x having values in Fx, extending % by 0 on

Zp-Zpx and denoting this function by the same symbol %, we have a locally con-

stant function % o n Zp. Then

L(-m,x-am+1%a) = (l-am+1%(a))(l-%0(p)pm)L(-m,x0),

where %0 = % if % is non-trivial and %0 is the constant function on Zp having

value 1 if % is trivial. Thus we have proven the first part of

Theorem 1. If % : (Z/pnZ)x —> Fx is a primitive character {when % is trivial,we agree to put n = 0), then

J%(x)xmdCa(x) = (l-am+1x(a))(l-x0(p)pm)L(-m,x0) for all m e N.

If % is an even character x (i-e- %(-!) = 1)>f ^ if x = i d ,

a W x ) \.(l-Z(a))p-nG(z)XbG(Z/pnZ)Xx-1(b)log(l-Cb) if X * id,

where G(%) = E ^ x C b ) ^ (the formula is independent of the choice of Q.

To finish the proof of the above theorem, we need to compute J%(x)x"1dâ(x). By

Corollary 2.3.2, if %(-l) = -1, the integral is identically 0 and is not included in

the theorem. Thus we assume that % is even. We first assume that | a-11 p < 1.

fi-ta]Writing O(t) = log-j k we can expand it formally as a power series in

fi-tal iT = t-1 in QP[[T]] because < > =a, which is in the domain of

[ 1 - t J t-1

convergence of the p-adic logarithm. Write this power series as G>. Then inside

Qp[[t-1]], we have from (6a) and (7) that [%W = t^[%]O = ( l + T ) ^ * ] * . Onthe other hand, since % is supported on Zp

x, the function x h-> x ' ^ x ) is a

continuous function on Zp. Thus we may consider the measure (p given by

xp = J(|)(x)x-1x(x)dCa(x) for all 0 e C(Zv;Op).

3.5. p-acfo DirichletL-functions 87

By the definition of O<p e O^tfT]], we know that Ox<p = t—g^. Therefore we

see that ( 1+T)^* 9 = [%]¥. Thus O r [ x ] O e Fp because Ker((l+T)~0 in

consists of constants. By (6a), the operation [%] kills constant functions,

and for the identity character id, [id]O(p = O(p and [id] [%] = [%]. This implies

. Therefore

Jx"1%(x)dCa(x) =

Now suppose that % | i+pn-izp * 1 (i.e. % is primitive modulo pn). Then by

Lemma 2.3.2, we see, writing £ = e(—), thatP

V fMrbIP"1 i f ? =

Z b

Therefore [%]<D(1) = P"nG(%)Xc6(Z/pnZ)XX-1(c)(Cc) for C = e(-±j). When

is the trivial character, we see that

= l - l if £ * 1.

Using the extended log in (1.3.9b), we can obtain the formula in the theoremwhen I a-1 I p < 1. The formula holds for general a because the p-adicL-function defined below is independent of a.

Let us define p-adic DirichletL-functions. Since %(x) is only supported on Zpx,

we may write

JZpXX(x)xradCa(x) = (l-am+1x(a))(l-x0(p)pm)L(-m,x0) for all m€ N.

We can decompose Zpx = |a. x W where

r4 if p = 2,W = l+pZ p for p = u .

F p IP otherwise,

and |i is the maximal torsion subgroup of Z px . Fixing an element

u G W-(l+ppZp), we have an isomorphism of topological groups Z p = W

given by Z p 9 s h) us e W. Let co : Z px -> |4, be the projection map.

Thus co(x) = Mm xpn if p > 2 and co(x) = ±1 according as x= ±1

mod 4Z2 if p = 2. This character co is called tha Teichmiiller character. Then

we define (x) for x e Z px by (x) = C G M ^ X . Then X H (X) is the

projection of Zpx onto W. We then fix a character % of (Z/prZ)x and define

the p-adic Dirichlet L-function with character % by


Then Lp(s,%) is a continuous function on Zp except when % = id, and in thisspecial case, £p(s) = Lp(s,id) is a continuous function defined on Zp-{1}.Moreover we have the following evaluation formula:

(8) Lp(-mjc) = (l-(%co-m-1)()(p)pm)L(-m,(xco-m-1)0) for all m e N,

where the right-hand side is the value of the complex L-function while the left-handside is the value of the p-adic L-function and the values are equal in the field Fcommon to ¥p and C. Since the right-hand side is independent of the choice ofa, we conclude from the density of N in Zp that Lp(s,%) is independent of a.

We now show that Lp(s,%) is a p-adic analytic function on Zp when % * id anda p-adic meromorphic function on Zp-{1} when % = id. We can define a p-adicmeasure £a,%

o n W (= Zp) in the following way. For any given continuousfunction <|> : W —» Op, we define

Jw(Ky)dUx(Y) = JZpX%co-1(x)0((x))dCa(x).

Then we can write Lp(s,%) = (l-%(aXa)1~s)"1JwY~sdCa,%(Y)- Identifying W with

Z p via i : Z p s s (-) us e W, we can associate with % a power series

3>a,%(t) G Op[[t-1]] as in (3) in the following way:

(l-%(a)(a)1-s)Lp(s,%) = J w y sdCa,x(Y) = JZpu"sxd(t*Ca,%)(x) = * q > ( O .

We define tt^t) by Q9(v'h). Then Lp(l-s,%) = (l-%(a)(a)s)-1Oaa(us). By

(1.3.6a,b), the exponential (resp. the logarithm) function converges on pZp

(resp. W). Thus we can write us = exp(slog(u)). This shows that Lp(s,%) is ameromorphic function on Zp whose pole comes from the possible zero of(1-X(a)(a)1"s) at s = 1. When %(a) * 1, this function (l-xCaXa)1"8) does nothave a zero at s = 1 and hence Lp(s,%) is an analytic function. Here note thatLp(s,%) itself does not depend on the choice of a because of the density of N inZp and the evaluation formula (8). Since Zp

x = |i.xW and W is topologicallygenerated by u and jn is cyclic (When p = 2, |i = {±1}, and when p > 2,[I is the group of(p-l)-th roots of unity), we can take a so that co(a) generates |LLand (a) topologically generates W. Then %(a) = 1 <=> % = id. Thus if% *id, Lp(s,%) is analytic on Zp. We see that (a)s = l+log((a))s+higherterms of s by (1.3.8a,b). This shows that £p(s) has a simple pole at s = 1whose residue is (1-p"1).

Summing up these considerations, we have

3.6. Group schemes and formal group schemes 89

Theorem 2. For each Dirichlet character % : (Z/prZ)x -* Fx with %(-l) = 1,there exists a p-adic analytic function Lp(s,%) on Zp-{ 1} such that

Lp(-m,x) = (l-(xco-m-1)0(p)pm)L(-m,(%co-m-1)0) for all m e N.

% w non-trivial, Lp(s,%) zs analytic even at s = 1

<9« r/ze other hand, £p(s) =Lp(s,id) to a simple pole at s = 1 vv/jose residue is

d-p-1).

When % is an odd character (i.e. %(-l) = -1), then xco111"1^!) = (-l)m becauseof co(-l) = - l , and thus Lp(-m,x) = 0 for all m e N by Exercise 2.3.3.This shows that Lp(s,%) is identically zero if % is odd. This is why we excludedthe case of odd characters %. Although non-trivial Lp(s,%) exists only for even%, the values of complex L-functions with odd characters show up as the specialvalues of the p-adic L-function Lp(s,%). In fact, when m is even, %CGml is anodd character.

§3.6. Group schemes and formal group schemesIn order to give an interpretation of p-adic measure theory on Zp using formalmultiplicative groups in the next section, we recall here briefly the properties ofaffine schemes and affine group schemes. For details of the theory, seeMumford's book [Mml, §11]. Let A be a commutative algebra with identity andR be an A-algebra. We consider the affine scheme G/A = Spec(R)/A- Thus Gis a covariant functor from the category J%C#/A of A-algebras to the category ofsets given by G(S) = HomA-aig(R,S) for any A-algebra S. For any algebrahomomorphism (p : S —> T, the functorial map G((p) : G(S) —> G(T) is givenby G(9)((()) = (po(|). The set G(S) is called the set of S-valued points (or simplyS-points) of G. For two affine schemes GA and G'/A = Spec(R')/A, amorphism of schemes f : G —> G' is a set of maps f(S) : G(S) -> G'(S)such that for every A-algebra homomorphism 9 : S -» T, the following diagramis commutative:

G'(S)

Let Scfi/A be the category of schemes over A [Ha, II]. If r\ : R' -> R is an

A-algebra homomorphism, then T|* : G —> G' given by ri*(S)((()) = §°r\ is

obviously a morphism of affine schemes. Conversely if f: G —> G1 is a

morphism of affine schemes, we in particular have a map f(R): G(R) -> G'(R).

We have two natural maps:


i : HoiriA-aig(R\R) -> H o m ^ C G ' ) given by r| h-> r|* andK : Hom^CCG') -» HoiriA-aig^R) given by jc(f) = f(R)(id).

By definition Ti*(R)(id) = id<>r| =r| . Thus 7uoi = id. Let T] = f(R)(id) for the

identity map ide G(R). We want to show f = r|* (i.e. 107c = id). We have a

commutative diagram (for any map <|>: R -> S of A-algebras):

G(S) f(S) ) G'(S)t Gfl>) T G'(4>)

This implies

This shows that

(1) 71 : Hom^(G,G') s HomA-aig(Rf,R) by n(i) = f(R)(id).

Let G = Spec(R), G1 = Spec(R') and S = Spec(B) be (affine) A-schemesand f : G —> S and g : Gf —> S be two morphisms of A-schemes. Weconsider the following universal property for an A-scheme X. Whenever we havea commutative diagram

T —*-* G

G- - ^ s ,

we have a unique morphism of schemes h : X -» T such that fo(()oh = gocpohand p = <|)°h and p' = cpog are independent of 9 and <|). This universality isthe contravariant version of the universal property defining the tensor productR<E>BR' (see §1.1). Thus such a universal object X is given by Spec(R®BR').We write X = GxsG'.

Now we suppose that R has the structure of an A-bialgebra; that is, there are threeA-algebra homomorphisms

m : R -» R® AR, e : R -> A and i : R -> Rsatisfying

(Gl) The diagram R —^—» R®AR is commutative;

R ( 2 > A R m®id


m m(G2) The diagrams R —> R®AR and R —> R®AR are commutative;

id \ j ^ id®e id \ ^ e®id

R Rm m

(G3) The diagrams R -> R®AR and R -> R®AR are commutative,

where JLL : R® AR ->

(G4) The diagram

AiRR

-> R® AR A->

= R Ris the multiplication:

m

R -> R®AR 3

mî 1

R®AR

= Ra®b h-> ab;

x ® y is commutative.

XR®AR 3 y®x

SinceGxAG(S) = HomA_aig(R®AR,S) = HomA.aig(R,S)xHomA.aig(R,S) = G(S)xG(S),the morphism m (resp. i) induces m* : G(S)xG(S) -» G(S) (resp. i* : G(S) ->G(S)), and e : R -> A induces e e G(S) by composing the original e withthe A-algebra structure: A —» S. Writing x.y e G(S) for m*(x,y)(x,y e G(S)) and x"1 for i*(x), we know that

(i) (x.y)-z = x.(y.z) (Gl), (ii) x.e = e.x = x (G2),(iii) x-1.x = x.x"1 = e (G3) and (iv) x.y = y.x (G4).

This shows that G(S) is an abelian group. In particular, we havee = e = i*e = e°i. Thus if R is an A-bialgebra, then G is a functor on Sch/Ahaving values in the category &6 of abelian groups. In this case, G is called acommutative group scheme defined over A. When R only satisfies Gl-3 but notG4, G(S) is a group (which may not be commutative). In this case, we just callG a group scheme. A morphism of group schemes G and G' is a morphism ofschemes which induces a group homomorphism between G(S) and G'(S) for allS. It is easy to check under the isomorphism (1) that such a morphism correspondsto a homomorphism cp of A-bialgebras, i.e., cp : R' -» R is a homomorphismof A-algebras which makes the following diagrams commutative: for themorphisms of bialgebras mf, e' and i1 (resp. m, e and i) for R' (resp. R)

R1 U R1

and I (p i cp

R -> R.

R

iR

mi .

9

• R ' ® A R f

icp®9R®AR,

R

iR

e1 - >

9

A

1A


Conversely if a functor T : Sch/A -> A& is given and if T(S) = HomA-aig(R,S)for every A-algebra S, we can define an A-bialgebra structure on R by (1) usingthe group laws (i)-(iv) above. Thus any functor from SCUJA to Sib represented byan A-bialgebra R is in fact a commutative group scheme. Let cp* : G —> G1 bethe homomorphism of group schemes induced by (p : R' —» R. We thenconsider R<p = R®R'A = R/(p(Ker(e'))R, where we regard R (resp. A) as anR'-algebra via cp (resp. e'). Then by the above compatibility of (p with thebialgebra structure, the bialgebra structure (m,e,i) of R induces the bialgebrastructure (m(p,eq),i(p) of R(p. Thus we have a commutative group schemeKer(cp*) = Spec(Rq>). We now have an exact sequence of groups for all S:

1 -> Ker(q>*)(S) -> G(S)

Exercise 1. (i) Give a detailed proof of the fact that R9 is an A-bialgebra.(ii) Prove the above sequence is exact.

Let us now give some of the most important examples of commutative groupschemes. Let A = Z and R = Z[t,t"1] for an indeterminate t. We definem : Zfot"1] -» Z[x,y,x"1,y"1] = R®AR by m(t) = xy, i : R -» R byi(t) = f1 and e : R —» Z by e(t) = 1. Then it is easy to check propertiesGl-4 and thus R gives a group scheme defined over Z. We write this scheme asGm (or GLi). Any algebra homomorphism of R to a ring S is determined byits value at t. Thus Gm(S) can be embedded into S. If x e Gm(S), thenx(t"1) = x(t)"1 G S and therefore Gm(S) = Sx. If x,y e Gm(S), thenx»y(t) = x(t)y(t) by definition, and thus Gm(S) is the multiplicative group Sx asa group. We consider the endomorphism [N]s of Gm(S) given by[N]s(s) = sN. This is induced by an endomorphism [N] of R taking t to t^.Then | iN = Ker([N]) = Spec(Z[t]/(tN-l)) and hence

M S ) = { ? 6 s x U N = i ) .

Let G/A = Spec(R)/A be a commutative group scheme. We may regard0 G R(8>AS as a function on G(S) having values in S for any A-algebra S inthe following way:

<|>(s) = s((|)) for s G G(S) = HomA-aig(R,S) = HomS-aig(R®AS,S).

We then consider the space DerA(G) of derivations on R over A having valuesin R. Thus DerA(G) is an R-module of A-linear maps D : R -> R satisfyingD(fg) = fD(g)+gD(f). The A-linearity of D is equivalent to the fact that Dannihilates A in R. The composition of two derivations is no longer a derivation,but we still can think of the algebra Diff(G) generated by derivations over A in

An element of Diff(G) is called a differential operator on G. A dif-


ferential operator D : R —> R is called invariant if the following diagramcommutes:

R-^R(2a) im im

R®AR -> R<8>AR.D®id

We write £(G) for the A-algebra of invariant differential operators. A differentialoperator D is invariant if and only if for any <|> e R, defining(|)x = (id®x)om(<|>) e R®AS for x e G(S) = HomA_aig(R,S), we have

(2b) D(<|)x) = (D(|))x for all x e G(S) and all S/A.

In fact, for y e G(S),

§x(y) = (y®id)o(id®x)om(<|>) = (y<8)x)om((|)) = <|)(y»x).

Thus (|)x as a function on G(S) is the right translation of <|>. Thus it is legitimateto call D invariant if (2b) holds for all S and all x. Let us prove the equivalenceof the conditions (2a) and (2b). The commutativity of (2a) implies

D((|)x) = (D®id)((id®x)om(<|>)) = (id®x)o(D®id)om(<|>) = (id®x)om(D<|>) = (D<|>)x.

Conversely if the above identity (2b) is true for all x, then it is obvious from (1)that the diagram (2a) is commutative and D is invariant. We write £teA(G) forthe space of invariant derivations on G/A. If D is a derivation on Gm/z over Z,it is determined by its value at t (because R = Z[t,t"1]). On the other hand,

D' = D(t>3r is a derivation satisfying D'(t) =D(t). This shows

) = R ^ and Derz(Gm/z)<g>zC = Derc(Gm / c) ,

and thus Liez(Gm) injects into £tec(Gm). Since a Gm-invariant first order dif-

ferential operator on Gm(C) = Cx in Cfot^br is, by (2b), a constant multiple

of tjjj: = dZ (t)» w e k n o w t h a t Liez(Gm) = z^f- T h u s

(3a) Derz(Gm) = R-|, LieZ(Gm) = Z t | and #(Gm)<g>zQ = ^

A polynomial on Z is called numerical if P(n) e Z for all n e N . We knowfrom Proposition 1.1 that the binomial polynomial gives a basis over Z of the ringof numerical polynomials. Thus we have by (3a) that

(3b) £>(Gm) = {P(tgjr) I P(T) is a numerical polynomial} = Z[3n]nGz,


where 3n is the differential operator in Lemma 4.1. Writing Dp = Pter), we

see easily that Dp is characterized by the fact Dp(tm) = P(m)tm.

We consider in this note a group scheme a little more general than Gm. Let M be

a free Z-module of finite rank. We consider the group algebra R = Z[M]. Thus

R is a commutative ring generated by a Z-free basis ta for a e M (t° = 1)

and tatp = ta+p. When M = Z as an additive group, R = Zfot'1]. Note that

any algebra homomorphism X : Z[M] —> S is determined by its value at tai for

a basis {cci} of M. Since tai is invertible in Z[M], ^(tai) has to have values in

Sx. Thus, writing T for Spec(Z[M]), we know that

(3c) T(S) = HomA-aig(Z[M],S) = Homgr(M,Sx) = Homgr(M,Gm(S)).

Thus as a functor, we know that T = Homgr(M,Gm). Since T is a groupfunctor, by (1), T is an affine group scheme.

Exercise 2. Make explicit the bialgebra structure of Z[M].

If we identify M with Zr choosing a basis {OCJ}, we have an isomorphism

This shows that T = Gmr non-canonically. We call a rational polynomial P on

M numerical if P(a) e Z for all a e M. Then by (3b), we see that

(3d) 2XT) = {Dp IP is a numerical polynomial on M),

where Dp is the invariant differential operator on T given byDP(ta) = P(a)ta for all a e M.

Let K be a finite extension of Qp, and assume A to be the p-adic integer ring ofK. Hereafter, we always consider T to be defined over A. Thus we change thenotation and write hereafter R for the coordinate ring A[M] of T over A. Foreach integer N, we define the endomorphism [N] which takes x e T(S) toxN G T(S). Then as a group subscheme of T, we define TN = Ker[N] in T.

Then TN = Spec(RN) for RN = Z[M/NM] = ®iA[ta\f ai]/(tN(Xi-l), and we seethat(4) TN(S) = HomA_aig(A[M/NM],S) = Homz(M/NM,|iN(S)) for all S.

We now restrict the category on which the group functor T is defined. Let

AcC = JZCC/A be the category of A-algebras S = lim (S/paS) for the maximal

ideal p of A. Let 9&C/A be the subcategory of &<£ consisting of A-algebras in


which p is nilpotent. Thus we may regard the category Ad of /?-adic algebras asthe category obtained from 9{iC adding projective limits of its objects. LetF = Spec(R')/A for an A-algebra R'. We consider the restriction of F to fAfif.Then for any S e Ob(fA#0> every x e F(S) = HoiriA-aig(R\S) has to factorthrough R7paRf for sufficiently large a (such that pa kills S). Thus

F(S) = HomA-aig(R!,S) = HomA-aig(R\S),

where R' = lim (R'/j^R') = lim (R7paR!) is the p-adic completion of R!.

Thus considering the restriction of F to 9& is studying the p-adic completion ofR' algebraically and is studying the germs (with infinitesimals) along the specialfiber at p of Spec(R'). For S e ObfcAfiO, we write Sred for the reduced part ofS, i.e., the residue ring of S modulo its nilradical n$. Returning to the original

R = A[M], we define a new functor T : 9{iC -» AS by

(5) f (S) = Kernel of the natural map T(S) -» T(Sred).

Since T(S) = Homgr(M,Sx) for any S, f (S) = Homgr(M,l+nS). Since p isnilpotent in S, for any x e ns, taking N so that x =0 , we see that

if p n > N . Since P 1 -> 0 as n -> oo if

I i I < N, we see that 1+ns is contained in |apn(S) for large n. On the otherhand, if £ e |ipn(S),

r nis divisible by p, because l+i-l)?11 and for 1 < i < pn are both divisible

by p. Thus (£-1) is nilpotent in S. Thus

(6) If p is nilpotent in S, |ip~(S) = lim M-pa(S) =a

This shows that, because p is nilpotent in S,

f (S) = Homgr(M,^poo(S)) = Homconti(Mp,Sx),

where Mp = lim (M/pa) and Sx is supposed to have the discrete topology,a

while Mp is equipped with the p-adic topology. Then the last equality followsfrom (4) because any continuous homomorphism of Mp to Sx factors throughM/paM for some a. Now we can extend this functor to M by

(7a) f ( S ) = Hm f(S/paS) = Homconti(Mp,Sx),a


where f (S/paS) = Homconti(Mp,(S/jpaS)x). Thus, by (4), on Ad, we see that

f (S) = HomA-aig(RP-,S) for

(7b) Rpoo = lim Rpn = lim <8>iA[ta\fai]/(tpn(Xi-l) = A[[ta i-l , . . . , ta r-l]].n n

In this sense, we may write T = Spf(Rp~)> which is the formal completion of T

along the ideal (tt t l-l, ..., tar-l) (see [Ha, II.9]). We thus call this functor theformal group (scheme) of T. To complete the proof of (7b), we need to show

lim A[t,t"1]/(tpn-l) = A[[t-1]] as compact algebras.n

Since A f c f 1 ] / ^ - ! ) = A[t]/(tpn-l) = A[T]/((T+l)pn-l) for T = t-1, we first

show that (T+1)^-1 e (p,T)n+1. When n = 0, this is obvious. We proceed byinduction on n. We see that

(T+l)pn+1-l = ((T+l)pn-l)(l+(T+l)+---+(T+l)p-1) = ((T+l)pn-l)(p+TQ(T))

for an integral polynomial Q of T, and hence by the induction hypothesis,

(T+l)pn+ -1 E (p,T)n+2 because (p+TQ(T)) e (p,T). Thus there is a natural

map lim Afcr 1 ] /^"-! ) -> Hm A[T]/(p,T)n+1 = A[[T]]. This map is con-n n

tinuous under the projective limit of the natural topology on both sides. By com-pactness, the image of the map is closed and contains a dense subset A[T]. Thusthe map is surjective. The injectivity of the map is obvious becausenn(p,T)n = {0}. This shows the isomorphism.

§3.7. Toroidal formal groups and p-adic measuresWe study here the relation between the space of p-adic measures on Mp and thecoordinate ring Rp~> of the formal group T. We will get a natural isomorphismRp«> = Meas(Mp;A) generalizing the isomorphism A[[t-1]] = Meas(7*v\K) givenin (5.1b). Here we keep the notaion of the previous section. In particular, thebase algebra A is the p-adic integer ring (with the maximal ideal p) of a finiteextension K/Qp. We continue to use the notation introduced in the previous sec-tion. For S e Ob(.#d/A), we can write

Meas(Mp;S) = {cp : C(Mp;S) —> S | cp is S-linear and continuous},

where we use the topology of uniform convergence in the space £(MP;S) ofcontinuous functions on Mp with values in S, while S is equipped with thep-adic topology. By definition, we have

3.7. Toroidal formal groups and p-adic measures 97

Rpoo = Mm Rpn = Urn A[M/pnM] = A[[tai-l,...,tar-l]]n n

and

RPoo<§>AS = Urn (Rpoo®AS/paS) = Mm S[M/pnM] = S[[ t a i - l , . . . , t a r - l ] ] .a n

For any finite group G, the group algebra A[G] has the obvious universalproperty: for any given group homomorphism 2; : G -> Sx for S e J%Cg/A,there is a unique A-algebra homomorphism £* : A[G] -> S making thefollowing diagram commutative:

A[G] —S—» S

T tG — ^ Sx

The algebra Rpoo®AS has a similar universal property for continuous homo-

morphisms £ : Mp -> Sx (i.e. £ e f (S) = Homcont(Mp,Sx)). Thus, for any

given £ e T(S), by continuity, ^ a : x H £(x) mod paS factors through

M/paM. Therefore, by the universality characterizing group algebras, we have the

S-algebra homomorphism

£a* : S[M/paM] = Rpa(g>AS -* S/paS

extending £a. By construction, {£a*} forms a projective system yielding

£* = lim £a* : Rpoo<§)AS -» S,

which extends £. Thus Rpeo<8>AS is sometimes written as S[[M]] and is calledthe continuous group algebra of M. Anyway we have recovered the canonical iso-morphism proved in the previous section:

f (S) = HomA_aig(Rpeo,S) = HomS-aig(RPoo<§>AS,S) (^ h* ?•).

In naive geometric terms, the evaluation at an S-rational point % of a variety givesan algebra homomorphism from its coordinate ring into S. Reversing thisprocess, in modern algebraic geometry, we define S-rational points to be algebrahomomorphisms from the coordinate ring (given without any reference to geo-metric objects) into S. From this point of view, the value f(£) of an element f inthe coordinate ring ?.s a function of S-rational points £ is nothing but the value ofthe algebra homomorphism ^ at f. Following this convention in algebraic

geometry, we regard an element f = f(ta i-l, . . . ,ta r-l) in the coordinate ring

Rp~<t>AS = S[[tai-l,...,tar-l]] of f as a function of £ e f(S) by putting


We now want to prove the following result of Katz [K6]:

Theorem 1. Let S be an object in Mj^, Then there is afunctorial isomorphism

of A-algebras between R P OO®AS (= lim (Rpoo<g>AS//7aS)) and f7tfea5(Mp;S)a

characterized by the following properties. Writing the correspondence as

RP<§>AS 3 f <-> |LLf e Meas(Mv;S), we have

(i) for each point £ e f (S) = Homcont(Mp,Sx), J^(x)d|if(x) = f(£);

(ii) for any numerical polynomial P : M -» Z regarded as a continuous func-tion on Mp,

where Dp is the differential operator in (6.3b);(Hi) for £ flttd P as above, and for any locally constant function <|>: Mp -> S,

/or F(t) = ([<|)]f) /or the operator [<|)] on RpooÂS satisfying [<\>]ta = (|)(a)ta.

Before proving the theorem, we recall briefly the isomorphism (5.1a), its con-struction and its properties. By Mahler's theorem (Theorem 2.1), to each measure(p e Meas(7jv\S), we assign a power series

(•) O(t-l) = X J(J)d<p(M)n = JtMcp e S[[t-1]],n=0

where in the last equality, we regard d|J, as a measure having values in S[[t-1]] inoo

an obvious manner, using the expansion tx = ^ ( n Vt- l ) n . Write O = O 9n=0

and (p = (p >. Then we have verified

(la) for each £ e Gm(S), J§(x)dq> = *9(?(1)-1) = ®9(§),

where in the last equality, we have used the convention described above the theo-rem. Since Z p is topologically generated by 1, we see that ^(x) =

x. Thus replacing y in (*) by ^(1), we know that

3.8. p-adic Shintani L-functions of totally real fields 99

We already know from (6.3a) that any invariant differential operator of Gm/z is

given by Dp = P(tjr) for a numerical polynomial P. Then we see from Exercise

5.1 and (la) that

db)

Finally, for any locally constant function <|>: Zp -» S, we have

(lc) J<Kx)P(x)S(x)d<p(x) = Dp[4>]O<p($),

where [(()] is the operator defined in (5.6a,b) satisfying

Proof. We may identify T = Gmr and M = Zr by fixing a basis {(Xi) of M.

Writing ti for tai, we already know that Rp~ = A[[ti-l,...,tr-l]] (6.7b). Thenwriting n = (nO for an r-tuple of natural numbers (i.e. n e Nr) and defining

(nJ = n i=i(n!) f° r x = (xi) G Zpr = Mp, we see in the same manner as in

§3.3 that

(2) Any f e C(Zpr;S) can be expressed uniquely as an interpolation series:

f(x) = Sn>oan(f)( n ) with an(f) e S satisfying Mm an = 0, where I n > 0\ii/ Inl—>©o

means £n=o*"^n=o and | n | = E i l n i l . Conversely, if a sequence {an}

satisfying l i m a n = 0 is given in S, then the infinite sum Sn>oan( n ) con"Inl—>°° \ii/

verges in S giving a continuous function on Zpr having values in S.

In fact, if we write £O(Mp;S) for the space of locally constant functions on Mp

with values in S, it is plain that LC(Zvr;S) = LC(Zv;S)®s--®sLC(Zp;S). Since

XO(Zp;S) is dense in O(Zp;S) under the topology of uniform convergence, we seethat C(Zp

r;S) = C(Zp;S)®s---®s£(Zp;S). Then (2) follows from Mahler'stheorem. Then all the assertions of the theorem are easy consequences of (la,b,c).

§3.8. p-adic Shintani L-functions of totally real fieldsWe fix an algebraic closure Q p of Qp and Q of Q. We also fix embeddingsof Q into Q p and C so that any algebraic number can be regarded both as acomplex number and as a p-adic number. We extend the absolute value I I p asassured in (2.5). Let F be a number field of degree d in Q. We write I forthe set of all embeddings of F into C. We assume that all the embeddings of Finto C actually fall in R, i.e., that F is totally real. We write R+ for the strictlypositive real line. We write FR = F®QR, which is canonically isomorphic to R1

via F 3 \ h-> (t,c)oeh We put FR+ = R+1. For any subset X of FR, we

100 3: p-adic Hecke IAunctions

write X+ for Xf!FR+. We write E for O+x = O T | F R + . Let v = {vi, ..., vr}be a set of Q-linearly independent element in F+. We write C(v) for the opensimplicial cone spanned by v, i.e., C(v) = ZiR+Vi in FR+. AS shown in §2.7(Theorem 2.7.1), there exists a finite set V of the v's as above such that

FR+ = UEG EUVG V£C(V) (disjoint union).

Let O be the integer ring of F. We fix an ideal a* {0} in O. For anyX e F+, we can replace V by XV = {Xv | v e V} without affecting the aboveproperty. In particular, we may assume that v is contained in a for all v e V.We consider the torus T/z associated with M = a defined in (6.3c). Let R bethe coordinate ring of T. Then R is a group algebra of a, i.e.,

R = Z [ t a | a G a]. Let Z p be the integral closure of Zp in Qp. We define

R ' P = { ^ § | P ( 0 , Q ( t ) e R<8>zZp and

Similarly, we define

^ | R<S>ZC and

We put R(v) = o+ntZvievXiVilo < xi< 1}. For each v e V, x e R(v)

and for £ e T(C) = Hom(a,Cx), we define

da)

(lb) When ^(vi) * 1 and | (vO I < 1 for all vi e v, we see that

(lc) If (vO is a non-trivial / m-th root of unity for a prime / prime to p

(m > 0), then | (vO-1 I p = 1 and hence fv ,x e R'p.

For the coordinate ring Rp°o of the formal group f, the argument which provesLemma 4.2 combined with (6.3b) shows that, for every numerical polynomial Pon a,

(2) R'pz>Dp(Rfp) and R'p is embedded into Rp-®zpZp .

Numbering the elements in I, writing the i-th conjugate of a e F as cc^ and

replacing ta by exp(-EiOC^Vi), we see that

(3) fv&x(t) corresponds to £(x)G(y,A,x,%) (see (2.4.3)),

where Li(y) = IjVi(j)yj, %i = £(vO and x = (xi) given by x = EiXiVi.Finally note that the norm map N : a —> Z is a numerical polynomial on a and

3.8. p-adic Shintani L-functions of totally real fields 101

hence we have an invariant differential operator D# e 2XT/z) (D^(ta) = N(a)ta).Then, directly from (2.4.8c), we get

Proposition 1. For each £ e Hom(a,Cx) satisfying \ £(v01 < 1 and

^(vi) & 1 for all vi in v, we have

(-n)l,A,x,x) = (DN)n(fv,^x) I t=i for all n e N.

Write ^ = {UvGvC(v)}ria. Then a+ = UeeE£î (disjoint union). For anyfunction <|) on a having values in C, we put

Let <|) : a/fa —» C be a function for an ideal f of O. Then c|) is a linear

combination of additive characters \\f e Hom(a//a,Cx). In fact, for any given

a G a/Zfl, we have

by the orthogonality relation (Lemma 2.3.1). Since $ =A (a)"1Z\j >(a)xa> $ is alinear combination of additive characters \|/. Write ([) = Z\| (t)V and put

This is legitimate because \|/!; e Hom(a,Cx) with | \|/^(vi) | < 1 ifI ^(VJ) | < 1 . Then, noting

we have with the notation of Theorem 7.1

has an analytic continuation to the whole complex s-plane andsatisfies ( D ^ f f ^ l ) = (DN)nWf^(l) = U^(-n) for all n e N andfor each function <|>: aJCa-^ C.

Now we fix a prime ideal f of F with O/C= Z//Z for a prime / ^ p. Theadditive group alia is a cyclic group of order /. We consider the function<|> : fl//a-> C given by (|)(x) = X\|/*idYM, where \|/ runs over all non-trivialadditive characters \\f : alta-* Q x . By the orthogonality relation (Lemma2.3.1), we see that


if x e aC,(5a) <>(x) = -J

Then

(5b) ffl,(J)(t) = "X\)/îdXv€ vXxe R(v)fv'V.x(O»

where \|/ runs over all non-trivial characters \|/ : alia-* Q x . We assume thatv\€ at for all v = (vi) e V and for all i. This is possible because there areonly finitely many ideals which violate this condition (such an ideal is a factor ofvia'1 for some i). Then by (lc), f^ e R'p. Thus we have from Theorem 7.1

Theorem 1. Let the notation and the assumption be as above. Then there exists

a p-adic measure \xa,ce Meas(ap;Zp[\ii\) such that for all locally constant

functions f : Op = lim alpaa -> Q and n e N,

Jf(x)N(x)ndMx) = Uf*(-n),

[ if x £ at,

if x G a£

§3.9. p-adic Hecke L-functions of totally real fieldsWe consider the ray class group ClF(pa) defined in §1.2. If a > (3 > 0, we see

from the definition that 2+(p^) 2 3?+(pa) and thus we have a natural map

ClF(pa) = I(p)/2V(pa) -» /(p)/^P+(pP) = ClF(pp), where

/(p) = { = "7 I » and ^ are integral and prime to p},

2>+(pa) = lP+n{a0 | a e FX, a= 1 modxpa} (see Exercise 1.2.1).

Thus we may consider the protective limit

(la) ClF(p°°) = Mm ClF(pa),a

which is a compact group. By definition, we have a natural group homomorphism

ia : (O/paO)x -> ClF(pa) given by a mod p a h-> the class of aO.

Here we have implicitly assumed that a is totally positive. Let E be the subgroup

of totally positive units in O*. Then it is obvious that Ker(ia) = the image of E

in (O/paO)x. The cokernel of i a is just C1F(1) = HT+. Taking the projective

limit, we have an exact sequence

3.9. p-adic HeckeL-functions of totally real fields 103

(lb) 1 -> E -* Opx -> ClF(p°°) -> Clp(l) -> 1,

where 0PX = lim (cypa0)x and E = lim Ker(ia) is the closure of E in OpX.

Note that Op = Ylp\pOp for prime ideals p and as seen in §3, we have forsufficiently large a that

log : l + p a 0 p - > Op

induces an isomorphism from the multiplicative group l+pa0p to an open additivesubgroup (of finite index) in Op. Since rankzp0p = [F:Q], we have

Opx = |ixZp[F:Q] as topological groups with a finite group \i. Since E is a

Zp-submodule of OpX, it is of finite rank over Zp. Since rankzE = [F:Q]-1 byDirichlet's unit theorem (Theorem 1.2.3), we know that

rankZpE < [F:Q]-1, i.e. Opx/E = j i 'xZ p

1 + 5

for a non-negative integer 8 and a finite group JLL*. Although the equality of theranks, rankzpE = rankzE, is conjectured by Leopoldt, this is still an openquestion except when F/Q is an abelian extension (a theorem of Brumer confirmsthis conjecture when F/Q is abelian [Wa, Th.5.25]). Anyway, G = C1F(P~) isa compact group, and we can decompose G = GtorxW non-canonically so thatGtor is a finite subgroup and W = Zp

1+5. The group /(p) of fractional idealsprime to p can be naturally regarded as a dense subgroup of G. On /(p), wehave the norm map N : /(p) -> Z(p) = ZpHQ. This map N coincides with theusual norm map on (P+ and hence a polynomial map. Thus Af: /(p) —» Zp iscontinuous with respect to the topology induced from G. Thus N extends to acontinuous character N : G -» Zp

x. Now take the completion Q of Q p anddenote by A the p-adic integer ring of Q. (Note that Q is substantially largerthan Qp; see [BGR, 3.4.3].) The space C(G;A) of all continuous functions onG with values in A is naturally equipped with the uniform normI f I p = SupxeG I f(x) I p. Let Meas(G\A) denote the space of bounded p-adic

measures on G with values in A:

) = HomA((:(G;A),A).

We can define the uniform norm on Meas(G\ A) by

|(p|p = Sup|f| =ilJGfd(p|p.


Theorem 1. For each element a e G = C1F(P°°), there exists a unique p-adic

measure £a on G such that for any character % : OFCP") -> Q x and any

n G N, we have

JGX(x)iV(x)ndCa(x) = (l-X(a)^V(a)n+1)np|p(l-Xo(pMf)n)^(-n,%o)^

where we denote by L(s,%) the primitive L-function (i.e. the L-function of

primitive character %Q associated with %) and hence the factor (l-%0(p)N(p)n) is

non-trivial if the primitive character % has conductor prime to p.

Proof. For each ray class c in C1F(1), we choose a representative a- c^ whichis prime to p. Then we choose V as in the previous section so that v iscontained in a for all v e V . We then choose a prime ideal C with a/aC= Z//Zfor a prime / in Z such that vi £ at for all i and all v G V. Among theprime ideals C with alia = Z//Z for a prime / in Z, there are only finitely manywhich do not satisfy the above condition for a fixed V. Thus by changing I ifnecessary, we have the measure (i^/ on av= Op as in Theorem 8.1 for allc e Clp(l). Since G = Uc^c^C^pÊ) (disjoint union), to each function<|) G C(G;A), we can associate a continuous function <|>c (c G Clp(l)) on Op

by < M x ) = { ^ " l x ) * X G °PX' Then we define17 1 0 otherwise.

This certainly defines a measure on G. We now compute

(2)

By Theorem 8.1, we see that

since a+ = UzeE£%a and the integrand is invariant under multiplication by E.Thus

J X a l)N(aa

3.9. p-adic HeckeL-functions of totally real fields 105

where oca * runs over all integral ideals prime to p which are in the same class asa"1 in Clp(l). This combined with (2) shows the desired formula when a = LThe uniqueness follows from the fact that the polynomial functions and locallyconstant functions are dense in 0(G;A). Now the ideals i for which the measures

K^t are constructed are dense in G by Chebotarev's density theorem (see §1.2).Thus for general a e G, we can choose a sequence {4} of such idealsconverging to a. Then by the evaluation formula, £n = C^ converges to £a inMeas(G\A), which finishes the proof.

Using the notation of §5, we now define the p-adic Hecke L-function with

character % : ClF(pa) -> Q x by

£P(s,X) =LF,p(s,%) = (l-%(a)(iV(a))1-s)-1JG%(x)co-1(iV(x)>(iV(x))-sdCa(x),

where co is the Teichmuller character of Zpx. Then Lp(s,%) is a p-adic analytic

function on Z p except when % = id, and in this special case,£p(s) = £p,p(s) = Lp,p(s,id) is a p-adic meromorphic function defined onZp-{1} and having at most a simple pole at s = 1. Moreover we have thefollowing evaluation formula:

"ô) for all m e N,

where we write simply co for co°iV, and the right-hand side is the value of thecomplex L-function while the left-hand side is the value of the p-adic L-functionand the values are equal in Q.

The value of the p-adic L-function at positive integers are unknown except thevalue at 1 for F abelian over Q (see §5). When F is abelian over Q,CF,P(S) = Ilx^Q,p(s,%) by class field theory for Dirichlet characters associatedwith the field F. (For class field theory, we refer to [N] and [Wl].) Thus weknow the residue at s = 1 of £p,p by the result in §5. In fact, for general F notnecessarily abelian, Colmez [Co] proved the following p-adic residue formula:

(4) Ress=iCF,P(s) =

where h(F) = (/:£) is the class number of F, w is the number of roots of unity

in O (thus w = 2) and choosing a basis {£i,...,er} of the unit group O*,

Rp = ±det(/og(ei^))ij=i r_i and Dp is the discriminant of F. Here e® is thej-th conjugate of e in Q p and "log" is the p-adic logarithm defined in a

"Pneighborhood of 1 in Qp. For the choice of the sign of -T-^- and the proof of


the formula, see [Co]. The Leopoldt conjecture for F and p is equivalent to thenon-vanishing of Rp. This formula is a generalization of Theorem 5.2 and is aninteresting p-adic analogue of Theorem 2.6.2 and Corollary 8.6.2.

Although we have only discussed p-adic abelian L-functions over totally realfields, we can also construct abelian p-adic L-functions over CM fields. We referto [K3], [K5], [dS] and [HT2] for the various constructions of such p-adic HeckeL-functions. The existence of abelian p-adic L-functions is the starting point ofIwasawa theory, which studies a subtle but deep interaction between such p-adicL-functions and the arithmetic of abelian extensions of the base field F. We referto [Wa], [L] and [dS] for basics of Iwasawa theory. The so-called "mainconjectures" in the Iwasawa theory have been proven recently in many instances.Although there are no books written on this subject yet, we refer for recentdevelopments to the following research articles: [MW], [Wi2], [MT], [R] and[HT1-3].

Chapter 4. Homological Interpretation

In this chapter, we will give a homological interpretation of the theory ofthe special values of Dirichlet L-functions over Q and will reconstruct p-adicDirichlet L-functions by a homological method (called the "modular symbol"method). A similar theory might exist for arbitrary fields, but here we restrict our-selves to Q. The modular symbol method was introduced by Mazur [MzS] in thecontext of modular forms on GL(2) as we will construct later, in §6.5, p-adicL-functions of modular forms (on GL(2)) by his original method. Basic factsfrom cohomology theory we use in this section are summarized in Appendix at theend of this book. We use standard notations for cohomology groups introduced inAppendix without further warning and quote, for example, Theorem 1 in the ap-pendix as Theorem A.I. If the reader is not familiar with cohomology theory, it isbetter to have a look at Appendix before reading this chapter.

§4.1. Cohomology groups on Gm(C)

We consider the space T = C/Z, which is isomorphic to G m (C) via

z h-> e(z) = exp(27ciz). Thus T = P ^ O - f O , ^ } , where P*(C) is the projec-

tive line. We apply the theory developed in Appendix to X = P1(C) and

Y = T. With the notation of Proposition A.5, S is first {0,°°} and later will be

M-NU{0,oo}, and So will be {0,°o}. We have 7Ci(T) = Z. Let A be any com-

mutative algebra. Let L(n; A) for 0 < n e Z be the subspace of the polynomial

ring A[X,Y] consisting of homogeneous polynomials of degree n. We let the

additive group Z act on L(n;A) by vP(X,Y) = P(X-vY,Y) for v e Z. Then

we define a sheaf Zln;A) = L(n;A) on T (with the notation in Theorem A.I) by

the sheaf of locally constant sections of the projection (CxL(n; A))/Z —> T, where

n e Z acts on CxL(n;A) by n(z,P) = (z+n,nP). We identify C/Z with

(-i©o,ioo)xR/Z and compactify it as T* = [-ioo,i«x>]xR/Z (i = >Tl). With the

notation of Appendix, we have T* = Ts° for So = {0,°o} = {±i~}. Let N

be a positive integer, and take out N small open disks around -^ ( re Z) from

T*. The resulting space we write as T^ for S = N^Z/ZU {+<*>} = |LINU{0,OO}.

We also consider TN~ ° (resp. T^°), removing the boundaries from T^

around s e So = {±°°} (resp. s e S-So). This notation fits well with

Proposition A.5. We write the boundary around s e S of T^ as dsT^. Note

that T^~S° is no longer compact. We want to study Hq(TN,£(n;A)) and the

compact support cohomology group H2(TJ5["So,.£(n;A)) on T^~S°. We first

compute the homology group U\(T^ ,9 Tjjf ,A) for d Tjjf = TooLJT.oo, where

T+oo = ±°ox(R/Z). Let cr be a small circle centered at r e S inside Tjjf and

108 4: Homological interpretation

cr be the vertical line passing through r e R . Wewrite Coo for the circle added at ©o. Then we seeeasily that

fHi(Tfp,3Tgp,A) = Ac°0{0rG(Z/NZ)Acr}( l a ) < rfc^/iM^,

[Hi(TN,A) = ACoo®{®rG(Z/NZ)Acr}.

S—S

We now compute fti(TN °). Fixing a base point x,

we draw the line from x to cr (r e Z or ±©°) and

turn around the circle in the positive direction and

return to x. This path will be denoted by 7tr. Then

F = 7Ci(T^~S°) is generated by 7Cr (r = ±<*> andr G Z/NZ), and there is only one relation among them: ô

where the product is taken in the increasing order for the index 0 < r < N. Byusing the exponential map e(z) = exp(27iiz), we can identify TN-S with

Gm-nN for ^ N = ( C e C X | C N = 1 } .

Identifying Gm-|iN with P jiiN-fO,©©}, we can identify F with TCi(Gm-|iN)-Anyway we have a natural action of F on L(m;A). Of course this action factorsthrough 7Ci(T) = 7CooZ = {7Coom I m e Z}. We write F^ for the stabilizer of£ G |IN« Then F^ = K^Z (where n^ = nT for C, = e(r/N)) acts trivially onL(m; A). We have a natural map

res: H ^ F ^ m j A ) ) -> H^r^LCmjA)) = H^Cr/N^CmjA)) = L(m;A),

which fits into the following exact sequence (Corollary A.2):

gf ,£(m;A))

. Then

(lb)

Let

because they are of the same homotopy type. Moreover TN does not have any

boundary and hence HC(TN,A) = A. By Proposition A.6, H2(TN,£(m;A)) is

dual to HC(TN,-£(HI;A)) = 0 as long as A is a Q-algebra, because there are no

compactly supported (locally constant) global sections except 0. This shows

H2(TN,£(m;A)) = 0. We give a different proof of this fact: We first prove

HJ}R(T ,C) = 0. We need to show that for any co = fdxAdy for z = x+iy,

4.1. Cohomology groups on Gm(C) 109

co = dT] for some r\. Defining F(z) = Jyf(x + it)dt, we have P- = f, and F

is a function on T. This shows that d(Fdx) = { ^ d x + ^ d y } Adx = -fdxAdy.

Thus d(-Fdx) = fdxAdy, which shows the result in this case. For general TN,

we take a small neighborhood U of each hole x = ^ isomorphic to a punctured

disk D at the origin. Then D is isomorphic to a cylinder by zh> log(z). Thus

any differential 2-form co can be considered to be a 2-form on a cylinder and

therefore is of the form drj. Thus by pulling rj back to U, locally any differential

2 form co is equal to drj on U for some rj. Taking a C°°-function on T which

is equal to 1 on a smaller open disk in U and vanishes outside U, we know that

the support of co-dr| does not meet the hole. Thus by changing a representative co

of the cohomology class in HJ) R (TN,C) , we may assume that co is well defined

on T (i.e. CO does not have singularities at the holes). Then as already seen,

co = dr| and hence HQR(TN,C) = {0}. Since C is faithfully flat over Q, we

see that H2(TN;Q)<g>QC = H2(TN,C) = {0}. Thus H2(TN,Q) =0, and hence

we see that H2(TN,A) = H2(TN,Q)®QA = {0}. Now we prove H2(TN,£(m,A))

= {0} by induction on m. We consider the exact sequence of F-modules

0 -> L(m;A) x Y > L(m+1;A) —2-» A -> 0,

where 7i(P(X,Y)) = P(l,0). This induces an exact sequence of sheaves

(*) 0 -> X(m; A) x Y ) £(m+l;A) —^-»A -> 0from which we get another exact sequence

0 = H2(TN,Am;A)) — ^ U H2(TN,Am+l;A)) ^ U H2(TN,A) = 0.

The vanishing of H2(TN,£(m;A)) follows from the induction assumption. Thisshows H2(TN,£(m+l;A)) = 0.

We see easily that H°(TN,iXm;A)) = L(m;A)r = AYm. Thus we have from (*)an exact sequence

0 -> A -> HKTN^mjA)) x Y ) H ^ N ^ m + ^ A ) ) — ^ H ^ T N . A ) -> 0.

Thus if A is a Q-algebra,dimAH1(TN,A)+dimAH1(TN,i:(m;A))-l =

Since dimAH^TNA) = N+l, we see thatdimAH1(TN,iXm;A))+N =

In particular, we get

(2) ^


Each inhomogeneous 1-cocycle u of F is determined by its values u(7C ) for

£ E JINU{°°} because of the relation: 7CooIIrG (Z/NZ)71*71-00 = *• Thus w e c a n

embed the module of 1-cocycle Zl(T,h(m;A)) into copies of L(m; A)

res : Z1(r,L(m;A)) <-> L(m;A)[Z/NZ]0L(m;A)

where L(m; A)[Z/NZ] denotes the module of a formal linear combination of ele-ments in Z/NZ with coefficients in L(m;A), which is in turn isomorphic toL(m;A[Z/NZ]) for the group algebra A[Z/NZ]. On the other hand, since K^ for£ e |IN acts trivially on L(m;A), res brings the submodule B^FJLOnjA)) ofcoboundaries into (7Coo-l)L(m;A)

resCB^JLteA))) = (7ioo-l)L(m;A).Thus we have(3a) rfCTttAmjA)) = tfîmA) = L(m;A)[Z/NZ]eL(m;A)/(7Coo-l)L(m;A).

We define Hp (T^,L(m; A)) by the kernel of the natural restriction map

(3b) res : H\TN,L(™;A)) S H 1 ^ 0 ,£(m;A)) -> H 1 ^ 0 ,£(m;A)).

Then we see from the relation that 7Toorire (Z/NZ)7^71-00 = ^

(3c) Hj>(TN,£(m;A)) = {u e L(m;A[Z/NZ]) | {u} G (7Coo-l)L(m;A)},

where {SrGZ/NZUrr} = SrGz/NZUr, and from (2)

(3d) rankAHp(TN,Z<m;A)) = N(m+1)-1, if A is a Q-algebra.

We now define Hecke operators T(n) for each integer n & 0 acting on the co-homology groups. We consider the projection map % : C/nZ —> C/Z. We put

V = TC"1(TN) and 7ti(V) = O as a subgroup of F.Since the projection map % : V —» TN is a local isomorphism, we have two nat-ural maps:

7t* : H!(TN,i:(m;Q)) -> H^V.^ toQ)) andTr : H ^ V ^ m j Q ) ) -> H ^ T ^ ^ m j Q ) ) .

The existence of the morphism TC* is obvious. We explain the construction of thetrace operator Tr. Since the projection n : V —> TN is a local homeomorphism,for each small open set U in a simply connected open set in TN, ^ ( U ) is iso-morphic to a disjoint union of open sets each isomorphic to U. Write simplyM = L(m;A). We write n* for the direct image functor, i.e. 7i*M is the sheaf

on TN generated by the presheaf U f-> M(n'l(\J)). We take an open subset Uo

in rc^CU) so that K induces Uo = U. Then by definition, we know that


= M(Uo)d for the degree d of K. This isomorphism is explicitly given

as follows. We may identify rc'Û) with the image of the disjoint union

Ui8i(Uo) in V, where {5i} is a complete representative set for O\F. Here we

have a commutative diagram for the universal covering H of TN:H n° ) C-N^Z/nZ—^L-* V —2-» TN

l&{ i&{ IH > C - N ^ Z / n Z > V

in which Si: H = H naturally induces Si: C X - | IN = CX-JIN- Then we iden-tify M(5i(Uo)) = M with M(Uo) = M via the map:

M(5i(Uo)) 9 X H Si^x e M(Uo) = M(U).

Now it is clear that 7C*M/s = Indr/<j>(M) on TN, where Indryo(M) is the in-

duced module M®z[]Z[r] and the F-action is given by y(m<8>a) = mây"1.

Note that the direct image of a flabby sheaf is flabby by definition and that 7t* is

an exact functor by (A.5a), because n is a local homeomorphism. Therefore any

flabby resolution of M/s gives rise to a flabby resolution of (TC*M)/TN Jus t ^v

applying re*. Thus we know that

J * M ) = H^(V,M) (Shapiro's lemma).

Now we define Tr : 7t*M/rN(U) -> M ( U ) / T N by Tr(x) = ZiSi^x. This in-

duces a morphism of sheaves. Obviously this is induced from the trace map of

algebras Tr : Z[T] -> Z[O]id<8>Tr : Indr/o(M) = M<8>z[<Da]Z[r] -> M.

Then Tr induces a map of cohomology groupsTr : HUV,M) = ^ ^

(I 0\For a n = L L we define an(z) = z/n. Then a n induces a morphism of

sheaves ocn : £(m;A)/v -» X(m;A)/TN by (z,P(X,Y)) H> {zln?((X,Y)lan)),which in turn induces a morphism of sheaves ocn*£(m;A) -> X(m;A)/v, wherethe inverse image an*^C(m;A) is the sheaf on V generated by the presheafU \-> X(m;A)(On(U)). Then we have a natural pull back map

an* : rf(TN,£(m;A)) -> ff(V,jC(m;A)).Then we put(4) T(n) = Troan*.

Note that the action of oc preserves boundaries at ±°o. Therefore we can define

the operator T(n) in the same manner on H'CT+oc^mjA)), HJ(3 T^0,L(m;A))


i S—sand HC(TN

0,£(m;A)). Then the boundary exact sequence is compatible with

the action of T(n). More generally, we can let an upper triangular matrixfa b\

a = e M2(Z) (ad * 0) act on C and L(m; A) by

<x(z) = (az+b)/d and aP(X,Y) = P((X,Y)V),

where a1 = det(a)a"1.

We insert here a computation of the Fourier transform on finite abelian groupswhich will be used later. Let <|) : Z/NZ —> C be a function. We define theFourier transform <j) = f(§) : Z/NZ -> C by

Then we see from Lemma 2.3.1 that, for J(<t>) | t(x) =

(5a) I I Z ^- Y A M Y e ( x ( t y " z ) ^ - I N * ( z / 0 if t l 2 *-2,yGz/NZ^W2.xEZ/NZe^ N ' ~ jo otherwise.

Now we apply this Fourier transform to a primitive Dirichlet character

% : (Z/CZ)X -> Cx for a divisor C * 1 of N. We write N = N'C. We ex-

tend the character % to a function % : (Z/CZ) —> C just defining its value to be

0 outside (Z/CZ)X. Then we compute

(5b) mod N

f— - IZ"1(x/N')G(x)N' if N ' | x ,( N

l ) " {A 0 o the rwise .

In other words, %(tx) with 0 < 11 N1 is supported on (N7t)(Z/NZ)x.

We now compute the action of T(n) on H ^ T N ^ ^ C ) ) using differential forms(see Theorem A.2). We consider differential forms with values in JJ(m;C) of thefollowing type. For an integer 0 < j < m

(6a) O)j(f) = f(z)(X-zY)m~jYjdz for any meromorphic function f on T.

In particular, we study the following explicitly defined meromorphic functions.Let |LL: Z -> {±1, 0} be the Mobius function and % be a primitive Dirichletcharacter modulo C. Then we put


(6b) f(z) = fid(z) = 0 ^ ,

f.,id(z) = Xo<t|s ^ W 2 ) = f><nz) for 0< s I N,n=l,(n,s)=l

(6c) fx(z) = X~=0X(n)e(nz) = X(-l)c'iyL2=ie^C - l C- l

j=0 j=0

Since FooanFoo = (Jn_0 Foo(xnj for an,i = L I, it is easy to see from defini-

tion that the action co i-> Zj(oCn,j*co) I cxnj on differential forms induces the opera-

tor T(n) on de Rham cohomology groups with coefficients in L(m;A) (see

Theorem A.2), where co I anj(x) = anjlco(x) (a1 = det(a)a-1) for the action of

a n j l on L(m;C) and oCnj*co is the pullback of co under the action of an j on

C. We now compute the action of the operator T(n) acting on C0j(fx):

o)j(g) | T(n) = ^ a ^ C D j t g ) I an>i

i-l V ^ n - 1 . - , . - r Z+i ^rvm-i /Z+iN , , i-1 1 +L

=n J E i (x + l Y - i r * n Y ) g ( ir ) d z = ffl<nJ

Then it is easy to see that C0j(g) | T(n) = ©j(g | jT(n)) for g | jT(n) given by

(7a) gljT(n)=nJ-1X":o1g(^)-

If g has a Fourier expansion g(z) = £°°_ a(m,g)e(mz) (a(m,g) e C), then we

see that(7b) | J

Since C0j(fx) gives a closed form, we have its de Rham cohomology class [cOj(f%)]

in HbR(TN,Am;C)) (see Theorem A.2). Then we have[co/g)] | T(n) = [coj(g) | T(n)] = [cOj(g I /T(n))].

We compute the action of T(n) on fx using the formula (7b). The functions in

(6b) have two Fourier expansions: one at °° and the other at -°°, which are given

as follows. Since f(z) = , \ \ , we may expand it into the geometric series

f(z) = ^ ^ e O n z ) if Im(z) > 0. Writing the same function as f(z) = , , , ,

we have the expansion valid on the lower half plane:

f(z) = - l -£"= 1e(-nz) if Im(z) < 0.

We now list the two expansions of f% for % * id, which are computed using theabove expansion of f and the definition of f%:


S~ X(n)e(nz) if Im(z) > 0,(8a) f*(z) = i n2

[-Sn=1%(-n)e(-nz) if Im(z) < 0.

We define g | t(z) = g(tz) for each divisor t of N. Since the Fourier expansionof fx with % T* id and f I t-f with 1 < 11N has no constant terms, the coho-mology classes [C0j(fx|t)] and [cOj(f-f|t)] actually fall in the parabolic cohomol-ogy group; that is,

(8b) [coj(f% 11)] G H^TNîr^C)) for % * id and 0 < 11 N\

[C0j(f-f|t)]€ Hj>(TN,i:(m;C)) for K t | N .

By (7b), we have

(8c) C0j(f% 11) | T(n) = nj%(n)o)j(fx 11), co0(l) |T(n) = coo(l) for n prime to N.

Theorem 1. The set of cohomology classes{[a>j(fx 11)] | 0 < j < m, 0 < 11 N'}U{[co0(l)]} (= {[(X-zY)mdz]})

forms a basis of H^TîXn^C)), where % runs over primitive characters mod Cincluding the trivial character mod I, and we have written N = N'C.

Proof. Let Q x = {[©j(fx 11)] | 0 < j < m, 0 < 11 Nf}. Writing Wx for the

subspace of H^TN.XOIIJC)) spanned by Q,%, we see from (8c) that W^flZ^W^

= {0}. When % & id, f% \ t has non-trivial simple poles at P t = ( x e N"XZ | Ctx

G (Z/CZ)X) by definition, where C is the conductor of %. Here we understand

C = 1 for % = id and (Z/1Z)X = {0}. This shows JCx©j(fx 11) is non-zero if

and only if x G Pt. Therefore Q% is linearly independent in Wx. Thus writingthe number of positive divisors of N as d(N), we see thatd i m c W x = d(N')(m+l). Moreover by computation, we know that

JCeoCGo(l) e (7ioo-l)L(m;C) (see (9b) below) and hence coo(l) e ZXWX. Since

we know that dimcH1(TN,^(m;C)) by (2), what we need to show is

5LdimcWx = N(m+1). Writing (ppr(C) for the number of primitive characters

modulo C, we have an obvious identity: £o<c|N9pr(C)d(N/C) = N f° r t n eEuler

function 9. This shows the desired dimension formula.

Exercise 1. Give a detailed proof of the fact that X0<c |Ncppr(C)d(N/C) = N.

We note that O)j(fx) only has poles at reduced fractions x = p and

(9a) JCxO)j(fx) = -


Similarly the value of JZ coo(l) at (X,Y) = (1,0) is

(9b) (JzZ+1coo(l))(l,0)=L

1 fZ+l

Noting that Hf)R(Coo,M) = M/(7Coo-l)M via CGH> Jco (1,0) for M = L(m;C),

(9b) shows that the cocycle yi-» J coo(l) is rational. Thus we have

Corollary 1. Let K be the field generated over Q by the values of all charac-ters of (Z/NZ)X. Then the following elements form a basis of H1(TN,£(m;K)):{G(%"1)o)j(fx | t)}j>t,xU{coo(l)} in the notation of Theorem 1. Moreover we have

Nm-JG(%-i)0) j ( fx) G n\TN,L(m;Z[xl)) for primitive %.

Since T^ (C, = eta)) acts trivially on L(m;A), we have a natural restriction map

res r : H ^ T N , L(m;A)) -> H 1 ^ , L(m;A)) s L(m;A).

For each closed form co, this map is realized by coh->Jc co. Then we define

(f>m = q>: tfCTN.XdmA)) -> L(m;A[Z/NZ]) by q>(x) = x T l " f ]resr(x)r.r=ovu x y

By definition (p isinjectiveon Hp(TN, ^(m;A)) and

<p : H^TN^mjA)) ={x = ^ x r r e L(m;A[Z/NZ]) | x r G (w

We define the action of T(n) for n prime to N on L(m;A[Z/NZ]) by

T(n)(]Trxrr) = ^^nx^rn" 1 ) ,

where rn"1 e Z/NZ is such that (rn-1)n = r mod N.

Proposition 1. We have (poT(n) = T(n)°9 for all n prime to N.

Proof. Since Hi(TN, A) is dual to H ^ T N , A), the operator T(n) onH X (TN, A) induces its dual operator on HI(TN, A), which we still denote byT(n). We first compute the action of T(n) on cr/N e HI(TN, A). By definition

T(n)(cr/N) = an(7C*(cr/N)) for n : V -^ TN.Then we see that

Inl-l

T(n)(cr/N) = X c(r/Nn)+(i/n)-i=l

Note that C(r/Nn)+(i/n) = 0 in the homology group except when r+Ni = 0 mod n.

If r+Ni = 0 mod n, C(r+Ni)/Nn = cn-ir/N for n-1r G Z/NZ. This shows

(10) T(n)(cr/N) = cn-ir/N.

Note that H1(TN,£(m;A))/Ker((pm) = {H1(TN,A)/Ker(q>o)}®L(m;A), because nâcts trivially on L(m;A). This isomorphism at the level of differential forms is


given by O)j(f) H> fdz®Xm"jYj = f ^ O)j(f)(z). Then it is clear that when

A = C, the action of T(n) is interpreted on the right-hand side of the aboveidentity by T(n)®an for T(n) on H^T^C) . Then the above formula (10)shows the desired result for A = C. The result for general Q-algebras A thenfollows from the result over C because H1(TN,X(ni;A))(8)AC = Kl

Let Hm(N;A) be the A-subalgebra of EndA^HTÂmjA))) generated by T(n)for all n prime to N. For any A-algebra homomorphism X : Hm(N;A) —» C,we define

H1P(TN^(m;A))[?i] = {x e H ^ T N ^ ^ A ) ) | X | h = X(h)x}.

All the X's are explicitly given by ^(T(n)) = n}%(n) for Dirichlet characters

% : (Z/NZ)X -» Ax and integers 0 < j < m. We call X primitive if the associ-

ated Dirichlet character is primitive modulo N. Then we get from Proposition 1

Corollary 2. (i) Hm(N;A) is isomorphic to the A-subalgebra inEndA(L(m;A))<S>A[Z/NZ] generated over A by an®(nl mod N) for all nprime to N. In particular, H0(N;A)= A[(Z/NZ)X].We now suppose that A is a Q-algebra. Then we have

(ii) Hp(TN,X(m;A))[A,]=L(m;A[Z/NZ])[A,] is free of rank one over A if X isprimitive;

(Hi) Suppose N is a prime. Then Hp(TN,X(m;A))[X] is free of rank one over A

if A,(T(n)) = 1 for all n prime to N;(iv) Hj,(TN,X(m;C))[X] = C(Oj(f%) if X is primitive and ?i(T(n)) = nj%(n)for all n prime to N;

(v) Suppose N is a prime. Then Hp(TN,£(m;C))[Aj = C(Do(f-f I N) if= 1 for all n prime to N.

Now we consider the natural projection map

p : H ^ T ^ ^ m j A ) ) ^ nl?(TN,L(m;A)).

We analyze Ker(p) as a Hecke module. By the boundary exact sequence (lb),

Ker(p) is the image of

H°(3Tgp ,£(m;A)) = H0(Too,£(m;A))eH°(T.eo,i:(m;A)) s H°(R/Z,X(m;A))2.

We see easily that H°(R/Z,£,(m; A)) = H°(Z,£(m; A)) = AYm if A is a ring of

characteristic 0. Then it is easy to see from the definition of T(n) that

Y m | T(n) = Xr="oan,iYm = n m + 1 Y m .Thus T(n) acts on the one dimensional space Ker(p) via the multiplication bynm+1. The eigenvalue nm+1 of T(n) does not appear in Im(p). Thus we have

^ ^ S ^ ( m ; A ) ) s Im(T(n)-nm+1)eKer(p)

4.2. Cohomological interpretation of Dirichlet L-values 117

as long as A is a Q-algebra, and therefore we have a unique section

(11) i : 4(TN )X(m;A)) -> H ^ T ^ .

satisfying loT(n) = T(n)oi for all n. Since [Gft'^N^cOj^)] for a primitive %is an integral element which is an eigenform of all T(n) and is cohomologous to acompactly supported form, we see that

i([G(x-1)NJ©j(fx)]) G

Since T(a)-am+1 is well defined over H*(TN~So,i:(in;Z[%])), we have for any

integer a > 1 prime to N

(12) (T(a)-am+1)i bring the image of HJ>(TN,4m;A)) in H1p(TN,iXm;A<g>zQ))

into the image of Hc(TN~s°,£(m;A)) in H*(TN~s°,£(m;A®Q)) for any sub al-

gebra A in C.

§4.2. Cohomological interpretation of Dirichlet L-valuesWe fix a primitive character % * id mod. N. We write X : Hm(N;Z[%]) -> Z[%]

for the Z[%]-algebra homomorphism given by X(T(n)) = %(n). We fill the hole at

0 of TN = T-N"1Z (resp. T^°) and call the new space YN (resp. XN). By

(1.10), T(n) for n prime to N still acts on H ^ Y N , £(m;A)). Since ©j(f%)

does not have a pole at 0, we may consider [co/f^)] e Hp(YN,£(m;C)). Let

8% = G(x"1)[co0(fx)]. By (1.9a) combined with Corollary 1.2, Nm8% generates

the A,-eigenspace Hp(YN, L(m;A))[X] defined in Corollary 1.2. We consider the

integral J 08% on the vertical line c° passing the origin. By (1.8a), we havem

(D Jc<Ac = -GCX- 1 )^

The embedding R —> YN given by y i-» iy induces a morphism

Int : E£(YN, £(m;A))-» H^(R,£(m;A)) =L(m;A) (co h-> f°° ©).

Here X(m;A) on R is a constant sheaf. By Corollary 1.1, for each integer aprime to N, the cohomology class (am+1-%(a))[NmG(%"1)coo(fx)] for primitive %is integral (i.e. is contained in the image of HX(XN, £(m;Z[%]))); moreover, by(1.12), it can be regarded as being contained inside the image ofHc(XN, L(m;Z[%])) in HC(XN, X(m;Q[%])). Then by a theorem of de Rham(Theorem A.2), we see that


(am+1-%(a))NmJc08x = Int(Nm(am+1-x(a))8x)

To include the identity character % = id, we modify a little the cycle c°. Insteadof c°, we take a small real number e > 0 and put c° = ce. Since c° and c°

are homologous in HI(XN,3XN;Z) , we get the same result: Joco = Jceco as long

as 0) is holomorphic at 0. We define Int' on Hc(TN~S°,£(ni;A)) using c° inplace of c° in the same manner as Int. The map pm : L(m;A) —» YL(m-l;A)given by pmdiaiX^Y1) = IîaiX"1"^ combined with Int' gives

Intm : H ^ Y N , L(m;A)) -> H*(R, YL(m-l;A)) = YL(m-l;A).The power z1 of z of the coefficient of X^Y1 in (X-zY)m kills the pole atz = 0 if i > 0, and thus we can compute the map Intm in the same way as Inteven for coo(f-f'q). Then for a prime to q,

(am+1-T(a))Intm(qmco0(f-f|q))

( ^ ) j 1 j J e YL(m-l;Z).

Thus this proves again the result we obtained as Corollary 2.3.2:

Theorem 1. Let % id be a primitive Dirichlet character modulo N and a beany integer prime to N. Then

J!NjG(x-1)(27iO;j-1(l-%(-l)(-l)j)(am+1-%(a))L(l+j,x)G Z[%] and(27cO"j'1(l-(-l)j)(aj+1-l)qj(l<j-1)j!C(l+j)e Z for all primes q.

By the functional equation, we have for j e N,

(2) L ( - j , x - 1 ) j J 1 j 1

§4.3. p-adic measures and locally constant functionsIn §3.3, we studied the structure of the space of p-adic measures on Zp in termsof interpolation series. Here we describe the space via locally constant functions.Let p be a prime and G be a topological group of the form G = |UxZp

r for afinite group (I. We put Gi = (p^p)1. We fix a finite extension K/Qp and writeA for its p-adic integer ring. We equip K a normalized p-adic norm | | p suchthat I p I p = p"1. For any topological space X, we write LC(G;X) for the spaceof locally constant functions on G with values in X. Thus a function(|) : G —> X is in LC(G;X) if and only if for any point ge G, there exists anopen neighborhood Vg of g in G such that the restriction of <|> to V is a con-stant function. By definition, it is obvious that for any locally constant function §and for any subset S of X, §A(S) = Ug^-i^Vg is open; in particular, <]) is

4.3. p-adic measures and locally constant functions 119

continuous. Since G is compact, G = UgGGVg implies that we can find finitely

many points gi, ..., gs on G such that G = UjS=1Vg.. By the definition of the

topology of G, a basis of open sets of G is given by {g+Gji g e G,

i = 0,l,---}. Thus for large i, Vg- z> gj+Gi, that is, $ induces a function

§i : G/Gi —» X and <|> = <J>iO7Ci

for the projection %[: G —> G/Gi. The space £(G/Gi;X) is made of all func-tions on the finite group G/Gi with values in X and is isomorphic to the setX[G/Gi] of formal linear combinations Sg G G / G ixgg with xg e X via

<|> »-> SgeG/Gi<|)(g)g. Thus we see that

(1) LC(G;X) = lim C(G/G{;X) = lim X[G/Gi].i i

For a topological ring R, we define the space of distributions 2fo<(G;R) by

(2) 0ist(G;R) = HomR(£C(G;R), R).

If cp e £fo t(G;R) and if Xs is the characteristic function of an open set S of

G, we write (p(S) for cp(xs)- Since %h+Gi = SgGGi/Gj%h+g+Gj for j > i , we

have the following distribution relation:

(3) (p(h+Gi) = XgeGi/c/Pdi+g+Gj) for all h e G and j > i.

On the other hand, if we are given a system cp assigning a value cp(g+G0 e R

for all g e G/Gi and for all i sufficiently large satisfying (3), we can extend cp

to a distribution as follows. For a given 0 e £O(G;R), taking sufficiently large i

so that cp(g+Gi) is well defined and <|> = <|)iO7q with <|>i: G/Gi —> R» we define

cpC<t>) = SgGG/Gi(|)i(g)cp(g+Gi). This is well defined because for any other choice

j > i of the index, we have

because 4>jCs) = <l>iCg) for all g € G.

Thus we have

Proposition 1. Let R be a topological ring. Then a function(p : {g+Gi I i > M, and g e G ) - > R is induced from a distribution if andonly if cp satisfies (3) for all j > i > M.

Let R be a closed subring of K. For any measure cp e Meas(G;R)y 9 induces a

distribution, again denoted by cp, by (p((|)) = j(|>d(p. Then | (p(<|)) | p < | I p is finite. Now we want to show theconverse. For any continuous function (J) : G —» R, we can find for each


positive 8 > 0 and g e G a small open neighborhood Vg of g such that

I <Kh)-<Kh') | p < e for all h and h1 e Vg. Thus G = UgGGVg. Since G is

compact, we can choose finitely many g i , . . . , g s e G such that UjS=1Vgj and i

large such that Vg- z> g+Gi for all g e Vg-. Then choosing a complete repre-

sentative set R for G/Gi and defining <])e : G/Gi -> R by (|)e(h) = (|)(g) if

h € g+Gi, we see that <|)e e LC(G;R) and | <|>e-<|> | p < e. Thus LC(G;R) is

dense in 0(G;R) and

( 4 ) I <|>e-(])e I p e-<|))+(4)-<t)e) I p ^ m a x ( I <t>e-<l> I p» I ^"ê I p) < m a x ( e , e ' ) .

Let cp is a distribution with bounded norm I (p I p. This is equivalent to sayingthat | <p(g+Gi) I p is bounded by | (p | p for all i > M and all g e G. Then (4)implies

I <p(<t>e)-<P(<l>£') I p ^ I 9 I P I <t>e-0e I P i/n) e R.n—>«»

Then it is easy to verify that cp e Meas(G\R). Thus we have

Proposition 2. For any closed subring R of K, LC(G\R) is dense in C(G;R).Any bounded distribution on G with values in R can be uniquely extended to abounded measure with values in R. In particular,

Meas(G\A) = <Dist(G;A) via the restriction to LC(G;A)for the p-adic integer ring A of K.

§4.4. Another construction of p-adic Dirichlet L-functionsWe reconstruct the p-adic measure which interpolates the values of DirichletL-functions via cohomology theory. This type of formalism (the formalism ofmodular symbols) was found by Mazur in [Mzl] and [MzS] where he applied it toL-functions of elliptic modular forms (see §6.5).

We fix a prime p. Let K/Qp be a finite extension and A be the p-adic integer

ring of K. Let N > 1 be a positive integer prime to the fixed prime p. Let XN

be the space obtained from TN° by filling the hole around 0. The inclusion:

cr —» TN° induces an A-linear map: Hlc(X^,L(m;A)) —» L(m;A), which we

write as \ \-> J r ^. Then we consider a map

(1) c : p - Z = UT ^ -» HomA(H'(XN,£(m;A)), L(m;K))

given by

4.4. Another construction of p-adic Dirichlet L-f unctions 121

For £, G Hj.(XN,£(m;K)), we write c^(r) = L f^. Here we let the mul-

tiplicative semi-group M2(Z)riGL2(Q) act on L(m;K) by aP(X,Y) =

P((X,Y)V) (a G M2(Z)), where a1 = det(a)a-1. Then c§(r+l) = c^(r) by

definition, and c^ factors through Qp/Zp = p"°°Z/Z. Supposing 2; | T(p) = ap^

with I ap I p = 1, we define a distribution <X>£ on Zpx by

(2) O^(z+pmZp) = ap-ml P

Q J c ^ ( ^ r ) for z = 1, 2, ... prime to p.

This is well defined because c^(r+l) = c^(r). We take G = Zpx and fix an iso-

morphism G = | ixZp for a finite group [i. Then |U = {C, e Zpx | ^9 ( p ) =1} ,

where cp is the Euler function and p = 4 or p according as p = 2 or not.Then the subgroup Gj = l+pxZp corresponds to pxZp. To show that O^ ac-tually gives a distribution, we need to check the distribution relation (3.3). Wecompute

p ° ) v (£*\-Y fp °](l "(j+x)o I L c ^ P > ^ J l o IJ o I

-xVl

This shows

The general distribution relation (3.3) then follows from the iteration of this rela-tion. By a similar argument, we see that

(3)

where |£ | p = Supxj | j(x) | p for the coefficient ^(x) in Xm-jYj of

with x running over p"°°Z. Thus O^ is bounded and, by Proposition 3.2, wehave a unique measure O^ extending the distribution <!>£. Projecting down to the

coefficient in ( • jXm~^ of O^, we get a measure (py. Now we want to show

dcp^j(x) = xMcpô- To show this, we may assume that I h, | p < 1 by multiply-ing by a constant if necessary. We follow the argument given in [Ki] whichoriginates with Manin [Mnl] and [Mn2]. For (|) G C(Zp

x;A), take a locallyconstant function fa : (Z/pn ( k )Z)x -^ A such that | fa-ty \ p < p"k andn(k) > k. Then we know that | O^((|)k)-O^((|)) | p < | O^ | pp"k < p"k and


z=l,(p,z)=l

Pn<k>-i

Iz=l,(p,z)=l

j=0 z=l,(p,z)=l P

pn<k)_!

= ap-n(k) X Wz)(X+zY)mô(^y) modpk

z=l,(p,z)=l

( 1 ) m " j YJ mod pk,

where ^j(x) is the coefficient of c(x)(£) in Xn"JYJ. Thus taking the limit making

k -» °o, we see that

(4) J<j>dq> j = J(|)(z)zJd(pô(z) for all ty e C(Zpx;K).

Let N be a positive integer prime to p. We take % = 8 r i for each primitive

character % modulo N (8 r i may not be p-integral but is bounded because

(am+1-%"1(a))53C-i is p-integral as seen in §1). Then we write O^ as Ox and com-

pute for any primitive character $ of (Z/prZ)x the integral ](|>d<l>x. F° r e a c n

fa b\triangular matrix a = 0 , , we let a act on C by a(z) = (az+b)/d, and for

any differential form co on C, we write cc*co for the pullback of co by the actionof a. We see that, if <|> * 1, then %$ is primitive and

f d O V x r(V 0Yl -V U I A U

Z(p)r[P0 jjG(x)G«l))Jcof(rt)-i(X-zY)mdz

j=0m

= -Z(p)rG(%)G((|))G«)Z)-1X N-jL(-j,x<|))(Ij1)xm-jYJ (see (2.2))j=0 v J 7

m

j=0

4.4. Another construction of p-adic Dirichlet L-functions 123

Here we have used the formula

(6) Gfe)G(« = E u m o d N , vmodpIX(u)4Xv)e(£+f)

= XumodN,

Now assume that <j> is trivial. Then we see that

This shows that

(7) J= - f N-J(l-X(p)pj)L(-j,x)(I°)xm-JYJ.

j=o X J '

Now we assume that % - id and N = q * p is a prime. Then we take^ = i(coo(f-f I q)) and write cpid for ty^. Replacing c° by c° introduced in §2 toavoid singularity at 0, we see from (1.12) that (jfy is bounded and

r fe(l-^1(q)q-J-1)L(-j^)(I|1)xm-JYJ if $ * id,(8) - L d * , , = i J V J

/ m X

[aXm+XJi:i(l-q-J-1)(l-x(p)pj)C(-j)(^)xm-JYJ if 0 = id,

for a suitable a e Q. Thus projecting down to the coefficient of ( i jXm in 3>x,

we get by (4) a measure qfy satisfying, for all characters (j): (Z/prZ)x -> Kx,

(9) -U(z)zidcp%(z) = / ^ N ) N ( l x 0 ( p ) p ) L ( j , ^ ) if X * id,K) J9W W; id^-l^-l-JXl^)^)^^) if % = id.

Here at first sight, the measure (p looks to depend on L(m;A) or m. Howeverthe evaluation formula (9) does not depend on m. Note that every function onf : (Z/prZ)x -> K can be written by Lemma 2.3.1 as

f(g) = cpCpO'^xfWX^^"^) = (pO'^WgMZxfWcKx-1)}.Thus every locally constant function is a linear combination of finite order charac-ters by (3.1). Since the space of locally constant functions is dense in the space ofcontinuous functions (Proposition 3.2), each measure is determined by its value onfinite order characters. This shows the independence of (px with respect to m.Summing up all these discussions, we get


Theorem 1. Let p be a prime and N be a positive integer prime to p. For eachprimitive Dirichlet character % ^ id modulo N, we have a unique p-adic measure<px on Zp

x such that for all finite order characters $ of Zpx and j € N, we

have

J(Kz)zJd(px(z) = ^(N)-1N-J(l-x^(p)pj)L(-j,%^).

As for the identity character, fixing a prime q prime to p, we have a unique p-adic

measure (pq on Zpx such that for all finite order characters <|> of Zp

x and

j e N, we

By the evaluation formula in the above theorem, we conclude that (pq = -£q-i on

Zpx for the measure £a defined in Theorems 3.5.1 and 3.9.1. We now define

the p-adic Dirichlet L-function for each primitive character % : (Z/NprZ)x —> Kx,

writing %N (resp. %p) for the restriction of % to (Z/NZ)X (resp. (Z/prZ)x), by

(x) if %N ^ id,(s Y)

=

H ^ ' J H if %N = id.Thus we get a generalization of Theorem 3.5.2:

xTheorem 2. For etfc/* primitive Dirichlet character % : (Z/NprZ)x -» K%(-l) = 1, f/zere exwtt a p-adic analytic function Lp(s,%) on Zp «/ % ^ idon Zp-{ 1} if % = id ^MC/Z r/zar

Lp(-m,%) = (l-%co-m-1(p)pm)L(-m,xco-m-1) /or all m e N.

Chapter 5. Elliptic modular forms and their L-functions

A modular form of weight k with respect to SL2(Z) is a holomorphicfunction on the upper half complex plane

# = {z G C | Im(z) > 0}satisfying the functional equation

*for all I J e SL2(Z).

Thus it is invariant under the translation Z H Z + 1 and has Fourier expansion:

f(z) = ^°°__oo ane(nz) for e(z) = exp(27tV^lz).

We always assume that an = 0 if n < 0. A typical example of such a modularform is given by absolutely convergent Eisenstein series

E'k(z) = J / ( m n ) (mz+n)"k for every even integer k > 2,

where Z' means that the summation is taken over all ordered pairs of integers(m,n) but (0,0). The Fourier expansion of E'k(z) is well known (we will verifythe expansion later):

_ , , ^2(27i>Ti)k\1T?l , , o.lrn , , v -

^k\z) - — ( \ r i \ t & k\z) ~ * SV^-K) + Z, i^k-H11)^ »

where am(n) = Z0 < d | ndm is the sum of m-th powers of divisors of n. In this

section, we study the complex analytic theory of modular forms. To each holo-

morphic modular form f = ^ anqn, we associate a Dirichlet series

and with each pair of modular forms f and g = ^°°_ bnqn, we also associate

another Dirichlet series

Then, we will study algebraicity properties of these modular L-functions later inthis chapter and Chapters 6 and 10.

§5.1. Classical Eisenstein series of GL(2) /Q

A subgroup of SL2(Z) is called a congruence subgroup if it contains all matrices

a = mod NM2(Z) in SL2(Z) for an integer N > 0. We study here

Eisenstein series for congruence subgroups of SL/2(Z). In this book, we aremainly concerned with modular forms with respect to the following type of con-gruence subgroups: for each positive integer N

fa b>1 SL2(Z) | c e N Z |

126 5: Elliptic modular forms and their L-functions

I"i(N) = jl c A e SL2(Z) | c e NZ and d = 1 mod Nk

In particular, we are interested in the case where N = pr for a rational prime p.We write A for one of these groups. A more detailed study of classical Eisensteinseries for general congruence subgroups can be found in [M, Chap.7]. Each ma-

fa b\trix Y = I d I w i t n det(Y) > 0 acts on the upper half complex plane

# = {z E C | Im(z) > 0}

via the linear fractional transformation z h-> Y(Z) = S^r- Then for any givenfunction f on tt> we define an action of Y on f by

f I kY(z) = det(Y)k-!f(Y(z))(cz+d)-k.For each positive integer k, the space of modular forms fA4(A) of weight k forA consists of holomorphic functions f on 9{ satisfying.the following conditions:(la) f | ky = f for all ye A;

(lb) For each oce SL2(Z), flkOC has the following type of Fourier

expansion of the following form: f | k&(z) = ^n^Qa(n,f | ka)e(nz), where

n runs over a fractional ideal aZ in Q.

A modular form f is called a cusp form if a(0,f |ka)=0 for all a e SL2(Z).We write as 5k(A) the subspace of ^ ( A ) consisting of cusp forms. Let% : (Z/NZ)X -> C x be a character. Then we write ^k(Fo(N),%) (resp.5k(r0(N),%)) for the subspace of MtfTiQ*)) (resp. 5k(Fi(N))) consisting offunctions satisfying the following automorphic property:

f l k ( c d) = ^ ( d ) f f o r (c d ) G r ° ( N ) '

Since the unipotent matrix I Q lj is contained in A, every modular form f in

is invariant under the translation by 1; that is, we have f(z+l) = f(z).

Since e(z) = e(w) if and only if z = w mod Z, we can consider f as a func-

tion of q = e(z). Thus the above condition (lb) means that f as a function of q

is analytic at 0 and has the following Taylor expansion f(q) = Z°° a(n,f)qn.n=0

A typical example of a modular form of even weight k > 2 is given by theEisenstein series which is defined by the absolutely convergent infinite series

Ek(z) = I ' ( m , n ) (mz+n)-\

where "L"' indicates that the summation is taken over all ordered pairs of integersexcluding (0,0). By definition, it is clear that E'k e iA4:(SL2(Z)) if the abovesummation is absolutely convergent, since

, az+b( +

5.1. Classical Eisenstein series of GL(2)/Q 127

We consider slightly more general series for any non-trivial primitive Dirichletcharacter % : (Z/NZ)X -» Cx:

This series is non-trivial only when %(-l) = (-l)k becauseX^-nX-mNz-n)* = x(-l)(-l)k%-1(n)(mNz+n)-k.

One can easily verify that this series is absolutely convergent if k > 2. We seethat

= Z'(m,n) X'1(n)((mNa+cn,mNb+dn)[j])-k(cz+d)k.fa b\fa b\

Rewriting (mNa+cn,mNb+dn) as (mN,n) for . e Fo(N) (because c is

divisible by N), we have

^ ; % ) = %(d)E'k(z;x)(cz+d)k.

Thus E'k(z;%) satisfies the automorphic property defining the elements of^4(Xo(pr)>%)- We now compute the Fourier expansion of E'k(z;%). We use theformulas (2.1.5-6):

Z"1 +Zr=i {(z+n)-1+(z-n)-1}=^cot(7Cz) = 7 iV : : l { - l -2X^ 1 e(nz)},where e(z) = exp(27C V-Iz). This series converges absolutely and locally uni-formly with respect to z, and hence we can differentiate this series term by termk-times by (27ci)~1 -^, and we get

Then we have

N

Xx" 1 ( r )XLi Xr=-~ (mNz+r+nN)-kr = l

<*> N

m=l j=l

= 2L(k)X-1) + 2N-k (-f^) k £ £ x-»0)i n"e(n(mz4^ - ^ l j l lm=l j=l n=l


We now compute Z]^1%"1G)e(fj)- If n is prime to N, then by making the

variable change nj \-> j in the summation,N . N

where we write G(%~1) = S ^ X ^ O M - N T ) - (If X is trivial, then 1+G(id) is the

sum of all N-th roots of unity and hence 0. Thus G(id) = -1.) If % isprimitive, then as seen in Exercise 2.3.5 and (4.2.5a,b), we have

Anyway G(%) 0. If n is not prime to N, then we have Sj=1%"1(j)e(j4) = 0

by Lemma 2.3.2, because % is primitive. Thus we know, for primitive %, that

E'k(z;x) = 2L(k,x"1)

^ l'- [m=0n=0,(n,N)=l

By the functional equation (Theorem 2.3.2), we know that

-1)^^}"1E'k(z;We put Ek(z;x)={2N-kG(x-1)^^}1E'k(z;x) and

where we agree to put %(d) = 0 if d has some non-trivial common factor withN. Then we have

oo

Proposition 1. We have Ek(z;x) = 2'1L(l-k,%) + ^ ak-i,x(n)e(nz) if % isn=l

primitive modulo N or % = id.

We note here that Ek(z)-pk"1Ejc(pz) has the following Fourier expansion:

pn=l

where ip is the identity character modulo p (therefore, ip(pn) = 0 for all nand ip(n) = 1 if n is prime to p). Although we have only proved thisproposition under the assumption that k > 2, the assertion for non-trivial % istrue even for k = 1 and 2. We will see this in Chapter 9.

5.1. Classical Eisenstein series of GL(2)/Q 129

We now want to compute a slightly different series for a Dirichlet character %modulo N (we allow here % to be imprimitive):

oo oo

G'k(z;%) = £(».„) X(m)(mz+n)-k = 2 £ x(m) X (mz+n)"k

= 2 £ X(m) £ ^ ^ 1 nk-1e(mnz) = 2 ^ ^ f ; a'k-i.x(n)qn.m=l n=—oo n=l n=l

where we write a'm>x(n) = Xo<d|n X(n/d)dm. We put

n=l

o -For x = , we see easily that

E'k(z;%) | kT = N-1z-kX(m,n) X ^ W d n N ^ f n ) " k = N ^ G U z ; ^ 1 ) .

We in particular have, for a primitive character %,

Ek(z;%) | kx = %(-l)Nk-2G(x)Gk(z;%-1),

where we have used the fact that G(%)G(%'1) = %(-l)N. We note this fact as

Proposition 2 (Hecke). Let % be a Dirichlet character modulo N with

Z(-1) = (-l)k. We have

Ek(z;%) = 2-^(1-^%) + Y^=l ^k-i,x(n)qn for primitive %,

Gk(z;%) = 5k,2(27cy)-1+5k,i2-1L(0,%) + £ " = 1 a'k.i,x(n)qn for any %,

8kj = 1 or 0 according as k = j or nof. Unless k = 2 and % zs1 r/zeidentity character, the functions Ek(z,%) and Gk(z;%) ar^ elements in

^k(ro(N),x). Moreover we have Ek(z;%) | kT = %(-l)Nk-2G(%)Gk(z;%-1).

Proof. We only need to prove that E'k(z,%) and G'k(z,%) satisfy (lb). Weprove this assuming k > 2 and complete the proof in Chapter 9 (see Theorem9.1.1 and (9.2.4a,b)) dealing with the case where k = 1 and 2. We consider aslightly more general series: for a pair (a,b) of integers

EfkfN(z;a,b) =

Then it is easy to see that E'k>N(z;a,b) | koc = E'k>N(z;(a,b)a) and

E'k(z;z) = E 1


G'k(z;%) = E X

Thus we only need to prove that E'k,N(z;a,b) has no negative terms in its Fourierexpansion. We see by definition that E'k,N(z;a,b) is equal to

5a In*n=b mod N

^ ^ , /mz+b V / i \k xf XTmsamodN,m>0n=-» \ / msamodN,m<On=-<»

where 6a = I 1 i f a ^ 0 mod N,[0 o therwise .

By (2), we have for a constant c ^ 0

Z°° (\m z±b Yk v^ k i / Am z ± b x

I N hn = c ^ n e ( n ( N )}'n=-ooV J n=l

This shows that the Fourier expansion of E'k,N(z;a,b) does not have terms ine(nz) with n < 0.

As a byproduct of the above calculation, we get

(3) The constant term of the Fourier expansion of Gk(z;%) | kOC vanishes for

every a = e Fo(p), if % is primitive modulo pr and k > 2.

We prove the following facts for our later use

(4a) / / k > 2, then

I < k-i,%(n) I ^ ^ ( k - l ) ^ " 1 and I o'k-i,x(n) | ^^ (k - l )^" 1 .

(4b) If k = 1 or 2, for any e > 0, there is a positive constant C such that

I ak_i,x(n) | < Cnk"1+E and \ a'k.i,x(n) | < Cnk"1+e.

We compute

where n = n p | npe(p)- Then we see that

-p1-k)SC(t-l)nk-1 if k > 2 .

Since I o\.\y%{n) \ < \ a'k-i,id(n) I = I C7k-i,id(n) I, the assertion is also true fora'. The above argument yields

Since

5.2. Rationality of modular forms

<C(s-l)ns-! for Gs_i,id(n) = Xo<d|ndS"1 i f s > 2'

I ai,id(n) | < Iai+ e , i d(n) | < C(l+e)n£ for all e > 0.

131

This proves (4b) for k = 2. When k = 1, we have

I aOiid(n) I = I I p | n(e(p)+D if n = I I p | npe(p ) .

Thus I ao,id(n)n-£ | 1, we see, if n > 2, that.+e(p)) 1 e(p)

pe(p)e ~ pe(p)e

If p > 21/£, then p£ > 2 andIn _ ^pe(p)E " 2 e ( p )

Combining these two inequalities, we see that

I G0,id(n)n-e | < 1/£exp(21/£/elog2) < Cn£.

§5.2. Rationality of modular formsWe first deal with the rational structure of the space #4(SL2(Z)). We introduceseveral modular forms with integral Fourier coefficients to study the rationalstructure in an elementary manner. For k = 4, 6, 8, 10 and 14, we put

Gk(z) = 2C(l-2k)"1Ek = 1 + C k ^ i ak.!(n)qn e fWk(SL2(Z)).

By the actual values of £(l-2k) given after Theorem 2.2.1, we have

k

ck

4240

6-504

8480

10-264

14-24

We could have defined G12 in a similar manner, but then it would have had the

prime 691 as the denominator of its Fourier coefficients. We further put

G0(z) = 1 and A(z) = g|(z) - 27g3(z) for g2 = G4/12 and g3 = G6/216.

The function A is called the Ramanujan's A-function (or the discriminant

function). Computing the constant term of A, we see that A e 5i2(SL2(Z)). It

is known that A does not vanish on !H and even has the product expansion

A(z) = qfJQ ^-q")24

132 Elliptic modular forms and their L-functions: Chapter 5

Let X = (SL2(Z)\#)U{~} (see [Sh, 1.4, 6.1] and [M, §4.1] for how to give a

/standard structure of Riemann surface on this space). Put J = G4/A. Then J

has the q-expansion of the form: q"1 + X°° cnqn with cn e Z. The function Jn=0

gives an identification of X with the projective J-line P^J) [Sh, Chapter 4]. Wedefine for each positive even integer k

r = = J[k/12], if k s 2 mod 12,l[k/12] + l, otherwise,

where [q] for a rational number q denotes the largest integer not exceeding q.Put s(k) = k-12(r(k)-l).

Lemma 1. The equation 4a+6b = k-12(r(k)-l) = s(k) has one and only onenon-negative integer solution for each even integer k.

Proof. If k = 2 mod 12, then k-12(r(k)-l) = 14; in this case, the unique solu-tion is (a,b) = (2,1), and if k 4 2 mod 12, then k-12(r(k)-l) < 12 and theuniqueness and the existence of the solution can be checked easily. We list all thesolutions in the following table:

k mod 12S

ab

0000

21421

4410

6601

8820

101011

We choose the unique solution (a,b) as above for a given k and define^ = (G4)a(G6)b+2(r'1' i)Ai e afk(SL2(Z)) for i = 0, 1, —, r-1 (r = r(k)).

Note that hi e 3Vfk(SL2(Z)), and hi has the q-expansion (q = e(z)) with co-efficients in Z of the following form:

hi = q1 + £~ = i + 1 bnqn w i t h b n ^ Z.

This shows that hi are linearly independent. For any subring A of C and eachcongruence subgroup F, we put

) = {fe Mk(T) I a(n,f) e A for all n}, Sk(T;A) =

Theorem 1. We have dimc(^k(SL2(Z)) = rankz(f7Hfk(SL2(Z);Z)) = r(k) andfor any subring A of C,

f*4(SL2(Z);A) = ^k(SL2(Z);Z)<g>zA, 5k(SL2(Z);A) = 5k(SL2(Z);Z)<g>zA.Moreover {ho, •••, hr.i) (resp. {hi, •••, hr_i}j form a basis of f^"k(SL2(Z);Z)(resp. 5k(SL2(Z);Z)j over Z.1

XI am indebted to Y. Maeda for the construction of the basis (hi).

5.2. Rationality of modular forms 133

Proof. If we know that the dimension of the space of modular forms is less thanor equal to r(k), then hi gives a basis of fA4(SL2(Z);A) over A and everythingwill be proven. Let us show the inequality

dimc(44(SL2(Z))) < r(k).

Put s(k) = k-12(r(k)-l). Recall that s(k) = 0,14,4,6,8,10 according ask = 0,2,4,6,8,10 mod 12. We put

Tk(z) = G14-s(k)(z)A(z)-r.

Then Tk is holomorphic everywhere on !H and we have the followingq-expansion of Tk (q = e(z)):

(1) Tk(z) = ck,rq~r+ ••• +ck io+--- with ck)jG Z (ck,r = 1).

Since the weight of Tk is given by 14-s(k)-12r(k) = 2-k, for each f efA4(SL2(Z)), fTk(z) is of weight 2. Thus the differential form CO = fTk(z)dz =

—?=fTkdq/q satisfies y*co = co for all y e SL2(Z) and is holomorphic on fH.

That is, co has a singularity only at infinity. By construction, the singularity of coat infinity is a pole of order at most r+1. Let C0m = JmdJ which has a pole of or-der m+2 at infinity and which is holomorphic outside infinity. Let 8 be a mero-morphic differential form on X whose singularity at infinity is a pole of order nand is holomorphic outside infinity. Then n > 2 since deg(8) = 2g-2 = -2 forthe genus g = 0 of X. In fact, this can be proven as follows. Since X is aRiemann sphere with coordinate J, at any point x e X , J-J(x) is a local pa-rameter at x. Therefore dJ has order 0 at x. On the other hand, at infinity, dJhas order -2 and hence deg(dJ) = -2. For any morphism f : X —» PX(C) ofalgebraic varieties (or Riemann surfaces), the numbers of points in the fiber at 0and oo are equal (counting with multiplicity). Thus

deg(f) = #(points over 0) - #(points over °o) = 0and therefore, writing 8 = fdJ for a suitable function f: X —> P!(C), we seethat deg(8) = deg(fdJ) = -2. Thus, subtracting suitable constant multiples of thecoi's from 8, we may assume that 8-(boCOo+-*-+bn-2COn-2) has at most a simplepole at infinity and is everywhere holomorphic outside infinity; that is,deg(8 - bocoo + ••• +bn_2C0n_2) £ -1 . This implies 8 = boCOo+---+bn_2COn_2.Applying this argument to co, we know that co can be written asboCOo+-**+br_iCOr_i. Via the map f H» co = fTkdz, we can embed f*4(SL2(Z))into the space of differential forms generated by coo, • • •, GVi and thus we get

dimc(*4(SL2(Z)))<r(k).


Since dimc(#4:(SL2(Z))) = r(k), the first r(k)+l q-expansion coefficients ofany f e f7Vfk(SL2(Z)) have to satisfy a non-trivial linear relation. We can makeexplicit this linear form:

Corollary 1 (Siegel [Si]). / / f = X~=oanqn€= *4(SL2(Z)), then

Ck.oao + "• + Ck,rar = 0 (r = r(k)) and Ck,o ^ 0,where Ckj are the Fourier coefficients of Tk in (1).

Proof. Note that com = JmdJ = -L-HL—dq and ai-rm +1 dq uq

jf = 27u'mqm), and hence the coefficient in q"1 of com is always 0. The

constant term of Tkfdq is given by Ck,oao + •** +«k,rar, which shows+—•- Ck,rar = 0. Now we prove that Ck,o ^ 0. Note that

fl nn=l n=l ^=0

Therefore the coefficients of A"r are all positive. On the other hand, the coeffi-cients of Go, G8, G4 are positive and hence Ck,o 0 when k = 2 mod 4.Now suppose that k = 0 mod 4. Then

s(k) = k-12(r(k)-l) = k m o d 4

and we can write s(k) = 4t with t e Z. Now consider co = (27tO"1Gi4A"1dq/q

= GuA^dz which has a pole of order 2 at infinity and is holomorphic outside

infinity. Therefore we can write co = cdJ with c * 0 and thus G14A"1 = C-T- or

G u = c A ^ because r(14) = 1. This shows that Tk = cGi4.s(k)A1"r(Gi4)'1^ .

Since fAfi4(SL2(Z)) is of dimension 1, we know that Gi4_s(k)Gs(k) = G14 andGS(k) = (G4)1, comparing the constant terms. Then, we know that

By replacing G4 by (AJ)1/3 in this formula (G43 = AJ by the definition of J),we have

X _ c^l-r-t/3j-t/3dJ = 3C ^l-r-t/sdJ1"

Note that JH t / 3 ) = (G4)3-tA(t/3)"1. Then we have

dz ~ A dz


d(G4)3-lA-r _ d(G4)3-tA(t/3)-1A1-r-t/3

""A dz + ( C J 4 ) A dzand

dz " rTherefore, we see that

T 3c d ,r3-tA.rx ^ (3r+t-3)cr3-tdA-r

T k = 3^dz"(G4 A ) + (t-3)r G4 "dT •

Since ^-(G4"tA"r) has no constant term, we look at the second term of the above

formula. Since the coefficient of qJ in —T—- for j < 0 is negative, and the co-

efficients of G4~l are all positive, we know that the constant term of the second

term is negative. This shows that ck)o < 0.

Applying the linear form in Corollary 1 to Ek(z), we have

Corollary 2. r ^ l - k ) = - C k ^ X ] ! ? ak-i(n)ckJ for all 2 < k e 2Z.

Let me briefly explain how Siegel applied Corollary 1 to show the rationality of thevalues of the Dedekind zeta function of a totally real field F of degree d. Let abe an ideal of the integer ring O of F and consider the following Eisenstein se-ries for z e M (see Theorem 2.7.3):

for even integer k.

Here N(mz+n) is the product Tlairxfz+n0) taken over all embeddings a of Finto R, and (m,n) runs over all equivalence classes of ordered pairs of numbers(m,n) in a under the relation (m,n) - (m',n') if m' = em and n' = en fore G Ox. This series is absolutely convergent if k > 2. We can verify thatEk(z;a) is a modular form of weight k[F:Q] with respect to SL2(Z), and we cancompute its Fourier expansion (see Theorem 9.1.1). Even if one replaces a byXa for X G Fx, each term of Ek(z;a)

NF/Q(a)kN(mz+nyk = NF/Q(MkA^mz+?tn)-k

does not change, and thus, Ek(z;a) depends only on the ideal class of a. Tosimplify a little, we define

Ek(z) = £aEk(z;a),where a runs over a set of representatives of ideal classes of F. Then we have


n=l

where i3- is the different of F, 2; runs over all totally positive elements in ft"1,

D = A^F/Q(I^) is the discriminant of F, £F(S) = Z ^ F / Q ( ^ ) " S is the Dedekind

zeta function of F and Ok-i(£) = Z

Corollary 3 (Siegel). For each even positive integer k, we

~n = ~ ckd,0 G ^

This fact also follows from Corollary 2.7.1 proved by Shintani's method by thefunctional equation of £F(s) (Corollary 8.6.1).

Now let us note another application of Corollary 1. Let p be an odd prime andconsider the Eisenstein series f = Ek(z;coa) e f7t4(ro(p),coa). Let b be the orderof coa. Then fb is an element of #4b(ro(p)). We choose a complete representa-tive set R for Fo(p)\SL2(Z) and define Tr(g) = EyeRg I kY forg e fA4(Fo(p)). Then obviously Tr(g) e 3tf"k(SL2(Z)). We now want to com-pute Tr(fb).

Lemma 2. We can take as R the set of the following matrices: the identity ma-(1 ft (0 -I)

trix I2 and 8j = 8 L A for 8 = L Q with j = 1, ••• , p. / /

g(z) = E°° Aane(nz) an^ g | k8 =n=u

Tr(g) =

Proof. Take y= I J e SL2(Z). If c is divisible by p, then y is in Fo(p)

and in the left coset of I2. Thus we may assume that c is prime to p. Take aninteger j in the interval [l,p] such that cj = d mod p and consider

cj-dwhich is divisible by p by definition of j . Thus y e Fo(p)8j. If y and

The entry at the lower left corner of this matrix is

y' = I , d, I are in the same left coset of Fo(p), then cd' = c'd mod p and

hence SL2(Z) = UjFo(p)8jUFo(p) is a disjoint union. Note that

I j U g I k5j = i ; = 0 b n lP = 1 e (^ ) = pb0 + S;=1pbnpe(nz). From this, the for-

mula in the lemma is obvious.


Recall that for the Eisenstein series Gk(z;%) = £°° G'k_i,%(n)e(nz) w ^

tf'm,x(n) = £o<d|n X(n/d)dm, we proved the following formula in §1:

Ek(z;%) | kx = (-l)kpk'2G(%)Gk(z;x-1),

(o -nwhere x = 0 and % = coa for the Teichmiiller character co. Note that

J and therefore Ek(z;x) Ik8 = (-l)V1G(%)Gk(|;x"1). Thus if we

put f = Ek(z;%) and g = ( - l ) k p- 1 G(x)G k (- ;x- 1 ) , then fb | k5 = gb =P

£°° bne(—-) where bn for all positive n are algebraic numbers. Write f as

X + F(q) with X = 2-1L(l-k,coa) and F(q) e qQ(coa)[[q]]. Then

fb = Xb + S°° an(X)qn, where an(X) is a polynomial of X of degree strictlyn=l

less than b with coefficients in Q(coa). Thus by Siegel's theorem, we have anequation with coefficients in Q(coa):

ckb,oXb + X j b ) ckb>j(aj(X)+pbjp) = 0,where the degree of aj in X is strictly less than b. Thus the above equation isnon-trivial, and we get another proof (valid only when k > 1) of the followingfact:

Proposition 1. 2~1L(l-k,coa) for k > 0 is an algebraic number ifcoa(-l) = (-l)k.

We now want to show that 2"1L(l-k,coa) e Q(coa). For that purpose, we intro-duce the transformation equations. For each modular form f in Mk(T), we fix arepresentative set R for F\SL2(Z) and define

P(f;X) = n T e R ( X - f I kY) = X<1 + s i ( f ) x d " 1 + - + sd(f)-Note that Sj(f) e 5tfkj(SL2(Z)) and P(f;f|ky) = O for any y e SL2(Z). For

each f E M^(T)y we formally define the conjugate f° = E°° a(n,f)aqn as ann=0

element of C[[q]] for any automorphism a of C. Since the map

defines a ring automorphism of C[[q]], if we put

= Xd+si(f)°Xd-1+-"+sd(f)a, then Pa(f;f°) = O in C[[q]].

Proposition 2. For f e !Wk(SL2(Z)), we have f° e ^k(SL2(Z)) for each

a e Aut(C). In particular, Sj(g)a e ^kj(SL2(Z)) for any ge


Proof. Write f = coho+---+cr_ihr_i for the basis hj as in Theorem 1. Then

fa = c00ho+-+cr . i°h r . i€ *4(SL2(Z)) because hj e Z[[q]].

Let fM"(C) = ©"T fAfk(SL2(Z)) and for any subring A of C, we put

= fW(C)DA[[q]]. Note that 0f(C) is a graded algebra andf = ®kfk H> f° = 0kfka for a e Aut(A) is a ring automorphism ofLet A(A) be the quotient field of

Proposition 3. If f e fMk(F) for a subgroup T of finite index of SL2(Z),then for any a e Aut(C), f° e ^k(A) for a normal subgroup A of finite in-

dex in SL2(Z).

Proof. Let F (resp. F°) be the set of all roots of P(f;X) (resp. Pa(f;X)). Eachroot g of Pa(f;X) is not only a formal Fourier series but it converges on m be-cause it is a root of a polynomial with function coefficients. Thus fory G SL2(Z), we can consider g | kY, which is a root of

Pa(f;X) | y = Xd + si(f)° I kYX*"1* ... + sd(f)a | kdy = 0.The left-hand side of this equation is equal to Pa(f;X) becauseSj(f)a e fA4j(SL2(Z)). Thus F a is stable under SL2(Z). Let A be the subgroupof SL2(Z) which fixes all elements of Fa. Then the action of SL2(Z) on F°gives a representation of SL2(Z)/A into the group of permutations of elements ofF a , which is a finite group. Thus A is a normal subgroup of finite index inSL2(Z).

The following fact is clear from Corollary 2.3.2, but we shall give another proof ofthe fact:

Theorem 2. The value L(l-k,%) for %(-l) = (-l)k is an element of Q(%), and

for each ae Aut(C), L(l-k,%)a = L(l-k,%a).

Proof. Let a be an element of Aut(C). Then for f = Ek(z;%), all Fourier coef-

ficients of f° but its constant term coincide with those of g = Ek(z;%a). Thus

C = f°-g is a constant, which is a modular form of weight k for a subgroup A

of finite index in SL2(Z) (if f° is modular with respect to O and g with respect

to F, then O/Ofir = OF/F, which is a finite group). Therefore for some

y = e A with c * 0 , CI kY = (cz+d)"kC = C. Since cz+d* 1, we

know that C = 0. Thus (L(l-k,%))a = L(l-k,xa). This shows the assertion.

The above proof of the rationality of the Dirichlet L-values using the action ofa e Aut(C) is due to Shimura.

53. Hecke operators 139

§5.3. Hecke operatorsIn this section, we shall introduce the Hecke operator T(q) as an endomorphism

n o\of ^k(Fo(pr)>%)- We consider the double coset F L F for primes q withF = Fo(N) for a prime power N = pr. Then we can decompose, for

l 0\fa = I 0 I, FaF = UiFai as a disjoint union; actually, an explicit left coset

decomposition is given by

; ;)(;;) to N .F a F =

IE. rl» pJA proof of the above decomposition is given as follows. Take any

7 = 1 d e M2(Z) with ad-bc = q. If c is divisible by q, then ad is divis-

ible by q, and thus one of a and d is divisible by q. We have

y = r q H Je SL2(Z) L J if a is divisible by q. If d is divisible

by q but a is prime to q, then by choosing u e [l,q] such that

ua = b mod q, YI I e SL2(Z). If c is not divisible by q but a is divis-

(0 -I)ible by q, we can interchange a and c by multiplication by 8 = L Q I on the

left. If both a and c are not divisible by q, then by choosing u so thatf

ua = -c mod q, we know that the lower left corner of I 1 Jy is equal to ua+c

and is divisible by q. This shows (la) for F = SL2(Z). The case of

F = Fo(pr) can be similarly treated (see [M, Lemma 4.5.6]). Note that for any el-

ement I d in Fo(pr)^Fo(pr)J c is divisible by pr and a is prime to p

(because every element of Fo(pr) is upper triangular mod pr).

Exercise 1. Give a detailed proof of (1) when F = Fo(pr)-

We define the Hecke operator T(q) on 3Wk(Fo(N),%), using the disjoint decom-

positions FaF = UiFai and F a T = UiFpi (a1 = det(a)a"1), by

(lb) f | T(q) = Ii%(ai)f | kai and f | T*(q) = liXi^f I kPi,


(fa bY)where jn = %(a) (a is always prime to N by (la)). Here we under-

stand that % is trivial when F = SL2(Z). Then obviously T(q) and T*(q)gives an operator acting on fA*k(F) if F = SL2(Z). If ye Fo(N), then

= Il iFai = IliFaiY and thus

f I T(q) | ky = Xi%(ai) f I kOCiT = X(y)"1Xi5C((XiY)f I kOCiY = X(yYlf I T(q).-1 i fa b\

Here note that %(y) = %(a) = %(d) for I d I e Fo(N) because

a d s 1 modN. This shows that f I T(q) e fMk(F0(N),%). Similarly, we can

show that T*(q) preserve #4(Fo(N),%). Now we want to compute the Fourier

expansion of f|T(q). When f e fl4(Fo(pr)>%) for r > 0 , then

f |T(p) = p"xX f ( ^ ) = P- 1E a(n,f)e(^)|; e(y) = £ a(np,f)e(nz).u=l n=0 u = i n=0

As for T(q) for q * p, we have

(As the contribution of the term J J ^ Fl Q I to the coefficient of e(nz), by the

fq 0)same computation as above, we get a(nq,f). For the remaining term Fl 0 j I, we

get %(q)qk-h(n/q,f). Thus we have for f e fMk(ro(N),%) (N = pr),

(lc) a(n,f | T(q)) = a(nq,f) + xCqJqâdi/q.f),

where we have implicitly assumed that a(n,f) = 0 if n is not an integer and%(q) = 0 if q IN. If q and r are two distinct primes, then

a(m,f | T(q)T(r)) = a(mr,f | T(q)) + x f r ^ a W I T(q))= a(mrq,f) + x(q)qk-1a(mr/q,f) + xWi^^ ^ V

which is symmetric with respect to r and q and hence

(2) T(r)T(q) = T(q)T(r) if r and q are different primes.

By the above formula, we have

(3) a(m,f | T(q)2) = a(mq2,f) + 2x(q)qk"1a(m,f) + %(q)2q2(k"

We now define the operator T(qe) for e > 1 inductively by

(4a) W + 1 ) - { T ( q ) 6 + 1 i f i l N ,[TCqFCq^-xC^qk^TCq6-1) otherwise,

5.3. Hecke operators 141

where we define T(l) to be the identity map. Then we see from (2) thatT(qe)T(/s) = T(/s)T(qe) for different primes / and q. More generally, we candefine T(n) for each positive integer n by

(4b) T(n) = Ylji(qeiq)) if n = I ^ q ^ fo r Pr i m e s ^

Then we can write down the Fourier expansion of f | T(n) explicitly (see [Sh,(3.5.12)]) as

( 5 ) a ( m , f | T ( n ) ) = £ o n) ^ V M

Define a semi-group A (depending on N) by

(6a) A = {a e M2(Z) I det(a) > 0} if T = SL2(Z) and

A = {a = L J e M2(Z) | p|a, pr | c and ad-bc> 0} if T = ro(pr)-

Let R be a complete representative set for I\{ a e A | det(a) = n}. Then it isknown that in fact

which is the original definition of Hecke (see [Sh, III] and [M, IV]). In particular,on 5k(ro(pr)>%) (r > 0), we can easily check that

(6b) f |T(ps)= X f | k PU = l ^

Let A be a subalgebra of C. By (5), fA4(I\%;A) and 5k(F,%;A) are stable un-

der the Hecke operators T(n) if A contains Z[%]. Here we write^k(r,x;A) = fMk(r,%)nA[[q]] and 5k(I\x;A) = 5k(r,%)nA[[q]].

Let V be an A-submodule of any one of the above spaces stable under T(n) forall positive integers n. Then we define the Hecke algebra d(V) by theA-subalgebra of EndA(V) generated by T(n) for all positive n. Note that T(l)gives the identity on V and hence /t(V) is a commutative algebra with identity.Hereafter we suppose that A contains Z[%] if we consider f7Vfk(ro(N),%;A) and4(TO(N)JC;A). We defineHk(r0(N),x;A) = fi(fWk(ro(N),z;A)) and hk(r0(N),%;A) =

Lemma 1. The space f7tfk(ro(N),%;C) is of finite dimension over C.

Proof. By replacing T = Fo(N) by the kernel of %, we may assume that % istrivial. We fix a representative set R for r\SL2(Z) and consider the A-linear mapIM • MkfX) -> CM* for each positive integer M given by IMCO = (a(n,f I ky))n,y,where (n,y) runs over all possible pairs with 0 < n < M and y e R for


which we have some modular form f with a(n,f I kY) ^ 0 and M* is the num-ber of such pairs (n,y). The number M* is smaller than NM[SL2(Z):F] becausewe always have n e N-1Z for the pairs (n,y) as above. Take M>r(k)+1and suppose that IM (f) = 0. Let si(f) be the coefficient of Xd-1 inP(f;X) = IIYeR(X-f IkY) for d = #R. Then we have a(n,Si(f)) = 0 ifn < r(ki) < i(r(k)+l). Hence by the proof of Theorem 2.1, si(f) = 0 fori > 0. Thus P(f;X) = Xd. Since 0 = P(f;f) = f1, we know that f = 0. Inparticular, we have

< (r(k)+l)N[SL2(Z):F].

We define for the quotient field K of A"%(ro(N),%;A)={fe ^k(F0(N),x;K) | a(n,f) e A if n > 0 } .

Clearly we know that /%(Fo(N),x;A) z> f7tfk(ro(N),x;A), but they may not beequal.

Theorem 1 (duality). Suppose that A contains Z[%]. Define a pairing

( , ) : Hk(r0(N),x;A) x mk(r0(N),x;A) -> A by (h,f) = a(l,f | h) e A.Then this pairing is perfect on /rck(Fo(N),x;A) and 5k(Fo(N),x;A); in otherwords, we have isomorphismsp

HomA(Hk(r0(N),x;A),A) = mk(r0(N),%;A),HomA(/%(Fo(N),x;A),A) = Hk(F0(N),x;A),HomA(hk(r0(N),x;A),A) = 5k(F0(N),x;A),HomA(5k(r0(N),x;A),A) s hk(F0(N),x;A).

Proof. Here we prove this theorem under the assumption that either N = 1 orA = C and later return to this problem for general N = pr and A. First assumethat A = C. Write F = r o (N) . Since H k ( I \x ;C) is a subspace ofEndc(flik(r,x;C)) which is of finite dimension, Hk(F,x;C) is of finite dimen-sion. Thus we only need to prove the non-degeneracy of the pairing. Sincea(m,f | T(n)) = I b | ( m , n ) X(b)bk"1a(mn/b2,f), we see that

<T(n),f) = a(l,f |T(n)) = a(n,f).

If (h,f) = 0 for all h in the Hecke algebra, then a(n,f) = <T(n),f) = 0 for allpositive n. Thus f is a constant. Since k is positive, f must be 0.Conversely, if (h,f) = 0 for all f, then

a(n,f | h) = (T(n),f | h) = a(l,f | hT(n)) = <h,f I T(n)> = 0.Thus f I h = 0 for all f and hence h = 0 as an operator. This proves the as-sertion for A = C. The above argument is valid as long as A is a field and bothHk(F,x;A) and mk(F,x;A) are of finite dimension over A. In particular, thetheorem is true for A = Q when F = SL2(Z) by Theorem 2.1. Now we wantto show the theorem for A = Z and F = SL/2(Z). Since A is a principal idealdomain, it is sufficient to prove one of the following two assertions:


HomA(Hk(r;Z),Z) s mk(F;Z), HomA(«k(r;Z),Z) = Hk(r;Z).

We shall show that HomA(Hk(r;Z),Z) = /%(F;Z). Since

mk(F;Z)<g)zQ = mk(F;Q), Hk(F;Z)®zQ = Hk(r;Q).

The natural map from mk(F;A) into HomA(Hk(F;A),A) is injective, because if(h,f) = 0 for all h, then f = 0 as shown already. If 9 : Hk(F;A) -> Z is alinear form, we can extend (p to a linear form on Hk(F;Q) with values in Q bylinearity. Then we can find f e /%(F;Q) such that (h,f) = (p(h) for all h.Then a(n,f) = cp(T(n)) e Z and hence f e 7%(F;Z). This proves the surjec-tivity. As for general A, since /%(F;A) = /%(F;Z)<8)ZA, by definition, we knowthat Hk(F;A) = Hk(F;Z)<g>zA. Thus

HomA(Hk(F;A),A) = Homz(Hk(F;Z),Z)®A =HomA(mk(F;A),A) 2 Homz(mk(F;Z),Z)®A 2 Hk(F;Z)(g>A 2 Hk(F;A),

which finishes the proof.

We consider the hermitian product (called Peters son inner product) on5k(To(N)jc) defined by

(7) (f,g)N =

where O is the fundamental domain of Fo(N). First of all, we know that

f (y(z) ) = f ( z ) ( c z + d ) k for y = ^ J j e F . T h u s gf(y(z)) = g f ( z ) I c z + d 1 2 k .

On the other hand,dxAdy = (-2i)"1dzAdz and thus y*(dxAdy) = I cz+d I "4(dxAdy).

By taking the determinant of the formulabYz z | _ ry(z) y(z)Ycz + d 0 ^dj[l lj " t 1 1 j{ 0 cz + dj'

we know that Im(y(z)) = det(Y)|cz+d|"2Im(z). Thus fgyk~2dxAdy for y =Im(z) is invariant under the action of F, and hence at least formally the aboveintegral is well defined. By the same formula, | f(z)yk/21 for f e 5k(Fo(N),%) isa continuous function on Y = Fo(N)W. Since as a function of q = e(z), fvanishes at q = 0, on a small relatively compact neighborhood V of 0,f(q) = qF(q) with a holomorphic function F on V. Thus on the closure of V,

(8a) I f(z)y^2 I is bounded on M and | f(z) I < O(exp(-2?cy)) as y -> 00.

Writing the bound of I f (z)Im(z)k/21 as Co, we see that


| a(n,f) I = I f f(x+/y)e(-n(x+zy))dx | < C

By taking the minimum of the function coe27111^"^2, we conclude that

(8b) I a(n,f) I < Cnk/2 for f e 5k(F,%),

where C is a constant independent of n. Thus by (8a) a* fgyk~2dx A dy fora G SL2(Z) is exponentially decreasing as Im(z) -» oo, and hence the integral(7) converges. Thus (, )N is a well defined positive definite hermitian producton 5k(Fo(N),%). More generally, for any subgroup F of finite index in SL2(Z),we can define (f,g)r for f ,ge 5k(F) by the same formula (7) replacing Fo(N)by T. Note that if F 3 F and if [r :F] is finite, then for co = fgyk"2dx A dywith f,ge 5k(F), y*co = co for all ye F. On the other hand, if F = Uf%

then F = UjYj^r1, and XIX = Llj Yj<D is a fundamental domain of F . In factfor any z€ ^ w e can find a unique y e F such that y(z) e O. Then we canwrite Y = YJ"18 for a unique j and 8 e F . Thus 8(z) e YjO and ¥ is afundamental domain of F . This shows that

(9) (f,g)r = EiJd,^"1)*00 = Jo® = (r : r ' )( f 'S)r fo r r

For any a e A, the group F' = FPloc^Fa is of finite index in F. Thus for

f5g e ^k(F), we have flkOce 5k(F') and

g(fla)yk~2dxAdy = ^det(a)k~lf(a(z)))(a9zykyk~2dxAdy

= a*(glal)f(z)yk"2dxAdy,

where we have written j( , ,z) = (cz+d) and a = det(a)a . Then we

have, for the fundamental domain *¥ of F ,

(f | a ,g ) r = JyCt *(glal)f(z)yk-2dxdy = Ja(xF)(glal)f(z)yk"2dxdy = (f,g |

because aQ¥) is a fundamental domain of a F ' = aF'a"1. Since

(F:F')vol(O) = vol(Y) = vol(aOF)) = ûnder the invariant measure y"2dxdy, (FiF1) = (FtaF'a"1). We now want tocompute (f I [FaF],g) for F = Fo(pr)- Decompose FaF = IIjFaj and write Ofor the fundamental domain of F in H Then we have

(f | [FaF],g) = XjX(ocj)(f I ajfg) =

Lemma 2. We suppose that #(F\FaF) = #(F\FalF) for F = F0(pr). Then we

may choose aj so that FaF = LIjFaj and Fa l F = IIjFaj1.


Proof. Since the involution i brings a right coset onto a left coset, byassumption, the numbers of the right cosets and the left cosets in FocF are thesame. If FocF contains F£, and r\T, then ^ = Srjy for 8 and y in F. Thenfor C = S'1^ = riy, TC, = FS ^ = F^ and £F = r\yT = T]F. Thus we canchoose (Xi so that F a F = UiFoci = UiOCiF. Since i is an involution, this

means that (FocF)1 = U

Note that

A = {a = (a " l e M2(Z) p|a, p r | c and ad-bc>0}.

aThen for a = I , I e A, ad= det(oc) mod p. Thus if det(oc) is prime to

p, then %(al) = %(d) = X"!(a)%(det(a)) = %"1(a)%(det(a)). Therefore by the

above lemma, if FaF = FalF, we have

^jiXiî)'1 E> I oci1 = X 1(det(a))î%(ail)g I ail = %"!(det(a))g | [FaF].

Thus we have T(q)* = %-1(q)T(q) if q is prime to N for T(q) on

5k(Fo(N),%). More generally, we have

(10a) T(n)* = %-1(n)T(n) if n is prime to N.

Note that x = r normalizes Fo(pr) and for a = L , xax = a1.

Thus TfCFaFlT"1 = F a l F and #(F\Fa lF) = #(F\FaF) for F = F0(pr).

Therefore we can choose the decomposition F a F = IIJFOCJ so that

Fa l F = IIjFaj1 is a disjoint union. Since FocT = LIjFxajT"1 is also a disjoint

union, by (lb) and x(Oj) = xCCiOjT'1)1), we have

f I T*(q) = Xj Xj ^

We write Tx(q) for T(q) on 5k(Fo(pr)5X)- Then for any f e 5k(F0(pr),X)

Thus we have

(10b) xT^(q)x4 = T*(q) = T(q)* for all primes q.

Theorem 2 (Hecke). If either % is primitive modulo pr or r = 0, then

Hk(ro(pr)>X;C) is semi-simple and ^4(ro(pr)5X) ^ spanned by common eigen-

forms f of all Hecke operators T(n) such that f | T(n) = a(n,f)f.


Although this fact is true for all k > 1, all Fo(N), and all % primitive moduloarbitrary N, we prove the result only when k > 2 and N = pr. (See [M,Th.4.7.2, Th.7.2.18] for the proof in the general case).

Proof. First we prove the theorem for SL2(Z). In this case, we see from (10) that(f|T(n),g) = (f,g|T(n)) and thus T(n) is a hermitian operator. Since the T(n)'sare mutually commutative, we can diagonalize T(n) simultaneously on 5k(F).This shows that hk(SL2(Z);C) can be embedded into a product of copies of Cand hence hk(SL2(Z);C) is itself a product of copies of C. Note thathk(SL2(Z);C) = Cr for r = dimc5k(SL2(Z)) because Homc(hk(F;C),C) =5k(F). Then each projection X[ of hk(SL2(Z);C) into C is a C-algebrahomomorphism, and {X\, ..., XT} form a basis of Homc(hk(F;C),C). Let Xbe one of the A,i's and f be the corresponding element in 5k(F). Thena(n,f) = <T(n),f> = A,(T(n)) and hence f = lT_X(T(n))qn. Moreover

a(m,f | T(n)) = (T(m),f | T(n)> = a(l,f | T(n)T(m))

= <T(m)T(n),f> = ^(T(m)T(n)) = X(T(n))a(m,f).

Thus f I T(n) = ^(T(n))f and f is a common eigenform of all Hecke operators

T(n) belonging to the algebra homomorphism X. We write £ = X°° î(T(n))qn.n=l

Then {fj}i gives a basis of 5k(F) over C. Now we show that Ek is a commoneigenform of all Hecke operators. We show more generally that

Ek(X) I T(q) = ak.if5c(q)Ek(x) for every prime q.

We thus compute a(n,Ek(%) I T(q)) = a(nq,Ek(x))+x(q)qk-1a(n/q,Ek(%)). If n isprime to q, then

°k-i,x(nq) = Xd lnqZW^ ' 1 = Xd|nZb|qX(bd)(bd)k-1 = ak.ii%(n)ok.i,%(q).

Therefore a(n,Ek(%) | T(q)) = ak.i>x(q)a(n,Ek(x)). If n is divisible by q, then

a(n,Ek(%) | T(q)) = a(nq,Ek(%))+%(q)qk-1a(n/q,Ek(%))

= X d i nq%(d)dk"1+%(q)cik"1X

On the other hand, we have


In fact, obviously, {d | d |nq} = {d | d|n}U{qb b | n } . The intersection

d | n}f|{qb | b | n} is given by

{d I d I n, d = qb, b | n} = {qb | b | (n/q)}.

Similarly we can show Gk(%) I T(n) = a'k-i,x(n)Gk(%). Anyway this shows thatfA4-(SL2(Z)) has a basis {fi, •••, fr-i, Ek} which consists of common eigen-forms of all Hecke operators T(n), and hence the desired assertion forf*4(SL2(Z)) follows from this.

Now we consider 5k(To(pr)>%)- By (10), -\/%(n) T(n) is self-adjoint if n isprime to p. Thus, we can find a basis of 5k(Fo(pr)>%) consisting of commoneigenforms of T(n) for all n prime to p. We shall show that if f ^ 0 is acommon eigenform of all operators T(n) for n prime to p and if % is primitivemodulo pr, then f is an eigenform even for T(p). We consider the following setof positive integers:

X = {n | pjn, a(n , f )*0}.

We first show that X is not empty. If this set is empty, then a(n,f) * 0 only if

n is divisible by p. Let a = L. and put\9 VJ

I fa b i fa M 1Then g satisfies g I A = %(d)g for G O = a T o ( p ) a and is also

(1 1 ifa b^ f a pbînvariant under U = I Q J . Note that a , a = , , and

(a b^ IO = { , M b € p Z , c e p r Z , a,dG Z , ad-bc = 1 } .

V J

To show that g = 0, we take integers m and n so that(mp^+ lXnp^+ l ) = 1 mod pr but %(rvpx~l+l) * 1.

Note thatr-l

and thus gl Y= JcCnp^+lJg but at the same time g |um = g, g r u ^ ' ^ g andg I un = g. Thus %(npr"1+l)g = g and hence g = 0. This shows that iff ** 0, X is not empty. Take any n in X. Then writing f | T(n) = a(n)f, wehave

0 * a(n,f) = (T(n),f) = a(l,f I T(n)) = a(n)a(l,f).Thus a(l,f) ^ 0 for any non-zero common eigenform f of Hecke operatorsT(n) for n prime to p. Since g = f I T(p) is also a common eigenform of allT(n) for n prime to p, a(l ,g)*0. Put h = a(l,g)f - a(l,f)g. By definition,


a(l,h) = a(l,g)a(l,f)-a(l,f)a(l,g) = 0.

Since h is again a common eigenform of all T(n) for n prime to p, and hencea ( l ,h )^0 if h * 0 . Thus h must be 0 and f|T(p) is a constant multiple off; namely, f is an eigenform of T(p). Thus 5k(Fo(pr)>%) has a basis consistingof common eigenforms of all Hecke operators.

Next, we shall show that ^k(Fo(pr)>%) is spanned by 5k(Fo(pr)>%) and Ek(z,%)and Gk(z,%). Since f^(SL2(Z)) = {0} (Theorem 2.1), we may assume eitherk > 2 or % 1 and k > 2. Consider the sequence

(*) o -> 5k(r0(Pr),%) -» ^ k O W X x ) - 2 - > c 2 -> o.

The last map <p is given by (p(f) = (a(0,f), a(0,f|k5)) for 8 = I x Q I. The

surjectivity of (p follows from the fact that (p(Ek(%)) = (2"1L(l-k,%),0) and

(p(Gk(%)) = (O.w'LCl-kjX"1)) with w' ^ 0, which is a consequence of

Ek(z;%) I k8 = wGk^x" 1 ) with w * 0 (see Proposition 1.1).P

Since by the functional equation (Theorem 2.3.2), L(l-k,%) is a non-zero con-stant multiple of L(k,%"1), which is non-zero because the Euler product convergesif k > 1. Thus (p is surjective. Now let us show the exactness of the middleterm. Pick a modular form f in fA4(Fo(pr),%) and suppose that cp(f) = 0. Bythe strong approximation theorem (see Lemma 6.1.1), for each element a of

(a b\SL2(Z), we can find y=\ e Fo(p ) so that either

(i) a = y8 for some integer j in the interval [l,pr],

or

( l °"lpr"s

( l °l(ii) y s. A for some integer i in the interval [l,pr"s] and s in [l,r].

Then, in case (i), a(0,f | a ) = %(d)a(O,f | 8) = 0. Now we deal with case (ii).

Write simply F = Ti(pr) and Y = F0(pr)- Then we see that T /F = (Z/prZ)x

via . H d mod pr, and Y/Y acts on equivalence classes of cusps of F :

C = F\SL2(Z)/Foo, where F o o = | ± H ™1 e T ' l Letting SL2(Z) act on the

r 0 (Acolumn vectors (Z/pZ) , we see that F is the stabilizer of the vector . Thus

we can identify C with F\(Z/p Z) . Then the cusp corresponding to s.

for i prime to p is given by the vector xi = s. . Then the orbit O(x0 of


under the action of (Z/prZ)x = F/F' is isomorphic to (Z/pr"sZ)x via (Z/pr"sZ)x B

i h-> Xj. Considering the function c[> : i I—> a(0,f I S. ) as a function on

(Z/pr-sZ)x, we see that <|>(di) = %(d)<t)(i) for d e (Z/prZ)x. Since s > 0 and% is primitive modulo pr, we can find d = l+jpr"s such that %(d) * 1. Since theaction of (Z/prZ)x on O(XJ) factors through (Z/pr"sZ)x, x(d)<>(i) = <|>(di) = <|>(i).This implies $ vanishes identically. That is, for the cusp in case (ii), the constantterm of all modular forms in ^k(ro(pr)>%) vanishes. This shows thatKer(cp) = 5k(r0(pr),%). Thus afk(r0(pr),X) is spanned by 5k(Fo(pr),X) andEk(z,%) and Gk(z,%). We have already verified that

Ek(X) I T ( n ) = ^k-i,x(n)Ek(Z) and Gk(%) I T(n) = a'k-i,x(n)Gk(X) for all n.This finishes the proof of Theorem 2. We record the following fact shown in theproof of Theorem 2.

Corollary 1. Suppose that % is primitive modulo pr and let f and g be ele-ments in 5k(ro(pr)>%)« Then if f is a common eigenfunction of T(n) for all nprime to p, then f is also an eigenfunction of T(p). If f and g have the sameeigenvalues for T(n) for all n prime to p and g * 0, then f is a constant mul-tiple of g.

Corollary 2. Let f be a common eigenform of all Hecke operators witha(l,f) = 1 in 5k(SL2(Z)). Then the field Q(f) generated by all the Fourier coef-ficients of f is a finite extension of Q. Moreover a(n,f) are all algebraic integersand for all ce Gal(Q/Q), f° is again a common eigenform in 5k(SL2(Z)).Proof. Let F = SL2(Z). We know that f = a = S°° k(T(n))qn for an alge-

n=l

bra homomorphism X : hk(F,%;C) = hk(r,%;Q)®QC —» C. Thus X inducesa Q-algebra homomorphism of hk(F;Q) into C. Note that A,(hk(F;Q)) is an al-gebra of finite dimension over Q which is generated by X(T(n)). Thus Q(f) =^(hk(r;Q)) and Q(f) is a finite extension over Q. The image X(h^(T;Z)) iscontained in the integer ring of A,(hk(F;Q)). In fact, by representing T(n) as amatrix using the basis hi in Theorem 2.1, we see that (ho I T ( n ) , - - ,hr_i|T(n)) = (ho,---,hr_i)A(n) with A(n) e Mr(Z). Thus A,(T(n)) is a charac-teristic root of integral matrix A(n) which is an algebraic integer. We know that

HomZ-aig(hk(F,Z),Q) s {f | f I T(n) = ?i(T(n))f, a(l,f) = 1}: X H> fx

Naturally Gal(Q/Q) acts on the left-hand side. The action is interpreted into

f h-> f° on the right-hand side, which shows the last assertion.


We will see the assertion of Corollary 2 also holds for 5k(Fo(pr),%) for primitive% in the following section.

§5.4. The Petersson inner product and the Rankin productIn this section, we study how we can explicitly construct a basis for the spaces ofmodular forms (as we have just done for F = SL2(Z)).

Theorem 1. Let a > 2 be an integer and let \|/ be a character modulo pr with

A|/(-l) = (-l)a- Suppose k>2a+2. Then for any primitive character % modulo

pr ( r > l ) with %(-l) = (-l)k, there exist finitely many positive integers n\, n2,•••, nr such that G^Y^k-^X) I T(ni) for i = 1, ---,r together with Gk(%)and Ek(%) span ^4(ro(pr),%) over C. The above assertion is true with Eaty"1)in place of Ga(\ |/4). Similarly f?Vfk(SL2(Z)) is spanned by Ek and EaEk_a|T(ni)

for some n{ and 4 < a e 2Z such that k > 2a+2.

A more general result is obtained in [Wil] (for example, the assertion of the theo-rem is true even if a = 1). The proof we will give later is basically the same as in[Wil] although it is a little simpler because of our assumption that a > 2. Foreach Dirichlet character %, we write Z[%] for the subring of C generated by allthe values of %. Before proving the theorem, we list several corollaries:

Corollary 1 (duality). If k > 6 and F = Fo(prp) and % is a primitive

character modulo p r p /<9r r>0 , then #4(F,%;A) = f&4(r,%;Z|xl)®z[x]A for

any Z[x\-subalgebra A of C, where p = 4 or p according as p = 2 or not.Moreover under the pairing in Theorem 3.1, we have the following isomorphisms:

HomA(Hk(F,%;A),A) = /nk(F,x;A), HomA(mk(r,z;A),A) = Hk(F,%;A),HomA(hk(F,x;A),A) = 5k(F,%;A), HomA(5k(F,x;A),A) = hk(F,%;A).

Proof. By the theorem, we can find a basis {fih=i r of ^k(r ,%) inîe(r,%;Z[%]). In fact, we take any character \\f modulo ppr with \|/(-l) = - l .Then taking a in the theorem to be 2, we find {fi}i=i,...,r amongGi(\\f-l)Ek.1(\\fx)\T(ni), Gk(%) and Ek(%) which form a basis of afk(I\x;K)over K, where K is the field Q(%,\|/) generated by the values of % and \|/.Then choosing a suitable element a* e K for each i, we can easily show that thefi = Tr(aif i) = Saaiafi'a give a basis desired with coefficients in Z[%], where aruns over Gal(K/Q(%)). Thus the natural linear map from ^4(r,%;Z[%])®z[%]C!into #4CT,%) is surjective. By our construction of fi we can find ni, ..., nr sothat det(a(ni,fj))ij=i r ^ 0. Then for any (() G fA4(F,%;Q[%]), we can solvesimultaneously the linear equations £jXja(ni,fj) = a(ni,<|)) (i = l,...,r) within

5.4. The Petersson inner product and the Rankin product 151

Q(%). Then we see that <|> is a linear combination of £ with coefficients in Q(%).Thus dimQ(x)fA4(r,%;Q[x]) < dimc(^k(r,%)). This shows fM"k(r,%;Z[%])<8)C= f^k(r,%) and mk(r,%;Z[%])®C = ^fk(r,%). In the same manner as in theproof of Theorem 3.1, we know that

s Hk(T,x;Z[x]),HomZW(Hk(r,x;Z[x]),Z[x]) = «k(T,x;Z[x]).

Obviously by definition, Hk(T,x;A) is a surjective image of Hk(F,%;Z[%])(8)A. Ifthe image of an element T in Hk(r,%;Z[%])®A vanishes in Hk(r,%;A), extend-ing scalar to C, it vanishes in Hk(I\x;C). Since a4(T,x) = *4(T,x;Z[xl)®C,we know that T = 0. Thus Hk(r,%;Z[%])®A = Hk(T,x;A). Then we have

wk(r,x;Z[x])®z[x]A s HomZ[X](Hk(r,5c;Z[x]),Z[%])(8)A= HomA(Hk(r,%;A),A) = /%(r,%;A).

This shows the assertion for A.

Now we note a byproduct of the above argument. We have an exact sequence

0 -> ftfk(I\x;A) -> mk(r,%;A) -> N(A) -> 0

for a torsion A-module N. Since A is flat over Z[%], by tensoring with A, wehave another exact sequence

0 -> *4O\x;Z[3c])<g>A -> mk(r,%;Z[%])(8)A -+ N(Z[xl)®A -> 0.

Since the middle terms of the above two sequences coincide and the last maps ofthese sequences also coincide under this identification, we have

;A) = *4(r,x;Z[x])®A.

By Corollary 1 and Theorem 1, we know that

hk(r0(ppr),%;C) = hk(r0(ppr),z;Q(x))®Q(%)c if k > 4 .

Then, similarly to the proof of Corollary 3.1, we know the following result when

k > 6 . The result is actually true for all k > 0 [Sh3, p.789]. We will later

show this result for k > 2 as Theorem 6.3.2.

Corol lary 2. Let f be a common eigenform of all Hecke operators in^k(ro(ppr),%) such that a(l,f) = 1. Then the field Q(f) generated by all theFourier coefficients of f is a finite extension of Q. Moreover a(n,f) are all alge-braic integers and for all c e Gal(Q/Q), f° is again a common eigenform in

To prove the theorem, we shall prepare with some lemmas.


Lemma 1. Let g be an element of 5k(F,%) and {f} be the basis consisting of

common eigenforms. We assume that F = Fo(ppr) if X is primitive modulo

pp r and F = SL2(Z) if % is trivial. If we write g = Zfc(f,g)f with

c(f,g) e C, then c(f,g) = (g,f)/(f,f), a«^ */ c(f,g) * 0 /or a// f, then g | T(ni)for some n* spara 5k(F,%).

Proof. Since the proof is exactly the same for SL2(Z) and Fo(pr), we only treatthe case of F = Fo(ppr). As already seen, hk(F,%;C) is semi-simple and henceis isomorphic to Cd for some d as an algebra. Let X[ be the i-th projection ofhk(F,%;C) onto C, which is an algebra homomorphism. Then X{ spansHomc(hk(F,%;C),C) = 5k(F,%). Let fi be the cusp form corresponding to X{.Then {fi} forms a basis of 5k(F,%). We see that a(n,fi) = (T(n),fi) = ^(T(n))(in particular, a(l,fi) = 1) and

a(m,fi I T(n)) = <T(m),fi I T(n)> = a(l,fi I T(m)T(n))= Xi(T(m)T(n)) = Ad(T(n))a(m,fi).

Therefore we have fi | T(n) = A,i(T(n))fi. If i * j , then we can find T(n) suchthat A,i(T(n)) ^ \j(T(n)) since the T(n)'s generate the Hecke algebrahk(F,%;C). Thus the chosen basis coincides with {£}. In the proof of the semi-simplicity of the Hecke algebras, we have shown that if f is a non-trivial commoneigenform of Hecke operators T(n) for all n prime to p, then f is also an eigen-form for T(p) and a(l,f) ^ 0. Moreover we have shown thatT(n)* = %(nYlT(n) for n prime to p, where T(n)* is the adjoint of T(n) un-der the Petersson inner product. Note that T(n)* commutes with T(p)*.Therefore T(n) commutes with T(p)* if n is prime to p. If f is a commoneigenform for all T(n), f | T(p)* is a common eigenform for T(n) for all nprime to p having the same eigenvalues as f. Therefore if f | T(p)* * 0, thena( l , f |T(p)*)*0 and h = a(l,f I T(p)*)f - a(l,f)f I T(p)* is a common eigen-form of T(n) for all n prime to p with a(l,h) = 0; hence h = 0. Hence f isalso an eigenform of T(p)*. Thus what we have shown is that if f is a commoneigenform for T(n) with n prime to p, then f is also a common eigenform ofT(n) and T(n)* for all n including p. If there are two non-trivial eigenforms fand g belonging to the same eigenvalues of T(n) for all n prime to p, thenagain by putting h = a(l,f)g - a(l,g)f, we know that h is a common eigenformof T(n) for all n prime to p with a(l,h) = 0. Thus h = 0 and g is a con-stant multiple of f. This shows that î is determined by the value at T(n) for nprime to p. Thus, if i ^ j , then we can find n prime to p such thatXi(T(n)) * Xj(T(n)). Since T(n)* =x(n)-1T(n)> T = (^cOO^Kn) is self-adjoint and

(V3C(n))"1î(T(n))(fi,fj) = (fi I T,fj) = (fi5fj | T) = (^X(n) )"Vj(T(n))(fi,fj).


This shows that (fi,fj) = 0 if i ^ j . Therefore, writing g = Ei c(fi,g)fi, then

(g,fi) = c(fi,g)(fi,fi) and the formula c(fi,g) = (g,fi)/(fi,fi) follows.

Let us now prove the last assertion of the lemma. Let M be the vector subspaceof hk(F,%;C) generated by T(n) for all n. Then the pairing (h,f) = a(l,f |h)is still non-degenerate on M (in fact, if (h , f )=O for all h e M,a(n,f) = (T(n),f) = 0 and f = 0). Thus

dimc(hk(r ,x;C)) > dimcM > dimc(5k(F,x)) = dimc(hk(F,x;C)).

Therefore hk(F,%;C) = M. (This fact is even true for any subring A of Ccontaining Z[%]: hk(F,%;A) = Z n AT(n).) Let li be the idempotent corre-sponding to the i-th factor C of hk(F,%;C); thus A,j(li) = 8ij. Then one can ex-press li = ZjC(nij)T(riij) with c(nij) e C and njj > 0. Then for g as in thelemma, we have

j

On the other hand, g I li = Ejc(nij)g | T(nij). Thus if c(fi,g) * 0, then £ can beexpressed as a linear combination of g | T(njj). By picking a basis out of{g | T(nij)}ij, we get a desired basis g I T(ni).

To prove Theorem 1, we compute for each common eigenform f,c(g,f) = (g,f)/(f,f) for g = Gaty'Êk-atyJC). Note that if a> 2, G&(\\fA) hasa non-trivial constant term only at the cusp 0 and Ek.a(\|/%) has a non-trivial con-stant term only at infinity. Thus g is a cusp form (if a > 2). We write £ for\|/%, / for k-a and h for Gaty"1). Then by definition

where (cpr,d) runs over all pairs of relatively prime integers with d > 0. Since(cpr,d) is relatively prime, we can find integers a and b such that ad-bcpr = 1

and y = r e ro(pr)« If we pick another 5 = r in Fo(pr) withVCP ^y vcP "• J

the same lower row, then y8 = L e Foo, where* *

r . = {ye ro(Pr) I y(-) = « j = {±^ ™j | m

Therefore, we know that

Thus at least formally


c(g,f)(f,f) =

where <& is the fundamental domain of Fo(pr). Note that

Y*(y"2dxAdy) = y"2dxAdy for all y e SL2(R), y(y(z))k = y(z)k| j(y,z) I "2k,

and fC/(i))h(Y(z)) = W)Kz%~\i) |j(Y,z) | 2kj(Y,z)-'.Therefore, we have

f(i)h(Z)^1(Y)j(Y,z)"/ y(z)k = f(Y(zl)h(Y(z))y(Y(z))k

and for a non-zero constant Co (see §1)

c(g,f)(f,f) =

Since L L g r \r (pVY* *s a fundamental domain of r « and the domain

{z = x+V-Iy I y > 0 and 0< x < 1} also gives the fundamental domainof F^, we have, for e(z) = exp(2rc V-Tz),

c(g,f)(f,f) = coLC/,^1) J ^ fh(z)yk-2dxdy

= coLC/,^1) J~ ^mna(m,f)ca(n,h)e((m+n)Vzry)£ e((n-m)x)dxyk-2dy,

where c denotes complex conjugation. As is well known,ri (1 if n = m,

e((n-m)x)dx = \~ .Jo v ' ' [0 otherwise.

Therefore,oo

c(g,f)(f,f) = coLC/,^1) J ~ £ a(n,f)ca(n,h)exp(-43my)yk-2dyn=l

= co(47i)-T(k-l)L(/,^1) £ a(n,f)ca(n,h)n-s | s=k.!.n=l

By the formula before Proposition 1.1, we know that

en - Nk-' ( k ' M ) !

We need to justify that we may interchange the summation and the integral in the

above computation. By (3.8b), we have I a(n,f) I < Cn1^2 and by Proposition

1.2 and (1.4a,b), la(n,h)| < C'na"1+e with any e > 0 for constants C and C

independent of n; therefore, the computation will be justified if k-1 >

2 + a + e, that is, k > 2a+2+2e. We now define the Rankin product zeta

function of f and h by

(2) D(s,f,h) = ]T a(n,f)ca(n,h)n"s.n=l


Here h is a general modular form, not necessarily GaCij/"1). If f and h are cusp

forms, then | a(n,f)ca(n,h) | < Cn(k+a)/2 and thus D(s,f,h) converges absolutely

if Re(s)> 1 + ^ r because D(s,f,h) is dominated by £(s-(k+a)/2). If h is not

cuspidal, then | a(n,f)ca(n,h) | < Cn(k/2)+a"1 and D(s,f,h) converges absolutely

if Re(s) > a+2 by the same reasoning. Thus Theorem 1 follows from

Lemma 2. For two common eigenforms h e ^&(To(ppT),\\f'1) and

f e 5k(r0(ppr),X), D(k-l,f,h) * 0 if k > 2a+2.

Proof. We compute the Euler product of D(s,f,h). We note the recurrence rela-tion for each prime q:

T(qr)T(q) = T(qr+1) + X(q)qk'1T(qr-1).Therefore for any common eigenforms f and h of all Hecke operators such that

a(l,f) = a(l,h) = 1,we already know that f | T(n) = a(n,f)f and h | T(n) = a(n,h)h. For simplicity,we write a(n) = a(n,f)c and b(n) = a(n,h). From the relation, we know that

a(qr)a(q) = a(qr+1) + xCqjV'Vq'"1) andb(qr)b(q) = b(qr+1) +X|/(q)"1qk-1b(qr-1)

if r > l , where \|/ (resp. %) is the character of h (resp. f). We take two roots

a, (3 of X2-a(q)X+%"1(q)qk'1 = 0 and a',p' of X2-b(q)X+\(/(q)-1qk-1 = 0.

We write formally P(X) = S°° a(qn)Xn. Then we haven=0

oo oo

P(X)(cc+p) = ^ a(q n + 1 )X n +ap^ a(qn-1)Xn = (P(X)-l)X-1+apXP(X).n=0 n=l

This shows that

P(X) = (l-(a+p)X+apX2)"1 = (l-aX)-1(l-pX)"1 = ^ —

anda(qn) = (a-fiy1^1-^*1), b(qn) = (a'-pTôc'^-p'11*1),

Now computingV 1 /_. n+l on+1 \/_,i n+1 Otn+lN-vrn

>.(a -p )(a -p )XQ(X) = V a(qn)b(qn)Xn = J>=2-

(a-p)(a'-p')1 l

+ l

(a-p)(a'-P1)Xll-aa'X 1-ap'X 1-Pa'X 1-PP'X.


l-gg'pp'X2

i-(X¥)'1(q)qk+a'2x2

(l-aa'XXl-ap'XXl-poc'XXl-pp'X)'

we find that

This infinite product converges absolutely at s = k-l if k> 2a+2, because,

writing A = a or p and B = a' or p \ I ABq"s | < q"1 if Re(s) > l+^+a-l

for q sufficiently large, and the Euler product of £(s) converges if Re(s) > 1.

Since f°(z) = f(-z) , then f° e 5k(F,%c) and (f,f°) = (f,f). Therefore we

have, for g = hcEk.a(\|/'1Z)»

c(g,f)(f,f) = co(47c)1-kr(k-l)L(2-k-a+2s,x-V)^(s,f,h) I s=k-i= co(47i)1-]T(k-l)L(k-a,%-V)^(k-l,f,h).

The convergence of the Euler product gives us the non-vanishing of c(g,f) and wenow have Theorem 1.

Corollary 3. Suppose that f and h are normalized common eigenforms of allHecke operators in 5k(Fo(N),%) and fAfa(To(N),\|/), respectively. Put

Suppose that %~l\f is primitive modulo N = pr . If k > 2a+2, then

T(k-l,f,h) e Q(f,h), where Q(f,h) is the finite extension of Q generated by the

Fourier coefficients of f and h. Moreover for all a e Aut(C),

T(k-l,f,h)° = T(k-l,f°,ho).

This is a special case of a general algebraicity theorem of Shimura (see Theorem10.2.1 and also Theorem 7.4.1). As already seen

NL(k-a,%-V)D(k-l,f,h)= T(k-l,f,h).

Then the assertion follows easily from this formula and the fact that

We now want to show that f°c = f°a for our later use. The n-th Fourier coeffi-cient a(n,f) of f is the eigenvalue of T(n). On the other hand,

5.5. Standard L-f unctions of holomorphic modular forms 157

^ T(n) is self-adjoint if n is prime to p. Thus (<^%(n)) a(n,f) is real

anda(n,f)c = x(n)"1a(n,f).

In particular, a(n,f)ca = x°(n)"1a(n,f0) = a(n,f°c). Since the p-th Fourier coef-ficient of f is determined by those for n prime to p as shown in the proof ofTheorem 3.2, we also have a(p,f)ca = a(p,f)ac. In fact fG-f°c is a commoneigenform of all T(n) for n prime to p and a(l,f°a-fac) =0, and hence

The above proof in particular shows that

(3) a(n,f)c = x'^nÔijf) for n prime to p if f is a normalized eigenform

in 5k(ro(pr),X)-

§5.5. Standard L-functions of holomorphic modular formsLet X : Hk(Fo(pr),x;C) —> C be a C-algebra homomorphism. We assume thatX is a primitive character modulo N = pr. We define the standard L-function ofX by

By Theorem 3.1, we have a common eigenform of all Hecke operatorsf e ^k(ro(pr),X) such that a(n,f) = X(T(n)) for all n. We then know from(3.8b) and (1.4a,b) that I A,(T(n))n-s I < Cn"Re(s)+(k/2) if f is a cusp form, andU(T(n))n"s| < Cn-Re(s)+k"1+e (for all £ > 0) if f is just a modular form.

When f is a cusp form, i.e. X factors through hk(ro(pr)>%)> then we see that

(2) L(s,X) converges absolutely if Re(s) > 1+-|.

If f is associated with X, i.e., f = Z°° X(T(n))qn, then by (3.10a,b)n=l

(3) (f \x) I T(n) = ^(T(n))c(f | x) for all n,

where c denotes complex conjugation. Supposing that x is primitive modulopr, we know from Corollary 3.2 that f | x is a constant multiple of f°:

Proposition 1. Let f = £°° X,(T(n))qn for an algebra homomorphismn=l

X : hk(r0(pr),X) -^ C. Define Xc : hk(r0(pr),%c) -* C by Xc = c°XoC. We

suppose that % is primitive modulo pr. Then we have


(4) f | x = pr(k-2)/2W(?i) £ MT(n))cqn,n=l

for a constant W(X) with W(X)W(XC) = %(-l) and I W(A.) I = 1. Here weagree to put Fo(p°) = SL2(Z) and % = id in this case.

Proof. We only need to prove the last assertion. By Corollary 3.2, W(X) * 0.

Since T 2 = - p r L °X we see that (f | x)(z) = p(k"1)rf(-l/prz))(prz)-k and

( f | x ) h = %(-l)p(k-2)rf. Thus W(X)W(XC) =x(-l) . Applying fM> f(=I) to

the formula (4), we see that W(XC) = %(- l )W(^)c . This shows that

I | = 1.

We now show the analytic continuation of L(s,X) and its functional equation.First assume that f e 5k(Fo(pr)>%) for a Dirichlet character % modulo pr (here

% may be imprimitive). In this case, by (3.8a), the integral J f(i'y)ys"1dy is ab-

solutely convergent for all s and gives an entire function of s. By the same type

of computation as in §2.2, we have, if Re(s) > l+w,

Thus L(s,X) has an analytic continuation to an entire function on the wholes-plane. On the other hand, if either % is primitive modulo pr or r = 0, we seethat

f | x ( / y ) y s - l d y

= i-kP"rJ0 f ^

This shows

Theorem 1. Suppose that % is primitive modulo pr. Then, for each algebra

homomorphism X : hk(F()(pr),%) "^ C, the L-function L(s,X) is continued to

an entire function on the whole s-plane. If either r = 0 or % is primitive modulo

pr, the L-f unction L(s,X) satisfies the following functional equation

rc(s)L(s,A,) = pr((k/2)-s)/kW(^)rc(k-s)L(k-s,A,c) for Fc(s) = (2TC)-T(S).

For any primitive character \\f modulo N, we can define

Then we see from a similar computation as in (4.1.6c) that

5.5. Standard L-functions of holomorphic modular forms 159

(5) fv = £ \|/(n)a(n,f)qn.n=l

For any algebra homomorphism X : hk(Fo(pr)>%) —> C and a primitive charac-

ter \|/ : (Z/NZ)X -> Cx, we define

£ \|/(n)MT(n))n"s.n=l

Thus we have, as a corollary to the proof of Theorem 1,

Corollary 1. For each algebra homomorphism X : hk(ro(pr),%) —> C and

each primitive character \\f: (Z/NZ)X —> Cx, the L-function L(s,?t<8>\|/) is con-

tinued to an entire function on the whole s-plane.

Proof. Although we proved Theorem 1 assuming that % is primitive modulo pr,

it is clear from its proof that J f(iy)ys"1dy is an entire function in s if f is a

cusp form. Thus what we need to prove is that fv is a cusp form for somecharacter £ of Fo(M). It is easy to check that

(6) if y = I , I e Fo(M) for the least common multiple M = [N,pr] of

fa1 bf>ifor Y1 = I c. d. I e Fo(pr) with d1 = d mod pr.

This shows that

f,lr-G<v-VsV<-<r=l ^r=l ^ ^ r=l

Thus we have f¥ e 5k(Fo(M),%\|/2).

Define an algebra homomorphism A,®\|/ : hk(Fo([N,pr]),X) -> C byf¥ | T(n) = (X®\|f)(T(n))fv (i.e. (X®\|/)(T(n)) = \|/(n)X(T(n))). Then if N isprime to p, we can compute W(X®%) explicitly by using W(X) (see for details[Sh, Prop.3.36], [M, §4.3] and [H5, (5.4-5)]).

Chapter 6. Modular forms and cohomology groups

In this chapter, we prove the Eichler-Shimura isomorphism between thespace of modular forms and the cohomology group on each modular curve. Thisfact was first proven by Shimura in 1959 in [Shi] (see also [Sh, VIII]). We shallgive two proofs in §6.2 of this isomorphism. The first one is the original proof ofShimura based on the two dimension formulas. One is the formula for the space ofcusp forms and the other is for the cohomology group. The other proof makes useof harmonic analysis on the modular curve. After studying the Hecke modulestructure of modular cohomology groups, in §6.5, we construct the p-adic stan-dard L-function of GL(2)/Q following the method (the so-called "p-adic Mellintransform") of Mazur and Manin in [Mzl], [MTT] and [Mnl,2]. Throughout thischapter, we use without warning the cohomological notation and definition de-scribed in Appendix at the end of the book. If the reader is not familiar with co-homology theory, it is recommended to have a look at the appendix first.

§6.1. Cohomology of modular groupsIn this section, we shall prove the dimension formula of the cohomology group ofcongruence subgroups of SL2(Z) following [Sh, VIII]. Let F be a congruencesubgroup of Slyz(Z) and suppose for simplicity that F is torsion-free. (Thegeneral case without assuming the torsion-freeness of F is treated in [Sh, VIII].)If N > 4, Ti(N) is torsion-free. In fact, if ±1 * £ e F satisfies £m = 1 form > 2, then the absolute value of the sum of two eigenvalues of £ is less than 2.Thus | Tr(C) I = 0 or 1. On the other hand, if £ e Fi(N), Tr(Q = 2 mod N.It is easy to check that it is impossible to have | Tr(^) | < 2 and

if N > 4 .

Exercise 1. If N > 4, why is Tr(Q = 2 mod N impossible?

Let R be a commutative ring, and for any left R[F]-module M, we consider thegroup cohomology group HÎ^M) (see Appendix for definition). Here we startfrom an open Riemann surface Y = T\tf. For each s e P!(Q) = QU{°°}, we

qconsider the stabilizer Fs of s in F. If s is finite, writing s = 7 as a reduced

fraction, we can find integers x and y such that qy-px = 1. Puttingfq x\ -

o s = a = e SL2(Z), we have G(©°) = s. This shows that a Fsa is|Y 1 m\ I 1

contained in ( ± 1 } X U ( Z ) for U(A) =j i I I m e Af- Thus defining the

distance from s by d(z) = Im(a'1(z))"1, we see that d(z) is well defined onFSW, and we put US)£ = ( z e FSV#| d(z) < e}. For sufficiently small e, Us<£

is naturally embedded into FSV# We identify two cusps if Us,enUt,e * 0 for

c d j " 10 i . m o d N K

6.1. Cohomology of modular groups 161

any 8 > 0. We see easily that two cusps are the same if and only if y(t) = s forsome y E F. Thus the set S of cusps is bijective to F\PSL2(Z)/U. Since T isa congruence subgroup, there is a positive integer N such that

r < N , . l l - j 6 S L 2 < Z , ' r - n r i 0 ^

Then it is obvious that #(S) < (SL2(Z):r(N)), which is finite. In the above proofof the finiteness of S, the following strong approximation theorem is provedimplicitly:

Lemma 1. Let {Ni)iGN be a sequence of integers such that Ni is a divisor ofNi+i for all i. Then SL2(Z) is dense in lim SL2(Z/NiZ). In particular, writing

i

Z = ] l p Z p = lim SL2(Z/N!Z), SL2(Z) is dense in SL2(Z).

Proof. We need to show that the natural map SL2(Z) —> SL2(Z/NZ) is surjec-tive for any positive integer N. Let

V = {v E (Z/NZ)21 the order of v is equal to N),where the order means the order of the subgroup generated by v in the additive

group (Z/NZ)2. Then the vector v is represented by for relatively prime p

and q. Thus we can find x and y in Z so that py-qx = 1 and for

a = E SL2(Z), avo = v mod N, where vo = L L Thus for any

oco E SL2(Z/NZ), we can find a E SL2(Z) SO that avo = aovo mod N.That is, ccU(Z/NZ) = oc0U(Z/NZ). The natural map U(Z) -> U(Z/NZ) isobviously surjective, and hence by modifying a from the left by an element ofU(Z), we can find a' E SL2(Z) such that a1 = ao mod N. This shows thesurjectivity.

Now we show that Y-USGSUS)£ = Yo is compact if {Us,e}SGS do not overlap.

We may assume that T = SL2(Z) because the above space is a finite covering of

the space Yo for SL2(Z). Since a = L E SL2(Z), we can take its fun-

damental domain in the strip B in O{ with | Re(z) | < TT. Since

y ( 5 ( z ) ) = y / ( x 2 + y 2 ) for z = x+zy and 8 = ^ ^ j e SL2(Z),

x2+y2 > 1 <=> x(8(z))2+y(8(z))2 < 1. Starting from a given z = zo e B withx +y < 1, we apply 8 to z and bring 8(z) back in B by tanslation by apower of a. We write this element in B as z\. We can now define a sequencezo, zi,..., zn,..., by repeating this process of first applying 8 and then translatingback into B by a power of a, that is, zn = ocm(8(zn_i)) for a suitable integer mso that zn E B. As soon as we get | zn |

2 = Re(zn)2+Im(zn)2 > 1, we stop the

162 6: Modular forms and cohomology groups

process. If the sequence is of infinite length, there is a limit point because the unitdisk centered at the origin is compact. Since Im(8(z)) = y/(x2+y2) > Im(z), andthe ratio Im(8(z))/Im(z) is given by l/(x2+y2). Thus the limit point has to be onthe intersection of B and the boundary of the unit disk. This implies that thereexist a positive number h such that a fundamental domain of SL2(Z) can betaken in {z e B | Im(z) > h}. In fact, a standard fundamental domain is givenas the intersection of B and the domain outside the open unit disk centered at theorigin; see [M, 4.1.2]. Thus y(z) = Im(z) is bounded below independent of zin this fundamental domain, and hence y(z) is bounded below and above in the in-verse image Oo of Yo in this fundamental domain. That is, Oo is relativelycompact. This implies that Yo is compact.

To compactify Y = FV^ we need only compactify Us,e. Since Us,e is isomor-

phic to Uoo,e, we compactify Uoo,e. Since Ho is a subgroup of U, there exists

N e Z such that Z H q = e(-sr) gives an isomorphism of Uoo,e into

Gm(C) = C x and y(z) -» °o <=> q -> 0 on Uoo,e. We then add the cusp °oto Uoc,e so that e : Uoo,£U{°°} —> C gives an isomorphism onto a neighbor-hood of 0 in C. We write this compactification as Uoo,e and the correspondingcompactification of USt£ as Us>e. Then X = YUS is a compact Riemann sur-face having Us,e as a neighborhood of s € S. Thus we get X and Y = X-Sas in Appendix. We make the triangulation of Yo = Y-Use sUs,e for small e andget a complex £ satisfying the conditions (Tl-3) in Appendix. We may assumethat the fundamental domain Oo of Yo in the inverse image of Yo in 9{ consistsof simplices of £ (we may need to take out some simplices from £ which overlapin Yo). Let Si be the set of all i-simplices of £ We use the notation defined inAppendix for X and S, especially, H$ denotes the various parabolic cohomol-ogy group with respect to S (see Proposition A.I).

Proposition 1. For any F-module M, let DM = Zy€r (y-l)M. Then we

have H2(r,M) = M/DM and H2(r,M) = 0.

Proof. We choose £ in a manner suitable to our proof. We triangulate X so thatby cutting along the curves representing a basis of the homology group of X, weget a polygon of 4g sides, where g is the genus of X. We may assume that allthe holes of Yo are inside this polygon. Fixing one vertex qo of this polygonand cutting the polygon from qo to each cusp of X, we have another simply con-nected polygon Oo with 4g+c sides, where c = #(S) is the number of thecusps of X. We may identify this Oo with a fundamental domain O of P.Thus F is generated by 2g+c elements (Xi, Pi, ..., ocg, pg and %\, ..., nc

with the sole relation


Here we have identified S naturally with {l,2,...,c}. We use the same symbol

OQ for the sum of elements in the set S2 of 2-simplices in Oo which represents

the domain Oo. Then, using the notation introduced in Appendix

X S ts + XJHere Sj (resp. s j , ts) denotes the face of Oo corresponding to Oj (resp. Pj,7is). We extend £ to a triangulation K of fH$ = Tr^Yo) for the projection% : J{ —> Y (as described in Appendix). Thus we have the chain complex:(Aj,9) associated to K. Let u be a cocycle having values in M, i.e.,u G A2(M) = HoiriR[r](A2,M) with fo3 =0. Since A2 is generated overR[F] by S2, u is determined by the values of the simplices of S2. Note thatocp-1 = (oc-l)(P-l)+(a-l)+(P-l). Since T is generated by CCJ, pj and TCC,

D M = Eses (^s-l)M + X-=1 {(arl)M+(pj-l)M}.

If u = 3w with 3w e BP(K,M) (see Proposition A.I for BP), then

u(O0) = XSGS w^s) + X-=i {(aj-Dw(Sj) + (pj-l)w(stj)} E DM,

because w(tc) e (TCC-1)M by the definition of Bp(K,M). Thus u H u(Oo)

gives a morphism of HP(K,M) into M/DM. This map is surjective because one

can assign an arbitrary value to u(<Do) because there is no restriction to be a

2-cocycle. Conversely, if u(Oo) e DM, we can write

Since Oo is a polygon, we may assume that the triangulation is given by the trian-gles which is spanned by two vectors emanating from the vertex qo on the poly-gon and ending at the two vertices of the edges tc, Sj, Oj(Sj), s'j and Pj(s'j):

We define a chain w e Ai(M) = HoiriR[r](Ai,M)so that w(ts) = (7Cs-l)ys, w(sO = xi5 w(s'i) = x'iand u = w°3. In fact, it is sufficient to define wfor the 1-simplices inside Oo. To do this, let us takea 1-simplex t connecting two vertices of OQ. Thust divides Oo into the union of two polygons a andb. Then we determine w(t) so that w(3a) = u(a)and w(3b) = u(b). This is possible, because all the

values of w at the edges of a or b except t are already given and the value of tis given by the formula w(3a) = u(a): w(t) = u(a) - XSeAw(s), where A isthe set of edges of a different from t. Similarly if we denote by B the set ofedges of b different from t, we have -w(t) = u(b) - Zs'€Bw(s'), because theorientation of t is reversed in b. Since

u(a)+u(b) = £s e Aw(s) + £s.eBw(s'),


w(t) is well defined. Thus we have a parabolic chain w on Oo such thatw°3 = u. Then we extend w to Ai by the R[F]-linearity to have w°3 = u on

Ai. Thus the map HP(K,M) -» M/DM given by the evaluation at Oo is injec-tive. This finishes the proof of the first assertion. As for the second assertion, weconsider the boundary exact sequence (Proposition A.2)

es€SH1(rWg,M) -» H?(r,M) -> H2(r,M) -» 0 (exact)\\i m

®sesM/(7Cs-l)M > M/DM.

The above diagram is commutative by definition, and the second horizontal arrowis the natural projection ©SeSms mod (TCS-1)M H> Zsms mod DM, which isobviously surjective. Then the exactness of the first row shows the vanishing ofH2(I\M).

By the above proposition, we know that HC(Y,A) = A via the evaluation of

2-cocycles at Oo- As is well known, the Poincare duality between H2(Y,A) and

H2(Y,9Y; A) for A = R or C is given by the integration over 2-cycles of closed

differential 2-forms, if one computes the cohomology groups using de Rham reso-

lution. Thus we have

Corollary 1. We have H2(Y,A) = A for all A, and if A = R or C,this

isomorphism is given by: HC(Y,A) 3 [co] f-> J Yco E A, where co is a closed

2-form representing the de Rham cohomology class [co].

Proposition 2 (dimension formula). Suppose that R is a field and M is offinite dimension over R. Let g be the genus of X. Then we havedim(H1

P(r,M))

= (2g-2)dim(M)+dim(H°(r,M))+dim(H^(r,M))+ ^s6Sdim((7Cs-l)M)).

Proof. Let ®o be the fundamental domain of Yo as in the proof of Proposition1. Then S2, Si- {ts I SG S) and So give a triangulation of Yo. Thus byEuler's formula, we have #(S2)-#(Sr{ts I s e S})+#(S0) = 2-2g, where #(X)denotes the number of elements in a finite set X. On the other hand,#(S0 = rankR[r](A0, and thus

dimR(Ai(M)) = dimR(HomR[r](Ai,M)) = #(Si)dimR(M).

Note that if we put Ap(M) = (UG Ai(M) I u(ts) e (JCS-1)M}, then

dimR(AXP(M)) = #(Si)dimR(M) - Xs

because of the exact sequence


0 -» Aj,(M) -> Ai(M) -> 0seSM/(7Cs-l)M -> 0.By definition, we have

Hj,(K,M) = Zp(K,M)/B1(K,M), H|(K,M) = A 2 ( M ) / B £ ( K , M ) .

Thus, writing d(i) (resp. d, d1) for the dimension of the i-th parabolic cohomol-ogy group (resp. dim(M), ZSES dimR((7Cc-l)M)), we see that

d(0) = dim(Ker(3 : A0(M) -> Aj»(M))) = #(S0)d-dim(B1(K,M)),

d(l) = dim(Z1P(K,M))-dim(B1(K,M)),

d(2) = dim(A2(M))-dim(B|(K,M)) = #(S2)d-(#(Si)d+dt-#(S)d-dim(Z1P(K,M))).

Thus we haved(0)-d(l)+d(2) = #(S0)d-(#(Si)d+d!-#(S)d)+#(S2)d = (2-2g)d+d\

That is, d(l) = (2g-2)d+d(0)+d(2)-df, which is the desired formula.

We compute more explicitly the dimension of cohomology groups for some specialF-modules. For a commutative ring R with identity, we consider the spaceL(n;R) of homogeneous polynomials of degree n with two indeterminates X andY. By definition, L(n;R) = L(n;Z)®zR. We define the action of the semi-groupM2(R) on L(n;R) by

where yl = Tr(y)-Y= dst(y)Yl. Note that i : M2(R) -^ M2(R) is an involu-tion, i.e., (ocp)1 = p la l for all a, P e M2(R). Over Q, this representation ofM2(Q) on L(n;Q) is equivalent to the symmetric n-th tensor representation of thenatural identification of M2(Q) with itself. To make the dimension formula inProposition 2 computable, we need to compute diiriR(M), dimR(H°(F,M)),dirriR(Hp(r,M)), dimR((7ts-l)M) for M = L(n;R). One can easily compute thesedimensions when R is a field K of characteristic 0. We know from the defini-tion that dirriR(M) = n+l (i.e. a standard basis is given by Xn, Xn"xY, ..., Yn).We claim that

n o rl if n = 0,(1) dimR(Hu(F,M)) = dimR(Hp(F,M)) = j

which follows from

Lemma 2 (irreducibility). The F-module L(n;K) is absolutely irreducible if Kis afield of characteristic 0 and F is a subgroup of finite index of SL2(Z).

Proof. By the density of SL2(Z) in SL2(Zp), we may assume that K= Q p ,the algebraic closure of Qp. In fact, V =L(n;Qp) = L(n;Q)® Q p is irre-ducible if and only if L(n;Q) is irreducible. Suppose that we have aSL2(Zp)-stable vector subspace W # {0} in V. We pick an element P(X,Y) inW whose degree j with respect to Y is maximal. Since u(Xn^YJ)


= (X+Y)n"jYJ for u = , j has to be equal to n. Since oca = I * _}

acts on X^11"1 by the scalar a11"21, writing P(X,Y) = liCiX^11'1 with c0 * 0,we see that a a P = LiCia11"2^^11"1 e W. Choosing distinct ai fori = 0, . . . ,n, we can write Yn as a linear combination of ocaiP; that is,Yn e W. Note that SiYn = (X+Y)n e W for the transpose lu of u. Since(X+Y)n involves ICY1'1 non-trivially for all i, ICY1'1 is a linear combination of{aai(X+Y)n)i c W. This shows that W 3 V, which finishes the proof.To compute dimK((7is-l)M) when n > 0, we insert a definition. A cusp s iscalled a regular cusp if G ^ I ^ G S is contained in U(Z). A cusp which is notregular is called an irregular cusp. Then, we have, for some 0 * h e Z

_i _ f uh if s is regular,l-u if s is irregular.

We see that M/(7TS-1)M = M/(±uh-l)M via P h ^ G ^ P . If either s is regularor n is even and positive, we see easily that M/(uh-l)M = K by P h-» P(l,0),because -1 acts trivially on M. Then

(2a) If either n > 0 is even or s is regular, dirriK((7ts-l)L(n;K)) = n.

Now we treat the remaining case where n is odd and s is irregular. In this case,ds'^s^s-l is invertible on L(n;R) unless R is of characteristic 2. Thus

(2b) If n is odd and s is irregular, dirriK((7vl)M) = n+1.

Summing up the above formulas, we have

Corollary 2. If T is torsion-free and K is a field of characteristic 0, then

(3) dimK(H1P(r,Ln(K))) = j(2g-2)(n+l)+n#(S)+5n#(Si), if n > 0,

1 2 g , / / n = 0 ,

where Si is the subset of S consisting of irregular cusps and 8n is 0 or 1 ac-cording as n is even or odd.

Remark 1. If the reader is familiar with the dimension formula of the space ofcusp forms 5k(O, he will notice the curious identity

(4) dimR(5k(r)) = dimKH1P(r>L(n;K)).

Since the dimension formula of Sn+i^X) is proven in various places (for example[Sh, Theorems 2.24 and 2.25] and [M, §2.5]) and since its direct proof withoutusing cohomology groups requires either the Riemann-Roch theorem of curves or

6.2. Eichler-Shimura isomorphisms 167

the Eichler-Selberg trace formula, we shall not give the direct proof here. In the

following section, we prove the Eichler-Shimura isomorphism 5k(F) =

Hp(T,L(n;R)) which gives an indirect proof of this fact. In fact, we will give two

proofs of this fact, and the first one actually uses the dimension formula (4).

§6.2. Eichler-Shimura isomorphismsNow we establish a canonical R-linear isomorphism between Sn+2(T) and the co-homology group Hp(F,L(n;R)). We give two proofs. First we repeat the proofgiven in [Sh, VIII] which is based on the dimension formulas for 5k+2(O andHp(F,L(n;R)), and the second one is based on the Hodge theory of manifoldswith boundary.

For each point z e fH, we consider the L(n;C)-valued differential form

8n(z) = (X-zY)ndz. Let e = ( ^ l\ Then for y = (*

can easily check (noting foeoc = det(a)e for a e GL2(C))

= f(X

Thus, putting co(f) = 2n V=4f(z)8n(z) for f e 5n+2(F), we have y*o(f)= yco(f); thus co(f) is a section of the sheaf of differential forms having values inthe locally constant sheaf £(n;C) associated to the F-module L(n;C). Referringthe details of L(n;C) to Appendix, let us briefly state the definition ofL(n;C) = L(n;C). For any F-module M, we can define T = IV£<M letting Fact on JHXM by y(z,m) = (y(z),ym). Then M is the sheaf of locally constantsections of the projection n : T -> Y = FV# Here we take L(n;R) as M. Wefix one point z in #"* = ÛP^Q) (which is naturally embedded in P^C))and consider the integral, for each y e F,

q>z(f)(Y) = JJ(Z) Re(co(f)) e L(n;R).

Since co(f) is holomorphic, the integral is independent of the choice of the pathbetween z and y(z). When z e P!(Q) (i.e. a cusp), we suppose that the pro-jected image of the path in X is a well defined path near the cusp z e S . Thenthe integral is convergent even if z is a cusp, because the cusp form is decreasingexponentially towards the cusp. For another point z1 e #*, we have

<Pz(f)(Y)-<Pz'(f)(Y) = \l Re(©(f)) - J 7 ^ Re(co(f)) = (1-y) J" Re(co(f))


and for y,5 e F

(pz(f)(Y§) = JzY5(Z)Re(co(f)) = J^z

()Z)Re(co(f))+9z(f)(y)==79z(f)(5)+9z^

Thus (pz(f) is a 1-cocycle with values in L(n;R) whose cohomology class is in-

dependent of the choice of the base point z. If 7i(s) = s for SG PX(Q) with

TIE T, then (ps(f)(rc) = 0. This shows that

9z(f)W = (1-TC) JS Re(co(f)) G (;c-l)L(n,R) for all T I E P .

Thus (pz(f) is a parabolic 1-cocycle and we obtain an R-linear map

9 : 5

Theorem 1 (Eichler-Shimura). Suppose that F is torsion-free. Then the map

(p is a surjective isomorphism.

Before going into the proof of this fact, we note an application. First of all, usingthe notation of Proposition A.I applied to the curve Y = T\H and its compactifi-cation X, for each commutative algebra R and for each R[F]-module M of finitetype over R, we see that Ai(M) = HomR[r](Ai,M) is of finite type over R be-cause Ai is free of finite rank over R[F]. Therefore H^(F,M) is of finite typeover R. Here "*" means either the usual cohomology, the compactly supportedone or the parabolic one. Note that we have exact sequences

0 -> Z^KJvl) -> HomR[r](Ai,M) -> HomR[r](A2,M),

HomR[r](Ao,M) -> B^KJVI) -> 0, and 0 -> B^K,]^) -* HomR[r](Ai,M).

If A is an R-flat algebra (or module), tensoring by A the above sequences overR, we have

and hence(la) H1(r,M®RA)=H1(r,M)®RA if A is R-flat.

Here the sentence "A is R-flat" is a terminology in commutative algebra meaningthat M®RA —» M'®RA —> M M ® R A is exact whenever M —> M' —> M" isexact as a sequence of R-modules ([Bourl, I]). By the exactness of

o -> H^r.M) -> HHr.M) -> e s e S H 1 ^ , ] ^ ) (S = X-Y),we see that

(lb) H1P(F,M)®RA = Hj,(r,M®RA) if A is R-flat.

We record here a corollary of the theorem which is an easy consequence of (la,b).


Corollary 1. Let Lp (resp. Lj be the quotient of Hp(F,i:(n;Z)) (resp.H1(F,i:(n;Z))j by the maximal torsion subgroup of Hp(r,£(n;Z)) (resp,H^F^njZ))). Then the maximal torsion subgroup of Hp(F,i:(n;Z)) (resp.H1(F,i:(n;Z))>) is finite, and Lp (resp. Lj is isomorphic to the image ofH^F^njZ)) (resp. Hî l feZ)) ) in H^F^feR)) (resp. H ^ F ^ R ) ) ) .Moreover LP®ZR = Hp(F,£,(n;R)) and L®ZR =H1(F,i:(n;R)). Thus we canidentify Lp with a Z-lattice of the R-vector -space 5k(F) for k = n+2 via (p.

The injectivity of (p. Here we reproduce the proof given in [Sh, VIII]. Wedefine a pairing on L(n;Z). Consider the symmetric or skew symmetric matrix

e = (8n.i,j(-i)i(51))o<i,j<n.

Then we see easily that

%

where =l(xn, xn4y,..., yn). Regarding each entry un"V (xn'Y) of

(resp. ) as a monomial of degree n of indeterminate u and v (resp.

x and y) and letting y e GL2(C) act on them as an element of L(n;C), we seethat

(*) Y(un,un-1v,...,vn)0t{y(xn,xn-1y,...,yn)} = det(y)n ^ j ^ j

We regard L(1;A) as the space of A-linear forms on the column vector space A2,i.e. the A-dual of A2. Now we have the pairing A2xA2 given by

[I I,I I] = detl . We identify An+1 with the symmetric n-fold tensor

product (A2)®n and consider L(n;A) =L(l;A)(S>n to be the dual of (A2)0n.Identifying u n V with ei®(n-i)®e2

0i for the standard basis ex = 1,0) and2 = \0,l), we have a pairing on (A2)®n whose matrix is given by 0. Since

L(n;A) is the dual of (A2)0n whose basis dual to {X^Y1}! is given by{ei0(n"i)(8)e2<8)i}, we can define a pairing <,) : L(n;A)xL(n;A) -* A by

(2a) ( X aiX^Y1,î=0 j=0 k=0

/n\"âs long as ( ) e A. In particular,

(2b) <(X-zY)n,(X-zY)n> = X (-Dn"k(k)zn"kzk = (z-z)n.k=0

We see from (*) that for any y e


[pc,y] = [x,fy] for x , y e (A2)®"and hence(2c) (yP,Q) = ( P ^ Q ) for P, Q e L(n;A).

This induces a pairing via the cup product (see [Bd,IL7])

(3a) < , ) : H1c(Y,i:(m;A))(8)H1(Y^(m;A)) -> H*(Y,A) s A.

The last isomorphism is the one in Corollary 1.1. The differential form realizing

the class (Re(co(f)),Re(co(g))) can be made explicit as follows. Writing any poly-

nomial P = Z^gaiX^Y1 as a column vector Xao,...,^ which we denote by P,

we know that (Re(co(f)),Re(co(g))) is represented by the closed form

O(f,g) = tRe(co(f))A0-1Re(co(g)).

Thus by Corollary 1.1, we have (Re(co(f)), Re(co(g))> = JYQ(f,g). Thus we get

a pairing for f,g e 5n+2CO given by A(f,g) = JY^(f,g). We can compute

A(f,g) in terms of the Petersson inner product: Note that co(f) = f(z)(X-zY)ndz.Thus, by (2b), we have, for the complex conjugation c,

tco(f)A0-1(cQ(g))c = -2Vzl(fg)(z-z)ndxAdy = ( ^ V ^ f

(k = n+2). This shows that

(3b) A(f,g) = -(-2VzT)n-1{(f,g)+(-l)n+1(g,f)}, A(f,V:4n-1g) = 2nRe((f,g))and A(f,V-lng) = -2nV=lIm((f,g)).

In particular, the pairing A(f,g) on 5n+2(D is non-degenerate. We are going toshow that if (p(f) = 0 in the cohomology group, then A(f,g) = 0 for allg e A+2(r). The injectivity of (p follows from this fact. We consider the fol-lowing function for a constant vector a e L(n;R) (later we will specify the vectora suitably for our computation) and for a fixed point z e CK

F(w) = fWRe(co(f)) + a.

Then we see that

F(y(w)) = JJ(w)Re(co(f))+a = J^)Re(co(f))+(Pz(f)(y)+a

= JzWY*Re(co(f))+9z(f)(y)+a = pn(y)F(w)+(pz(f)(Y)+(l-y)a.

Since <p(f) = 0, we can find b e L(n;R) so that q>z(f)(Y) = (Y-l)b. Thus bytaking -b as a, we have F(y(w)) = yF(w) for all w. On the other hand, we see


that dF = co(f), where d is the exterior differential operator. Similarly, we de-

fine G(w) = JW Re(co(g)). Then we have dG = Re(co(g)) and

Q(f,g) = 'dFAS-MG = dCF-e^dG) = dCF-G^ReCcotg))).Let Oo be the fundamental domain of Yo as in the proof of Proposition 1.1 (see(Tl-3) in Appendix). Then using the notation introduced there, we know that theboundary of Oo is of the form:

2®o = £ s e S t. + YU Kaj-Dsj + (Pj-Ds'j}.

Thus we have, shrinking the holes of Yo (i.e. Yo -> Y)

A(f,g) = lim LdC'F.ReCcoCg))) = limYo^XJ Y0-»X

Since F(a(w)) = aF(w) for all a e F, we see for any 1-simplex A that

= jAta*F.0-1a*Re(co(g))

Thus we have

I J; = 0,because F(w) is bounded near cusps and Re(co(g)) is rapidly decreasing at eachcusp. This shows the vanishing of f and the injectivity of (p.

By the dimension identity (6.1.4), we conclude that cp is also surjective.However this proof of surjectivity requires the Riemann-Roch theorem (the di-mension formula (6.1.4)) which comes from algebraic geometry. We now give asketch of a purely cohomological proof of the surjectivity, which is a version of thetreatment given in [MM], [MSh] and [Ha] in our special case:

The surjectivity of (p. We show the surjectivity of the scalar extension of (pto C:(4) 5 k ( r )05 k ( r ) c = Sk(T)®RC = ^ ^

Here 5k(F) = {f(z) |f e 5k(F)} with denoting complex conjugation,

and Hp(Y,Z,(n;C)) is the natural image of the sheaf cohomology groupH^(Y,.£(n;C)) of compact support in the usual sheaf cohomology groupH^YjiXnjC)). We resort to the Hodge theory of Riemannian manifolds withboundary to prove the theorem. We refer for technical details to standard texts indifferential geometry. The boundary exact sequence of Corollary A.2 combinedwith the isomorphism between the sheaf cohomology group H^Y^njC)) andthe group cohomology group H1(F,L(n;C)) (see Cor. A.I and Prop.A.4) tells usthat Hp(Y,Z<n;C)) is naturally isomorphic to Hp(r,L(n;C)). We can compute


Hl(Y,L(n;C)) using the de Rham cohomology theory (i.e. H ^ Y ^

= Hj)R(Y,X(n;C)); see Theorem A.2 and Proposition A.4). We have already

defined the differential form co(f) attached to f e 5k(F). For f e 5k(F)c, we

define co(f) = f(z)(X-zY)ndz. Then co(f) for f e 5k(F) is holomorphic and

for f e 5k(F)c is antiholomorphic. Anyway they are closed forms, which define

cohomology classes in H^R (Y,£(n;C)). Thus we have a map

Identifying Hl(Y9L(n;C)) with H^FJLOnjC)) by the canonical isomorphism, itis not so difficult to show that Re(O(f)) = (p(f) for f e 5k(F) tracking down allthe isomorphisms between various cohomology groups presented in Appendix.Thus by the first proof, O is injective. The point here is to prove the surjectivityof <& onto Hp(Y,£(n;C)). First we shall show that O takes values inHp(Y,£(n;C)). If f e 5k(F)05k(F)c, we define for each cusp s e P^Q),

Fs(z) = JsZco(f).

This integral is well defined because f is decreasing exponentially towards thecusp s. As already seen, dFs = co(f). Note here that Fs(z) is not invariant un-der F; thus 7rlFs(y(z)) may be different from Fs(z). However Fs does behavenicely under F s = {y e F | y(s) = s}, i.e. Y^FgCyCz)) = Fs(z) for y e Fs.Thus Fs is a section of L(n;C) on Us-{s} for a small neighborhood Us of son X. Taking a C°° function (|)s such that its support is contained in Us and(|)s = 1 on a still smaller neighborhood of s, we define F = XseS^sFs. Then Fis a smooth global section of £(n;C) and co(f)-dF is compactly supported. Thusthe cohomology class of o(f) falls in Hp(Y,£(n;C)).

Now we construct a Laplacian acting on the sheaf of smooth differential p-forms$ on Y with values in X(n;A) (for A = R and C). Since % : SL2(R) -> 0<given by X H X ( V - T ) induces an isomorphism TC : SL2(R)/SC>2(R) = 9{, wecan consider a new covering space £'(n;C) = I\(SL2(R)xL(n;A))/SO2(R), wherethe action is given by y(x,P)u = (yxu,P | u) for (y,u) e FxSO2(R) with theright action of SL2(R) given by PI x(X,Y) = P((X,Y)lx). We claim:

(5) We have an isomorphism of covering spaces of Y: £(n;A)/y= L'(n;A)/Yinduced by the map: (x,P) H» (X,P | X) on SL2(R)xL(n;A) for A = R and C.

Let us prove (5). The map is well defined because y(x,P) corresponds to(yx,yP I yx) = (yx,P) = (y,l)(x,P). It is obvious that the map induces an isomor-phism at each fiber and hence, it is an isomorphism globally because both coveringspaces are locally trivial.


Hereafter we identify L\n; A) with L(n; A) by the map (5). The merit of the newrealization is that it is easy to give a canonical hermitian pairing to each fiber. SinceSC>2(R) is a connected compact group, the image of SC>2(R) in GL(L(n;R)) iscompact, which is thus contained in a compact orthogonal group of a positive def-inite symmetric form S on L(n;R). In fact, we can take as S the symmetric n-thtensor of the standard hermitian inner product: (P,Q) = P(ei) Q(e1)+P(e2) Q(e2)on L(1;R), where ei = \lfl) and e2 =

l(0,l) make up the standard basis.Since S(P| U , Q | U) = S(P,Q) for u e SC>2(R), this product induces a positivedefinite hermitian product on each fiber of £'(n;C) (= L(n;C)). On 9{, we haveas SL2(R)-invariant Riemannian metric y~2(dx2+dy2) = y"2dz<8>d z. Take anypair of C-valued differential forms {coi} 1=1,2 on a simply connected open set Uin Y which gives an orthonormal basis at each fiber on U under the Riemannianmetric. For example, coi = y-1dx and 0)2 = y~*dy are a good choice. Thenwe define *0>i = C0j ( i ^ j), *(COIAOL>2) = 1 and *1 = CO1ACO2. Extend this

operator C-linearly to the space of differential forms defined on U. Then weknow from de Rham that the operator "*" is independent of the choice of the basis

(0)1,0)2) and hence extends to a global operator on the sheaf AQ1. Let us write Ffor £(n;C) and consider the sheaf J4$l of smooth differential forms with valuesin F. Since the sheaf £(n;C) is locally constant, the same procedure of definingthe "*" operator works well for J%pl. For 0),r| e ^ F ! ( Y ) , writing

*co I u = <t>i(z)coi+(|)2(z)o)2 and r| = (pi(z)o)i+(p2(z)o)2,

we define S(T],co) = XijS(<|)i,(pj)CQiA*G)j. Then S(r|,co) patches up together

well to give a global 2-form. We define (r|,co) = JYS(r|,0)). Then ( , ) is a

positive hermitian form defined on the subspace of square integrable forms in. ^ ( Y ) . Similarly we can define a positive pairing ( , ) between .%°(Y) and-%2(Y). Let d1 (resp. d") be the holomorphic (resp. antiholomorphic) exteriordifferential operator. We then define the formal adjoint 5' and 8" of d1 and d"respectively. That is, (d'co,r|) = (CO,8'TI) (resp. (dM0),r|) = (co,8"r|)) for allsquare integrable co with square integrable d'o) (resp. d"co) and all squareintegrable r|. We put 8 = S'+S". Then 8 is the adjoint of the exteriordifferential operator d. We define the Laplacians A = d8+8d, A' = d'S'+S'd'and A" = d"8M+8"d". It is easy to see that A = A!+AM. We call 0) harmonic ifAco = O. Write co = Sjfj(X-zY)n'j(X-zY)jdz+Ijfj(X-zY)n-J(X-zY)jdz. Thenwe have, for y e F,

fj(Y(z)) = fj(z)j(Y,z)n-J+2j(y, z)J and fj(Y(z)) = fj(z)j(Y,z)n-Jj(Y,z)J+2,fa b^

where j(Y,z)=cz+d for y = I. If co is square integrable and harmonic,

by the spectral theory of the unitary representation of SL2(R) on the L2-spaceL2(I\SL2(R)), it is known that each fj(z) and fj(z) and all their derivatives areexponentially decreasing as Im(oc(z)) -» ©o for all a e SL2(Z) [Ha]. The


description of the spectral theory would take us beyond the scope of this book.Thus we just admit this fact and conclude the proof. By the exponential decaytowards the cusps, if CO is square integrable and harmonic, it is easy to showA'co = A"co = 0. In particular, d'co = d"co = 0 and 5'co = 5Mco = 0 if cois harmonic and square integrable. Thus if co is harmonic and square integrable,co is a sum of holomorphic form and anti-holomorphic form. Thus we mayassume that co is holomorphic (f j = 0). We now show that co = co(fo) for aholomorphic cusp form fo on T of weight n+2 if co is holomorphic. To dothis, we compute 8' for co = Xjfj(X-zY)n'J(X-zY)Jdz. Here we have writtendown co as a section of L(n;C). The section of L'(n;C) corresponding to co is

(1=^=1 , g € SL2(R)).

Writing (|) = Zk(|)k(X-zY)n-k(X-zY)k for a C°°-global section of £(n;C), wehave fk(y(z)) = fk(z)j(y,z)n-kj(Y,z)k and

k=o dz Z i y k=oThen the C°°-section of L\n;C) corresponding to d'cj> is given by

k=o °z Z i y

"W 2>(X-zY)n-kl(X-zk=0

We write 8n.k for the differential operator — + . Noting the fact:dz 2iy

S((X-iY)n~j(X + iY)j,(X-iY)n"k(X + iY)k) = 2n5j>k and

we have

It is easy to see that (see the proof of Theorem 10.1.2), for e = y2—,dz

JYfk8n-k4>kyndxdy = JYifkyn-2dxdy.

Thus we have

5'co = £ (efk+^fk+1)(X-zY)n-k(X-zY)k

k=0~ 3 k

By a direct computation, we have for 8k = -—dz 2iy

d"co = j ; (5kfk+^fk+1)(X-zY)n-k(X-zY)k,k=o

6.3. Hecke operators on cohomology groups 175

where we agree to assume that fn+i = 0. By the fact: d"co = 8'co = 0, weknow that

f - -M - f - •*<••This implies fi = 0 and fo is holomorphic. Again using d"co = 0, weconclude fj = 0 for j > 0. Thus co = co(fo) for foe 5k(F). If co isantiholomorphic, co = co(fo) for fo e 5k(F)c.

Now we want to define Hodge operators. Let <|) be a square integrable smooth1-form. Consider the L2-space L of 1-forms with values in L(n;C). Let *F bethe L2-closure of (A(|) | Supp((|)) is compact} in L. Writing Tc(Aipl) for thespace of compactly supported global sections for the sheaf .%*, we take, for anycompactly supported co e j^p1» a unique form \\f e W such thatI co-\|/ I = ^((o-\\r,(o-\\r) is minimal, i.e. \j/ is the orthogonal projection of CO

in W. Then we put Hco = co-\}/. Then by definition (Hco,Ay) = 0 for all com-pactly supported y. This implies Hco is a harmonic current. Since A is an el-liptic operator, any harmonic current is in fact an analytic function (cf. [DuS, §3]or [Ko, Appendix]). Then it is well known that we can find a smooth solution ofthe equation A|i = co-Hco because on *F the spectrum of A does not vanish.Hence A has a formal inverse defined on *F, which we write G [Ko, Appendix,§7]. Thus ji = Gco and HG = 0. That is, AGco = (l-H)co. Since d com-mutes with A, G commutes with d, and hence Hco is cohomologous to co if cois closed. As already shown, Hco is in the image of O. Thus O is surjective.

§6.3. Hecke operators on cohomology groupsLet A = {a e M2(Z) | det(a) *0}. We can let the semi-group A act on H asfollows. If det(oc) > 0, a acts on H by the usual linear fractional transfor-

(Tmation. If a = j = , we put j(z) = -z . If det(a) < 0, we can de-

compose a = ajj with det(aj) > 0 and define oc(z) = (ocj)(j(z)). One caneasily check that this action is well defined (i.e. associative). Actually, identifyingtH with SL2(R)/SO2(R) = GL2(R)/O2(R), this action is the left multiplication by

elements of GL2(R). Let a e A and ( r ,F ,a l ) be the semi-group in A gen-

erated by a1 = (let(a)a"1 and two congruence subgroups F and F .

For any (r,F,al)-module M, we define the Hecke operator [FocF1] with

det(oc)>0 acting on H^rjVI) as follows. First decompose FaF1 = LIiFc^.

For each y e T\ we can write a[j = yiOCj (yiâ* = ocjy1) for a unique j


with Yi e F. Then for each inhomogeneous cocycle u : F -^ Ln(A) (see

Appendix about cocycles), we define v = u I [FaF1] by v(y) = Xi 0Cilu(Yi),

where a1 = det(a)a'1. For y,8 e F, we define yi and Si as above. Now let

us check that v is a cocycle. Note that aiyS = YiSjOCk for some k. Thus

v(y8) = liCCiVyiSj) = Ii{ailyiu(5j)+ailu(yi)}

= Si(yr1ai)lu(5j)+v(y) = Ii(ajr

1)lu(5j)+v(y) = yv(5)+v(y).

This shows that v is a 1-cocycle of F'. If u is a 1-coboundary, i.e.u(y) = (y-l)x, then

v(y) = 2>il(Yi-l)x = Xj(aJr

1)lx - Xiailx = (y-l)]T ^ xand v is a 1-coboundary. This shows that the operator [FaF1] is a well defined

linear operator on H^^M) into H^F^M). Now if u is parabolic, by replacing

y as above by any parabolic element % e P, we know from the above computa-

tion that if u(rc) = (7t-l)x, then V(TC) = (7C-l)Zi(Xilx. Thus v is again parabolic

and [FaF1] sends Hp(F,M) into Hp(F',M). We define the multiplication in the

abstract Hecke ring R(F,A) as in [Sh, 3.2] and [M, §2.7]. We see easily that

[FaF]o[rpr] = [ (Far ) . ( rpr ) ] and hence R(F,A) acts on the cohomology

group if M is a A-module. When F = F' = Fo(N) for a positive integer N

and M is a A'l-module for

A' = {T h]e A | det(a) ?fc 0, c e N Z and dZ+NZ = Z } ,

we define the Hecke operator T(n) for each integer n > 0 by the action of

Z[FaF], where F a F runs over all double cosets in {a e A'11 det(a) = n}.

For a fixed point z e Oi, let us compute (pz(g)(Y) = J Re(co(g)) for

g = f I [FaF] = Xif | ai (FaF = Ili FaO in terms of f e 5k(F). Note that8n(a(z)) = det(a)(cz+d)-n-2a5n(z).

Since scalar t of A acts on L(n;C) via the scalar multiplication by tn, we obtainalRe(co(f))°cc = Re(co(f I a)) , where f I a = det(a)k"1f(a(z)) j (a ,z)-k

(k = n+2). From this fact, we know that

<Pz(g)(Y) = 5 > l F ( Z ) ai*Re(co(f)) = X i a i l H W Re(co(f)).

As in the proof of Theorem 2.1, we put F(w) = JW Re(co(f))+a. Then the

1-cocycle u(y) = F(y(w))-yF(w) represents the cohomology class of cp(f). Thenwe see that


<Pz(g)(Y) = E i a

for x = ZiOCilF((Xi(z)), since a ^ = yaj1. This shows that the isomorphism cpas in Theorem 2.1 is in fact an isomorphism of Hecke modules (i.e. compatiblewith Hecke operators). Similarly the isomorphism

O : 5 k ( r ) 0 5 k ( r ) c = H}>(r,L(n;C))

is an isomorphism of Hecke modules.

Let N be a positive integer and % : (Z/NZ)X -» Ax be a character with values

in a ring A. We define a new All-module L(n,%;A) as follows. We take L(n;A)

as the underlying A-module of L(n,%;A) and define a new action of A'1 by,fa b\

writing the original action of y=\ € A1 on L(n;A) as y»P,vc aJ

f P = JC(d)yl.P.When we regard L(n,%;A) as a left ro(N)-module, the action is given by

YP = %(d)y.P for y = [* *) e ro(N).

Theorem 1. For any positive integer N, we have natural isomorphisms of

Hecke modules O : 5k(ro(N),x)e5k(ro(N),x)c = H^roCNXUn^C)),where % denotes the complex conjugate of x and

5k(ro(N),x)c = { f ^ l f e 5 k ( r 0 (N) ,x)} .

Proof. We can define the cocycle (pz(f) for f e 5k(ro(N),x)®5k(ro(N),x)c by

cpz(f)(Y) = JJ(Z) co(f).Then it is easy to check that (pz(f) in fact has values in X(n,x;C) (not just in£(n;C)). Thus we have a natural map associating the cohomology class of (pz(f)to f:

<D : 5k(ro(N),x)©5k(ro(N),x)c -> H^roC^.LCn.XiC)).

Now we take a small normal subgroup T of Fo(N) of finite index in Fi(N). Wemay assume that Y is torsion-free. Then % as a character of Fo(N) factorsthrough the finite quotient G = ro(N)/T. The double coset n 2 r 0 ( N ) = ro(N)defines two trace operators:

Tr : 4(r,L(n;C)) -+ 4(ro(N),L(n,x;C)),Tr : 5k(r)05k(r)c -^ 5k(r0(N),%)e5k(r0(N),x)c.


We have dropped the symbol % from the left-hand side of Tr, not only because

L(n,%;C) = L(n;C) as F-module but also because the action of G given by the

operator [FaF] = [Fa] for a e FQ(N) is defined relative to the action of a

on L(n;C). Write "res" for the map given by restricting cocycles of FQ(N) to

the smaller subgroup F. Then it is obvious that Tr°res is multiplication by the

index [Fo(N):Fi(N)]. This shows that res induces an isomorphism from

4(T0(N),L(n,x;C)) (resp. 5k(Fo(N),x)®5k(Fo(N),x)c) onto

H^(r>L(n;C))[x] = (x e H^FJLfeC)) | x | g = X(g)x for g e G} (resp.

{5k(F)05k(F)c}[x] = {x G 5k(F)05k(F)c | x | g = %(g)x for g e G}).

Thus we have a commutative diagram:

I inclusion si res

O : {5k(r)e5k(r)c}[%] > H^TJLfo

Since the lower horizontal arrow is a surjective isomorphism of Hecke modules byTheorem 2.1, so is the upper top line.

Theorem 2. For every positive integer N and every character% : (Z/NZ)X -» Cx, we have, if k > 2

5k(F0(N),x;A) = 5k(Fo(N),x;Z[x])®z[%]A and

HomZ[X](hk(Fo(N),x;A),A) = 5k(F0(N),x;A) by $ h^ ^ = 1 ^)(T(n))qn

for any Z[%]-algebra A inside C or Q p .

Proof. By Theorem 5.3.1, we have

Homc(hk(ro(N),x;C),C) = 5k(Fo(N),x) via $ h^ ]T~=1 ^)(T(n))qn.

On the other hand, by Theorem 1, hk(Fo(N),x;Z[x]) leaves stable the image L

of H1p(T0(N),L(n,x;A)) in 5k(Fo(N),x)e5k(Fo(N),x)c. This implies

hk(F0(N),x;C) = hk(F0(N),x;Z[x])®z[X]C,and therefore

Homc(hk(Fo(N),x;C),C) = HomZ[X](hk(Fo(N),x;Z[x]),Z[x])®z[%]C.

The image of HomZ[X](hk(Fo(N),x;Z[x]),Z[x]) in 5k(F0(N),x) is exactly thespace 5k(Fo(N),x;Z[x]), and hence the assertion follows for A = C. We candeduce the assertion for general A from that for C in the same manner as in theproof of Corollary 5.4.1.


Now assume that N = pp a for a prime p. We now want to describe the similarresult for M(x) = f^k(Fo(N),%). The map O is well defined even on^k(Fo(N),%) and gives a commutative diagram for F = Fo(N) whose rows areexact:

0 -> S(T,x)®S(T,x)c ^ M(x)®S(T,x)c -» Coker(i)lit O 1<3> I <!>'

0 - » H1P(T,L(n,x;C)) - » H 1 (T,L(n ,x ;C)) -> e r e S H I

1 ( r . J L ( n f z ; C ) ) .

We now claim, for any field K containing Q(x) , that if a > 0

( l )H 1 ( r 0 (pp a ) s ,L(n ) x ;K))=(K. i f « is equivalent to either - or 0,[U, o the rwise .

When s = 0 or °°, the assertion follows from the argument which proves

(1.2a,b), since Fs is generated by either 7ioo = L - or no = a modulov° v VPP U

the center {±12}. As seen in §1, S s ro(paq)\SL2(Z)/Foo = {ideals of Z/papZ}given by

The last isomorphism follows from the strong approximation theorem (Lemma1.1) and the fact that the image of ro(ppa) in SL2(Z/papZ) is the subgroup ofupper triangular matrices which fixes one line in (Z/papZ)2. This implies thateach cusp s which is equivalent to neither ©o nor 0 is represented by a u =

for uepZ-ppaZ. Then *. = a ^ ^ = [ ^ l + ]±j E ro(pap)

for some h e Z. Since 7CS is a generator of ro(pap)s, h is determined by the

condition that u2h e p a pZ and | h | p is as large as possible. Thus we know

that I h I p = max( | p a p I p I u | p"2,l) > I p a p I p. Then by the primitivity of %,

%(l+uh) * 1 and 7ts-l is invertible on L(n,%;K). This shows the vanishing of

the cohomology because of H1(r0(pap)s,L(n,%;K)) =L(n,%;K)/(7Cs-l)L(n,%;K).

If k > 2, we already know from Proposition 5.1.2 that the Fourier expansion of

Ek(X) (resp. G(%)'1Gk(%)) has non-trivial constant term at 00 (resp. 0) and has

no-constant term at 0 (resp. 00). We will see in Chapter 9 the same assertion for

k = 2 provided that % * id. This shows that the map O' of Coker(i) to

0SEsH1(Fo(pa)s,L(n,%;R)) induced from O is in fact surjective. Comparing the

dimension, O1 is an isomorphism. Thus we have


T h e o r e m 3. Let N = p p a with a > 0, k = n+2 and % be a primitive char-acter modulo N. Then we have the following commutative diagram whose rowsare exact:

0 -> 5 k ( x ) 0 5 k ( % ) c -^ 44(X)®5 k (%) c -> Coker( i )in O in O m

0 -* 4 ^ HNow we deal with the case of SL2(Z). When k > 2, the argument is completelythe same as in the case of Theorem 3. When k = 2 (i.e. n = 0), the smallcircle around oo in X is bounded by Yo. Thus the natural restriction map:

H1(SL2(Z),C) -» ft\{±l}\l(Z)£)

[71 m^ I ]for U(Z) = j L I m e Z i s a zero map. This in particular means that the

constant term of any modular form of weight 2 for SL2(Z) vanishes. Thusaf2(SL2(Z)) = 52(SL2(Z)) = {0}, and we have

Theorem 4. We have the following commutative diagram if k = n+2 > 2:

5k(SL2(Z))05k(SL2(Z))c — ^ f * 4 ( S L 2 ( Z ) ) 0 5 k ( S L 2 ( Z ) ) c -> Coker(i)in O IU <D in

0 -> H J » ( S L 2 ( Z ) J L ( I I ; C ) ) > H 1 (SL 2 (Z) ,L(n;C)) -> HL(L(n;C)) -> 0,

where HL(L(n;C)) = H ^ f t l J U C Z X L f o C ) ) . WAen k = 2, we have the fol-lowing commutative diagram:

52(SL2(Z))052(SL2(Z))C s ^ k ( S L 2 ( Z ) ) e 5 k ( S L 2 ( Z ) ) c

Mi 4> iu O

Hj,(SL2(Z),C) s UHSL2(Z),C).

Then in the same manner as in the proof of Theorem 2, we have

Corollary 1. For every prime power N = p a and every primitive character% : ( Z / N Z ) X -> C x , we have, if k > 2,

Let N > 4 be an integer. Then T = Fi(N) is torsion-free. We can consider for(I(

primes q the double coset action T(q) = [FaF] for a = Q on modular

forms f on Fi(N), which is given by f | T(q) = f | k a i for the decomposition

F a F = UiFa i . In fact, we can choose oci so that Fo(N)aFo(N) = IIiFo(N)ai.Therefore we have two commutative diagrams:


ri(N))4T(q)

andH1(r0(N)>L(n,x;A)) c

lT(q) 4T(q)H1(r0(N)>L(n)%;A)) c H1(ri(N),L(n;A)).

The finite group (Z/NZ)* = Ti(N)\ro(N) acts on 314(1^ (N)) by

Similarly the operator (d) acts on the cohomology via the action of [F . F].

Now we define T(n) for general positive integers n by

a(m,f | T(n)) = X b | (m,n) ^-^(mn/b^f | <b»,

where b runs over all common positive divisors of m and n prime to N. Thisaction again coincides with the action of ZrarfFaF] where FaF runs over alldistinct double cosets in

{a G A I det(a) =n and oc= modNM2(Z)}

(see [Sh, III] or [M, §4.5]). Anyway we can define the Hecke algebraHk(Fi(N);A) (resp. hkQTi(N);A)) for any subring A of C as an A-subalgebraof the A-torsion-free part of H^riCNJJLfaA)) (resp. HJ>(ri(N),L(n;A)) forn = k-2. Then, by the Eichler-Shimura isomorphism, these algebras act faithfullyon f^k(Fi(N)) and 5k(Fi(N)). Note that we have not proven that fAfk(Fi(N);A)is stable under Hk(Fi(N);A) although it can be proven using a geometricinterpretation of modular forms due to Katz [K5] (see also [HI, §1]). Anyway wehave

(2a) Hk(Fi(N);A) = Hk(r1(N);Z)®zA and hk(Fi(N);A) = hk(Fi(N);Z)®zAfor k > 2.

There are natural A-algebra homomorphisms

(2b) Hk(Ti(N);A) -> Hk(Fo(N),%;A) and hk(Ti(N);A) -> hk(Fo(N),%;A),

which take T(n) to T(n) and hence are surjective. These homomorphisms areobtained by restricting T(n) for Fi(N) to the space of modular forms for Fo(N).

Now we want to describe the action of Hecke operators in terms of sheaf coho-

mology groups H£(Y,£(n,x;A)), HJ>(Y,Z,(n,x;A)) and H ^ Y ^ n x A ) ) . To in-

clude the most general case, we take a (F,F',a)l-module M for congruence sub-


groups F and F and a e A. We consider the locally constant sheaf M asso-

ciated with M. We can in fact split the operator [FaF'] on H^FjM) into three

parts: [FaF1] = Trr/d>ao[OaOa]oresr/^, where O = a F ' a ^ n r , <Da = a"!Oa

and Trpyâ = [F'lÔ ] for I2 = L 1 I. This can be checked as follows.

Note that

Ff = UiOa5i => a ^ F a F ' = Uia^FaSi => F a F ' = UiFa8i.

Moreover if the first decomposition is disjoint, then the other two are also disjoint.

Then by definition, it is clear that [FocF] = Trry<&a°[OaOa]°resrv.

Exercise 1. Give a detailed proof of [FaF1] = Trryd>a°[®aOa]°resrv.

We now construct the corresponding morphism of sheaves. The operator[<J>ocOa] is easy to take care. We write Y(O) for OW Then the map

a* : 2/xM s (z,m) h-> ( a ' ^ z ) ^ ^ ) G 9{YM

induces a morphism a* : M/Y(<E>) -> M/Y(#a) because

oc*(y(z,m)) = a^yao^fem).

Thus we have [<X>ocOa] : H^(Y(O),M) -> H*(Y(Oa),M)- The restriction map is

induced from the projection M/Y(O) ~-> M/Y(r> Now we define the trace op-

erator. Since the projection % : Y(Oa) -> Y(F') is etale (i.e. it is a local

homeomorphism), for each small open set U in a simply connected open set in

Y(F), TC^CU) is a disjoint union of open sets each isomorphic to U. Thus taking

an open subset Uo in 9-C isomorphically projected down to U, we know that

rc*M(U) = M(Uo)^r:â^ This isomorphism is given as follows. We may iden-

tify Tt'Û) with the image of the disjoint union Ui8i(Uo) for a disjoint decom-

position F' = Ui<£a8i. Then we identify M(5i(U0)) = M with M(U0) = M

by M(8i(Uo)) 9 X H 8ilx e M(U0) = M(U). Now it is clear that

where Indry<i)a(M) is the induced module M®z[<2>a]Z[F] with the F'-action

given by y(m(8)a) = mây1. Since the direct image of a flabby sheaf is flabby by

definition and TC* is an exact functor because n is a local homeomorphism, any

flabby resolution of M/Y(<&«) gives rise to a flabby resolution of (7C*M)/Y(n Jus t

by applying 71*. Thus we know that E£(Y(r),7C*M) = H^(Y(Oa),M). This

gives in particular a proof of Shapiro's lemma in group cohomology asserting thatfor any pair of groups G D H and H-module M

(3) HkG.Indo/HCM)) = ff(H,M).


Now we define Tr : 7t*M/Y(r)(U) -» M(U)/y(r) by Tr(x) = Zi5ilx, which in-

duces a morphism of sheaves. Obviously this is induced from the map

tr : Z[r f] -> Z [O a ] :

Tr = id®tr : Indrvâ(M) = M®Z[a]Z[r] -> M.Then Tr induces a map of cohomology groups:

Tr : t ^ i

We define [ T a r ] : H^YCD.M) -> Hi(Y(rf),M) by Trryao[OaOa]oresr/.It is tautological to check that this action of the Hecke ring is compatible with thecanonical isomorphisms between sheaf cohomology and group cohomology.

Let us now assume that M is also an A-module for a commutative ring A. Then,writing M* for the A-dual of M, we have a pairing by the cup product

< , >r : H^(Y,M)®H1(Y,M*) -> H*(Y,A) = A (Y = Y0T) = T\H).

First we suppose that A is a Q-algebra. Then this pairing is non-degenerate (forexample, extending scalars to C and then it is clear for M = L(n,v;C) = M*(see (2.3a)); see [Bd,IL7] for a proof in general). We have a commutative dia-gram for a subgroup O of F,

Indr/d>(M®AM*)

ii \ i < , >

< , ) : Indr/^(M)®AM* -> A[O\T]

i Tr(8>id 4 Tr

( , ) : M®AM* > A.

Note that Tr : A[O\T] —» A induces an identity on

A = H ^ Y ^ J . A ) = HjCY^.AtOVn) ~> Hj(Y(D,A) = A.This is an easy consequence of Proposition 1.1 and its proof. Then the above dia-gram induces another commutative diagram:

Tid®res "

< , ) : H ^ Y ^ M ) ® ^ " ^ ^ ) , ^ ) -> A

( , ) : H^YCn^DiSH^WnÎi) -» A.We thus have

(4a) (Trr/<D(x),y)r = (x,resr/d)(y))<D if F D O


We see easily that

(4b) (x | [OaOa],y)(Da = <x,y | [Oaa*O])<D for O a = aÔa .

In particular, when M = L(n,%;A), we identify M* with the dual lattice under

<,) in L(n,%~1;A®Q). From (2.2c), (ax,y> = <x,aly>, we see that

(4c) (x | [OaOa],y)d>a = (x,y | [OaalO])o for O a = a ' ^ a .

This implies, for x e HJ.(Y(r),M) and y G H ^ Y t r X M i ) ,

(4d) (x | [r<xr],y>r = <x,y I [r<xlr]>r.

Now we specialize our argument to the case where M = L(n,%;A). We writeL*(n,%-1;A) for the dual module in L(n,%"1;A<8>Q) under ( , ) . It is easy to seefrom (2.2a) that

(5) L^n.xÂ) = XLo A (i ) X n " i Y i in L^'X"1^)-

Since r o (N)a l r o (N) = Tro(N)aro(N)x-1 for % = (° ~*\ if we modify the

pairing ( , ) and define a new one ( , ) by

(6a) (x,y) = (x | T,y),we have(6b) (x | T(n),y) = (x,y | T(n)).

Since ( , ) is non-degenerate for any field of characteristic 0, we see fromTheorem 5.3.2 and the Eichler-Shimura isomorphism

Theorem 5. Suppose that either % is primitive modulo p a or a = 0. Then

H^(Yo(pa),i:(n,x;K)), H1P(Yo(p

a),i:(n,%;K)) and H1(Y0(pa),An,%;K)) are all

semi-simple Hecke modules provided that K is a field of characteristic 0.

Proof. Let K/F be a field extension of characteristic 0. Then the assertion for Kis equivalent to that for F. If the result is known for K = Q, then the assertion istrue for all fields of characteristic 0 by extending scalars to K. To prove theassertion for K = Q, extending scalars to C, we may assume that K = C. ByTheorem 5.3.2 and the Eichler-Shimura isomorphism (Theorems 1-4), we knowthat the assertion for Hp and H1. Then the assertion for the compact supportedcohomology group follows from the duality (6b) compatible with the Heckemodule structure on H1.


We continue to study the Hecke module structure of cohomology groups. Weknow that the following isomorphisms of Hecke modules:

*4(ro(pa),X;K) s HomK(Hk(r0(pa),x;K),K),

5k(r0(pa),x;K) = HomK(hk(r0(pa),x;K),K).

Since 5k(r0(pa),x;Kc)c = 5k(r0(pa),x;K) as Hecke modules via f <-> f,

Hl?(To(v

a)Mn,X;K)) s HomK(hk(r0(pa),x;K),K)2.

Since hk(ro(pa),%;K) is semi-simple, we know that

(7) HomK(hk(r0(pa),x;K),K) = hk(r0(pa),%;K) as Hecke modules,

because any semi-simple algebra S over K is self-dual under the pairing

(x,y) = TrK/Q(xy). Thus HJ)(r0(pa),L(n,%;K)) is free of rank two over

hk(r0(pa),x;K). Thus we have

Corollary 2. Suppose that K is a field of characteristic 0 and % is a primitive

character modulo p a . Then Hp(ro(pa),L(n,%;K)) is free of rank two over the

Hecke algebra hk(ro(pa),%;K). Moreover, we have the following isomorphisms

as Hecke modules: if either % ^ id or k > 2

H1(r0(pa),L(n,z;K)) = ^(ToCp^LfottK))

= hk(r0(pa),x;K)eHk(r0(p

a),x;K)and if % = id and k = 2

^ X K ) = H^(SL2(Z),K) s h k ( r 0 ( p a ) , x ; K ) 2 = {0}.

Since f^k(r0(pa),x;K) = HomK(Hk(r0(pa),x;K),K) s Hk(r0(pa),x;K) and5k(r0(pa),x) = HomK(hk(r0(pa),x;K),K) = hk(r0(pa),x;K) as Hecke moduleand since these Hecke algebras are all semi-simple, we see that

(8) Hk(r0(Pa),x;K) = hk(r0(p

a),x;K)eEk(r0(pa),x;K)

for an algebra direct summand Ek(To(pa),x;K). Let E be the idempotent ofEk(Fo(pa),x;K) in Hk(ro(pa),x;K). To get each of the following exact sequencesin one line, we drop FoCp06) from the notation of the following cohomologygroups in (9b,c) and (10) and also denote by Hg1 for the i-th cohomology groupfor ro(pa)s (s G S). Then the exact sequences of Hecke modules

(9a) 0 -> 5k(r0(pa),x;K) \ f^k(r0(pa),x;K) -> Coker(i) -> 0,

(9b) 0 -> Hj>(L(n,x;K)) -> HÛn.zsK)) -> e s ^ H s H U n ^ j K ) ) -^ 0,

(9c) 0 -> 0SGSHso(L(n,x;K)) ^ H^(L(n,x;K)) ^ H ^L f r j c* ) ) -> 0,


are all split as Hecke modules by the idempotent E unless % = id and k = 2for the last two sequences. In the special case of % = id and k = 2, we have

(9d) H1(SL2(Z),K) s H*(SL2(Z),K) = H^SL^Z^K).

Now we consider the action of j = on Hp(ro(pa),L(n,%,K)). Since j0 l,

normalizes ro(pa), j acts on Hp(ro(pa),L(n,%,K)) via [ro(pa)jro(pa)]. Thenj 2 = 1. Since j{a e Atl | det(a) = njj"1 = {a e A'11 det(a) = n}, j com-mutes with T(n). Thus the eigenspaces of j

(10) H^LOi.x.K))* = {x e H1P(L(n>x,K)) | x | j = ±(-l)n+1x}

is naturally a Hecke module. When K = C, we see that the action of j is givenby co f—> jl(j*co) at the level of differential forms. This is just interpreted in termsof modular forms as f(z) h-» f(-z). Thus j brings holomorphic modular formsonto anti-holomorphic ones. Then the Krull-Schmidt theorem tells us thatH^(ro(pa) ,L(n,x,C))± is free of rank one over hk(r0(pa),%;C). By thesemi-simplicity of hk(ro(pa),%;K), the same assertion is true for all fields K ofcharacteristic 0. Thus we have, for all fields K of characteristic 0, that

(11) H1P(ro(p

a),L(n,%,K))± is free of rank one over hk(r0(pa),%;K).

§6.4. Algebraicity theorem for standard L-functions of GL(2)In this section, we prove the algebraicity result for the Mellin transform of holo-morphic modular forms which are called the standard L-functions of GL(2). Inthe following section, we construct the p-adic standard L-function of GL(2) at-tached to modular forms of weight k > 2. Thus, in the rest of this chapter, wefix a prime p and embeddings Q —» C and Q -> Qp. For simplicity, weonly deal with modular forms in 5k(ro(pa),%) for a primitive character % modulopa. This restriction is caused by our neglecting to cover the theory of primitive (ornew) forms of arbitrary level N. Since such theory is fully expounded in [M], it isstrongly recommended to the reader to carry out our construction for primitiveforms of arbitrary level (using the theory in [M]).

Here we understand that % = id and Fo(p0) = SL2(Z) when a = 0. As seenin Theorem 5.3.2, hk(ro(pa),%;C) is semi-simple, and 5k(ro(pa),%) is spannedby common eigenforms of all Hecke operators T(n). Let f be one of thesecommon eigenforms. We also know that, if we take the Q(%)-algebra homo-morphism X : hk(r0(pa),%;Q(%)) -> C given by f I T(n) = A,(T(n))f, then f

6.4. Algebraicity theorem for standard L-functions of GL(2) 187

is a constant multiple of IT A,(T(n))qn. The form f with a(n,f) = A,(T(n)) isn=l

called a normalized eigenform. We fix such a X and its normalized eigenform f.We write Q(X) (resp. Qp(?0) for the subfield of Q (resp. Qp) generated by^(T(n)) for all n over Q (resp. Qp). Let O be the p-adic integer ring of QP(X)

and put V= OTIQ(^). By Theorem 3.2, the field Q(X) (resp. QP(X)) is a finiteextension of Q (resp. Qp). We now consider a new compactification X* ofY = ro(pa)V# Since FSW is a cylinder isomorphic to the space T in §4.1, weadd to Y a circle S1 at each cusp S E S as we did in §4.1 for T at /<*> andwrite this compactification X*. This type of compactification is called theBorel-Serre compactification of Y. Then for each r e Q, the vertical line cr

connecting r and the cusp °° is a relative cycle in H1(X*,3X*;Z), where3X* = UsesS1. Identifying cr with R+ = {x e R | x > 0}, we then have anatural morphism induced from R+ = cr —> X,

(1) Intr : H* (Y,£(n,x;A)) -> H* (R+,£(n,%;A)) = L(n;A).

This morphism is realized by the integration co h-» Jcrco for closed forms co.

When ro(pa) has non-trivial torsion, we just take a normal subgroup T of finite

index of ri(pa) and we define H* (Y,£(n,%;A)) to be the image of the restriction

map in

We have a natural surjection n : H*(Y,£(n,%;A)) —> Hp(Y,£(n,%;A)). We

want to show that there exists a section (defined over K = A<8>zQ)

i : Hp(Y,j£(n,x;A)) -» Hlc(Y ,L(nf%;K)) which is compatible with Hecke op-

erators. We already know from (3.8) that n has a unique section of Hecke

modules if A is a Q-algebra. Now we let A be a subalgebra of Q or Q p

containing the integer ring of Q(X) and write K for the quotient field of A.

Thus writing E for the idempotent of the Eisenstein part Ek(ro(pa),%;A) in

Hk(ro(pa)OC;A), we can find 0 * r | E A such that K]E e Hk(ro(pa),%;A).

Since the splitting of K over K is given by this E (3.8c), we know that i = 1-E:

Hp(Y,X(n,z;A))-> Hc(Y,X(n,%;K)) satisfies 7ioi = id, and r|oi has values in

H^(Y,£(n,%;A)). Now suppose that A is a principal ideal domain. We put

H1p(Y,£<n,x;A))±[^] = {x e H^Y.^njCjA))1 | x | T(n) = X(T(n))x}.

Then this module is free of rank one over A because its scalar extension to K isfree of rank one over K (3.10). Now we choose a generator 8±(X) of the aboveA-module. Now we define a complex quantity (called a canonical period)


* (A,) e C x as follows. For f = X~_ MT(n))qn e 5k(r0(pa),x),

co(f)±(-l)n+1co(f)ljco±(A) = 2 •

define

Then 0 * <Q±(X) G H ^ Y ^ n ^ C ) ) ^ ] , and we define Qr(X) = Q*(k) by

(2) ±

For each field automorphism a of C, we always agree to choose 8+(A,°) to be5±(Xf, where Xc : hk(r0(pa),%a;Q(%)) -> C is the Q(%a)-algebra homomor-phism given by ?to(T(n)) = A,(T(n))° for all n. Anyway the period Qr(k) isdetermined up to units in A. We just fix one so that the above compatibilitycondition holds for the Galois action. Since co+(A,) is exponentially decreasing ateach cusp, we know that

(3a) Intr(i(G>±(A.))) = Jcrco±(X) = Q±a)Intr(i(8(^))) e Q±(?i)L(n;K).

Note that

(3b) ^ ( c o ^ ) ) ) n

Exercise 1. Give a detailed proof of (3a).

Now we compute the value (3b). Note that(O±(k) = 2"1(27C-V:^r)(f(z)(X-zY)ndz±f(-z)(X-zY)ndz).

This shows that

Into(co±(?i)) = 2-1iJ

This shows in particular that (-j—• :—^—e TJÂ for j = 0,1,..., n, where

the sign of QîX) is given by the sign of (-iy. Let \}/ be any primitive Dirichletcharacter modulo N. Then a computation similar to (4.1.6c) (see also Corollary5.5.1) shows

This shows

6.5. Mazur's p-adic Mellin transforms 189

Theorem 1. Let X : hk(Fo(pa),x;Z[%]) -> C be a Z[%]-algebra homomor-

phism. Suppose that either a = 0 or % is primitive. Then for each Dirichlet

character \\f and each integer j with 0 < j < n (n = k-2J, we have

and for every a e Gal(Q/Q),

where the sign of QHX) is given by the sign of (-l)V(-l). Moreover, the p-adic

absolute value I S(j,X<S>\|/) I p is bounded independently of \j/ and j .

§6.5. Mazur's p-adic Mellin transformsWe are now ready to construct p-adic standard L-functions. Such L-functionswere first constructed by Mazur for weight 2 forms in [Mzl] and [MzS]. It wasthen generalized to higher weight modular forms by Manin [Mnl,2]. For furtherstudy and conjectures concerning the materials here, we refer to the paper ofMazur, Tate and Teitelbaum [MTT]. We give an exposition of the constructionusing the method of modular symbols following the formulation in [Ki], which isquite similar to the one we have already given for abelian L-functions in §4.4.

We shall use the same notation introduced in the previous section. In particular,

X : hk(To(pa),%;Z[%]) —» Q is a Z[%]-algebra homomorphism for a primitive

character % modulo p a (we assume that % = id if a = 0). Let O be the

p-adic integer ring of QP(A,). We put K = Q(X) and A = OflK. Then A is a

discrete valuation ring and Q T ( ^ ) is well defined. We assume the following

ordinarity condition necessary to have a good p-adic L-function of X:

(Ordp) U ( T ( p ) ) | p = l .

The algebra homomorphism satisfying this condition is called "ordinary" or"p-ordinary". We can construct a standard "p-adic L-function" without assuming(Ordp) (see [MTT]). However, the function obtained is not an Iwasawa function(i.e. is not of the form O(us-1) for a power series O e 0[[T]]).

For the moment, we assume that a > 0. Recalling that cr is the relative cycle

represented by the vertical line from r e Q to /«> on the Borel-Serre compact-

ification X* of Y = Fo(pa)V^ we consider the map

(1) c : p - Z = Ur=iP" l z -> HomK(H1c(Y,i:(n,x;K)),L(n;K))

given by c(r)(co) = Intr(co).


We define for each COG H*(Y,£(n,x;K)), c© : p'°°Z -> L(n;K) by

Ca>(r)=(J 7 ) c ( r ) ( C 0 ) -

Then c<o(r+l) = Cco(r) by definition, and c© factors through Qp/Zp = p"°°Z/Z.Supposing co I T(p) = apco with | ap | p = 1, we define a distribution O© on

(2) *(B(z+pmZp) = ap-mô j)c«(-ij) for z=l ,2 , . . . prime to p.

This is well defined because c(r+l) = c(r). If we write G = Z px and fix an

isomorphism G = |ixZp with a finite group JJ., we see that

M- = (C ^ Z p X l C ^ ^ l }

where 9 is the Euler function and p = 4 or p according as p = 2 or not.Then the subgroup Ga = l+papZp corresponds to paZp. Thus (2) is tanta-mount to giving the value of the distribution O© on the standard fundamentalsystem of open sets. To show that <D actually gives a distribution, we can checkthe distribution relation (4.3.3) in exactly the same manner as in §4.4:

= [Q JXJc(x/p)(co | T(p)) = apCco(x)

and

XjTi1 ^«(x+jpm+pm+1Zp) = Ott(x+pmZ).

This shows the necessity of assuming I ap | p = 1 to have a measure, not just adistribution. By a similar argument, we see that

(3) |0) t 0(z+pmZp)|p= |ap-mfp

om j W l ) | p = | r™ ~ZJlntz/pm(co)|p is

bounded independent of z and m.

Thus <J><o is bounded, and by Proposition 4.3.2, we have a unique measure Oo,

extending the distribution &&. Projecting down to the coefficient in ( :


of Oo, we get a measure (pcoj. Now we want to show dO(o,j(x) = xMcp .o- To

show this, we may assume that co is integral (i.e. co has coefficients in

L(n,%; A)) by multiplying co by a constant if necessary). We follow the argument

given in [Ki] which originates from Manin [Mnl,2], For each <|> e (T(Zpx;A),

take a locally constant function ^ : (Z/pn(k)Z)x -> A such that | fa~ty I p < p"k

and n(k) > k. Then we know that

I P (z)zjd(Pa),o(z)(Ij1)xm-JYJ mod pk,j=o

where Cj is the coefficient of c in Xn"JYJ. Thus taking the limit makingk —> oo, we see that

(4) J W c o j = J(|)(z)zjd(pG),o(z) for all $ e C(Zpx;K).

Now we take co = i(8±(^)). Then we write O© as O 1 = O~ and compute the

integral jcjxiO1 for each primitive character ty of (Z/prZ)x. We see that

(5)


Thus projecting down to the coefficient in ( • jXm of -O*, we get by (4) a

measure cp* = cp~ satisfying, for all characters <|): (Z/prZ)x —» Kx,

(6) J ^ z f d ^ z ) = M T ( p ) ) V j G ( ^ ) ^ ) ; ^ " 1 ) if 0 < j < k-1

and <|)(-l)(-iy has the same sign as that of (p*, and J(l)(z)zjd(p±(z) = 0 if

the sign of <|>(-l)(-iy does not match.

Now we suppose that a = 0. By the ordinarity assumption, we know that one

of the roots of X -^(T(p))X+pk"1 = 0, say a, is a p-adic unit and the other one

is non-unit, because k > 2. We write b for the other root. Then we define

f (z) = f(z)-bf(pz) for f = S~ k(T(n))qn. Then f e 5k(r0(p)). It is easy ton=l

verify that f | T(p) = af and f | T(n) = k(T(n))f for all n prime to p.

Exercise 1. Give a detailed proof of the above fact. (Note the T(p) of level 1and T(p) of level p are different.)

We now use co = 8f±(A,) = ^±(^)'1co(f) to construct a measure. By the samecomputation, we get for each integer j with 0 < j < k-1

(7) J<Kz)zMcp*(z) = a / W 1 ^ ^ , ^ } i f t h e «*» o f

cj>(-l)(-iy is equal to the sign of (p1, and J(|)(z)zM(p±(z) = 0 if the sign of

(IK-lX-iy does not match the sign of (p1.

Summing up all these discussions, we get

Theorem 1. Let p be a prime, and X : hk(ro(pa),%;Z[%]) -> Q be a Z[%]-

algebra homomorphism for a primitive character % modulo p a (we a/fow that

% = id if a = 0). Then we have two p-adic measures (f>\± on Zpx satisfying

the evaluation formulas in (6) and (7).

We now define the p-adic L-function for X and a primitive character \j/ modulo

pP as follows. We take the transcendental factor either £2A+(^) or QA (X) ac-

cording to the sign of \|/(-l). We also take the measure either <p%+ or (px again

according to \|/(-l). We write our choice of Or (resp. (p*) as Q v (resp. (py)-

Then we define

= Lxv^zXzrMcpv.


Corollary 1. Let the notation be as above. In particular, let \|/ be a primitive

character modulo pp. Let X : hk(r0(pa),%;Z[%])-> Qfor k > 2 be a Z[%]-

algebra homomorphism for a primitive character % modulo pa. Then we have an

evaluation formula: if either a > 0 or $>0,then

a = (3 = 0, then

where a is the unique p-adic unit root of the equation X

For further study of this type of p-adic L-functions, see [Ki] and [GS].

Chapter 7. Ordinary A-adic forms, two variable p-adicRankin products and Galois representations

A typical problem of p-adic number theory is the problem of p-adic inter-polation, which can be stated as follows:

For a given complex analytic function f(s) whose values at infinitely many integerpoints k are algebraic numbers, is there some p-adically convergent power seriesF(s) (with coefficients in a p-adic field) of p-adic variable s such that F(k) = f(k)for all integers k such that f(k) is algebraic?

Many successful answers to this problem have already been discussed in Chapters3, 4 and 6. Instead of taking complex analytic functions, we take complexanalytic modular forms here and consider the problem of p-adic interpolation. Wepresent, in this chapter and Chapter 10, some recent developments in the theory ofp-adic modular forms, in particular, (i) p-adic analytic parametrization of classicalmodular forms, (ii) p-adic L-functions attached to each p-adically parametrizedfamily of modular forms and (iii) Galois representations of these families ofmodular forms. Let us briefly explain what the p-adic family of modular forms is.As already studied in Chapter 2, a typical example of modular forms is given bythe absolutely convergent Eisenstein series (see Chapter 5):

Ek(z) = 2-1C(l-k) + Xr=1CTk-i(n)qn (k> 2),

where crm(n) = X0<d | n^m *s ^ e s u m °^ m~t n powers of divisors of n. Wemodify tfk-i(n) by removing the (k-l)-th powers of the divisors d divisible byp. Then the modified coefficient G^Vifa) = Zo<d|n,(d,p)=idk~1 depends p-adiccontinuously on the weight k; in fact,

if k EE k' = 0 mod p^Cp-l) , then G(p)k_i(n) = G(p)

k--i(n) mod p a .Thus for a fixed integer n, we can consider the function k h-> o^pVi(n) as therestriction of a continuous function o^s.i(n) of the variable s which varies in thep-adic integer ring Zp. Actually, this dependence on the weight k is p-adicanalytic; that is, the continuous function which induces s h-» a(p^s_i(n) can beexpanded at each s e Zp into a power series p-adically convergent on aneighborhood of s. Then the formal Fourier expansion

may be considered as a solution to the problem of p-adic interpolation for theEisenstein series Ek, where £p(s) is the p-adic Riemann zeta function given inTheorem 3.5.2. Note that E(k) = Ek(z)-pk"1Ek(pz) because of the modificationto a(p)

k_i(n) from ok_i(n).

7.1. p-adic families of Eisenstein series 195

For the moment, let us define naively a p-adic family of modular forms {fk} to bean infinite set of modular forms parametrized by the weight k whose Fouriercoefficients depend p-adic analytically on the weight k. Later, we shall give amore precise definition. In fact, in the case of {Ek}, the coefficients are integersand thus can be considered as complex numbers as well as p-adic numbers auto-matically. In the general case, the coefficients of fk may not be just integers, andin fact there are many examples of such families with algebraic Fourier coef-ficients. With this general case in mind, we fix, once and for all, an algebraic clo-sure Q p of the p-adic field Qp and an embedding of Q into Qp. Thus wecan discuss the p-adic analyticity of Fourier coefficients of fk relative to k.

To each modular form f = XI=o an<ln> w e n a v e associated in Chapter 5 an

L-function L(s,f) = XI=i anfl~s, and for each pair of modular forms f and

g = SI=o bnqn, we have another L-function Z)(s,f,g) = XI=i ant>nn~s- Oncesuch a p-adic family of modular forms {fk} is given, it is natural to ask theproblem of p-adic interpolation of the values (or more precisely their algebraic part)of {L(m,fk)} and {D(m,fk,f/)} by varying the weight k. We shall treat thisproblem for D(s,f,g) later in this chapter and in Chapter 10 (as for the treatment forL(m,fk), see [Ki] and [GS]). As for the p-adic interpolation of Galois repre-sentations, often one can canonically attach a Galois representation n^ (ofGal(Q/Q)) into GL2(QP) to each element fk in the family. Then we may alsoconsider the problem of p-adic interpolation of the function k h^ 7Ck(a) e Mfe( Qp)for any fixed a e Gal(Q/Q). If one succeeds in interpolating the functionk h-> Ttk(tf) for every a, one may eventually obtain a big Galois representationinto the matrix ring over the ring of analytic functions on Zp. We shall formulatethis problem more clearly later, in §7.5.

§7.1. p-Adic families of Eisenstein seriesHere, we study the p-adic family of modular forms given by Eisenstein series. Wewrite F for Fo(N) or Fi(N). Since we have already fixed embeddings of Qinto Q p and C, any algebraic number in Q can be regarded as a complexnumber as well as a p-adic number. We fix a base ring O, which is the p-adicinteger ring of a finite extension of Qp. Sometimes we need to consider the com-pletion £1 of Q p under | | p (which is known to be algebraically closed). Wefix a character \|/ = coa of (Z /pZ) x (p =4 when p = 2 and p = potherwise) for the Teichmiiller character co. We mean by a p-adic analytic family

(of character \\f) an infinite set of modular forms {fk}°° for some positive in-k=M

teger M satisfying the following three conditions:

196 7: A-adic forms, Rankin products and Galois representations

(Al) fk(A2) a(n,fk)e Q for all n,(A3) there exists a power series A(n;X) e O[[X]] for each n > 0 such that

a(n,fk) = A(n;uk-1) for all k> M,

where u = 1+p (which is a topological generator of the multiplicative groupW=l+pZ p ) . The family {fk} is called cuspidal if fk is a cusp form for almostall k (i.e. except finitely many positive k). Note that uk-l = (l+p)k-l isalways divisible by p, and hence |uk-l lp < 1. The convergence of A(n;uk-1)follows from this fact.

We now introduce the space of p-adic modular forms. First, we already know (seeTheorem 5.2.1, Corollary 5.4.1, Theorem 6.3.2, Corollary 6.3.1) that if k> 2

(1) *4(ro(ppa),x) = ^k(ro(ppa),%;Z[x])®z[X]C and

The assertion (1) holds for any subring A of C containing Z[%]. Thus we can

define the space fWk(ro(ppa),%;A) for any ring A (inside Q or Qp) by the

right-hand side of (1). Since ^4(ro(ppa),%;Z[%]) is naturally embedded into the

power series ring Z[%][[q]] via q-expansion, we can regard #4(ro(ppa),%;A)

as a subspace of the power series ring A[[q]]. For each fe fA4(ro(ppa),%;A),

its q-expansion will be written as f(q) = Xr=o a(n,f)qn.

Let A be the one variable power series ring O[[X]] with coefficients in O. We

call a formal q-expansion F(q) = Z°° A(n,F;X)qn e A[[q]] a A-adic form of

character \j/ if the following condition is satisfied:

(A) the formal q-expansion F(uk-1) gives the q-expansion of a modular form

in f^4(ro(q),\|/co'k;0) for all but finitely many positive integers k.

This is the definition of A-adic forms given in [Wil]. A A-adic form F is called aA-adic cusp form if F(uk-1) is a cusp form for almost all k. We will see later thatthere exists a A-adic cusp form which specializes to a non-cuspidal form at k = 1.By our definition, to give a p-adic family of modular forms {fk} is to give asimultaneous p-adic interpolation of their Fourier coefficients by the power seriesA(n;X). That is, by evaluating the p-adic analytic functions A(n;us-1) at integersk > a, we get the n-th Fourier coefficient of the modular form fk. When wedefined a p-adic analytic family {fk}, we required an extra condition thatfk = F(uk-1) is a classical (complex analytic) modular form for almost all k.


Thus a A-adic form F gives rise to a p-adic family {F(uk-1)} if F(uk-1) is aclassical form for all but finitely many k.

Now we want to construct an example of a p-adic family out of the set ofEisenstein series {Ek}. Since the n-th coefficient of Ek for positive n is a sumof (k-l)-th powers of divisors of n, we first show, for a positive integer a primeto p, the existence of a power series O(X) such that O(uk-l) = ak (u = 1+p)for integers k. We consider the binomial power series:

As seen in Chapter 3, (1+X)S is a power series with coefficients in Zp. Thispower series converges in the interior of the unit disk. Thus we can define thep-adic power 7s = (l+(y-l))s (for y€ l+pZp = W) with exponent s e Zp

and a morphism from the additive group Z p into the group of one-unitsW = l + p Z p by

()

As seen in §1.3, the p-adic logarithm function log induces an isomorphismW = pZp. As a power series, we have an identity log((l+X)s) = slog(l+X);thus, also as a map, log(us) = sZog(u). Note that I log(z) I p < I p I p for anyz G W and I log(\x) I p = I p I p * 0. Therefore, for any given z e W, byputting s(z) = log(z)/log(u), we have s(z) e Z p . Thus s : W = Z p .Therefore we can write z = us(z) = (l+(u-l))s(z). Thus if an integer d satisfiesthe congruence d = 1 mod p, then we can write d = us(d), and for the powerseries

Ad(X) = d

we have Ad(uk-1) = d"V(d)k = dkA. Therefore Ad(X) has the desired prop-

erty for d when d= 1 mod p. To treat the case of d not congruent to 1mod p, we use the decomposition Zp

x = WX|LL introduced in §3.5 and the as-sociated projection X H ( X ) = © ( X ) " ^ of Zp

x to W. For each integer dprime to p, we put

Then we know that Ad(uk-1) = d V ^ * = d-1<d)k = ©(d^d*"1. In particular, if

k = 0 mod (p(p), then Ad(uk-1) = dk~l. We now define for 0 < n e Z andfor each even Dirichlet character \|/ = coa of [i

Av(n;X) = £ 0 < d , n (pd)=1 V(d)Ad(X).


This power series Av(n;X) is quite near to the power series we wanted to find,since Ay(n;uk-1) = G^k-i(n) if k=amod(p(p) for the Euler function (p.More generally, we have

A¥(n;uk-1) = Xo<d|n, (P,d)=iV^"k(d)dk-1 = ck-i,v<D-k(n) for all k > 0.

Now we consider the p-adic interpolation of a constant term. We consider theDirichlet L-function

As seen in §3.6, we have that

(2a) There exists a power series OV(X) in Zp[[X]] for each character \j/ of

(Z/pZ)x with \|/(-l) = 1 such that for all integer k > 1

k | ( l - \ | /co-k(p)pk-1)L(l-k,Vco-k) if V * id,V " t<E>id(uk-l) = (uk-l)(l-co-k(p)pk"1)L(l-k,co-k) if \j/ = id.

Let A = O[[X]] and define the A-adic Eisenstein series E(\|/)(X) e A[[q]] foreach even character \\f = coa with 0 < a < p-1 by

where Av(0;X) = O¥(X)/2 if y = coa * id and Aid(0;X) = Oid(X)/2X.

Proposition 1. For each positive even integer k > 2 with k s a mod 9(p),we have

E(\|/)(uk-l) = Ek(z)-pk-1Ek(pz)=Ek(lp)€ fMk(ro(p)) in Q[[q]],

where ip w f/ze trivial character modulo p. M<9r generally, for each non-trivial

Dirichlet character % : (Z/papZ)x -^ Q x w/f/z % \ ^ = \\f and for k > 1, we

have E(\i/)(%(u)uk-l) = Ek(%co"k) e f^k(r0(pap),%co-k).

This proposition shows that we get the classical Eisenstein series even from thespecialization at %(u)uk-l with %(u) ^ 1, which is not included in the definitionof p-adic analytic families and A-adic forms. When \|/ = id, E(\|/) is not a A-adicform because it has a singularity at X = 0. However XE(\j/) is a A-adic form.

Proof. Assume that k = a mod cp(p) for the Euler function (p. Write

Ek(z) - ^ X~Then we have

( lp^^CClk)^ = (\^-l)L(\kM?(i>*)l2 = Ava0 = (l-p^^CCl-k)^ = (\-^-l)L(\-k,M?(i>*)l2 = Av(0;uk-l).


As already seen, an = crk_i(n) = Av(n;uk-1) if n is prime to p. When n isdivisible by p, then

= Av(n;uk-1).

This shows the identity of the power series as in the lemma. Note that

pk"1Ek(pz) = E k | k [ 0 A. Thus Ek(z) - pÊkCpz) is a modular form for

OVi (p

One sees easily that T contains Fo(p). The general case of non-trivial character%co"k is much easier. In fact, writing s(d) = s((d)), we see from %(u)s^d)

= %((u»s(d) = X«d» thatAv(n;x(u)uk-1) = Xd-V(d)(l+x(u)uk-l)s(d) = X x ^ d ^ " 1 = ok.1)Xt0-k(n).

0<d|n 0<d|n

As seen in Theorem 3.5.2, if %co"k is non-trivial and % I \i ~ V» w e n a v e

(2b) Av(0;x(u)uk-l) = 2"1L(l-k,xco"k).

This shows the assertion in the general case. Strictly speaking, we have onlyproven the proposition when k > 2 because the Fourier expansion of theEisenstein series was not yet computed for k = l and 2 in §5.1. This will bedone in Chapter 9.

Now we can produce many p-adic families of cusp forms by using E(\j/). In fact,

we take a modular form f e Mm(ro(pap),%;0) for a character % having values

in O and make the product fE(\j/)(X) inside A[[q]]. Then we have

fE(\|/)(uk-l) e f kH

Lemma 1. For u e Ox and v e O with I v | p < 1, the substitution

A(X) \-> A(uX+v) gives a ring automorphism of A = O[[X]].

Proof. For A(X) = £" = Q anXn, we see that

A(uX+v)= S i c X-{£:=m a ^ v - Q } .The inner infinite sum is absolutely convergent p-adically, since v is divisible byp. Thus A(uX+v) is a well defined power series. The substitution of u^X-u^vfor X gives an inverse map, and hence A(X) h-» A(uX+v) is a ringautomorphism.

Let f e ^ m (F 0 (pp a ) ,x ;O) . Expanding ffi(\|/)(X) as £1=0 an(X)qn and

defining a new series (called the convolution product of f and E(\|/)) by


F(X) = f*E(\|0(X) = X r = o an(%-1(u)uwe know that

F(uk-l) = fE(\|/)(x"1(u)uk-m-l)e ^k(ro(pap),XoVCO-k) for all k > m ,

where %0 is defined by writing % as the product e%0 for characters e of W

and %Q °f M- I*1 particular, if f is a cusp form of level p, we obtain A-adic cusp

forms in this way.

So far, we have constructed A-adic Eisenstein series using Ek(\|/). We now wantto do the same thing using Gk(\|/). This is easier in fact, because Gk(x) does nothave a constant term usually (see Proposition 5.1.2). We then define

(3a) B\|/(n;x) = S0<d|n>(dp)=1\|/(n/d)Ad(X) for n prime to p

and G(\|/)(X) = £1=1 B¥(n;X)qn.

Then, when % | p. = \|/, we have Bv(n;%(u)uk-1) = Gâ-kin), and we have

another p-adic analytic family of modular forms G(\|/) such that, for each charac-

ter % modulo pap,

(3b)

= Gk(z;x©-k)-pk-1Gk(pz;x©-k) e fA4(ro(pa+1p),coa-k) if k> 1.

§7.2. The projection to the ordinary partIn this section, we first define the idempotent attached to T(p) which gives theprojection to the "ordinary" part. We have defined for each characterX : (Z/pa Z) x -> Ox

5k(r0(pa),x;o) = %

We may regard these spaces as the O-linear span of f^k(ro(pa),X»Z[x]) orA(ro(pa),X;Z[x]) in 0[[q]]. Then the Hecke operators T(n) act on these spacesand satisfy the formula (5.3.5) describing their effect on coefficients of qn. Inparticular, we have the O-duality between the Hecke algebras and the spaces of

cusp forms. When x is primitive modulo p a or F = SL2(Z), the Hecke alge-

bra Hk(ro(pa),x;Q(X)) is semi-simple and hence

Hk(r,x;QP(x)) = Hk(r,x;Q(x))®Q(%)Qp(x)

is again semi-simple. Thus Hk(F,x;Qp(x)) = IIxQp(^), where X runs over

conjugacy classes in HomQp(%)_aig(Hk(r,x;Qp(x)),Qp) under Gal(Qp/Qp(x)).

7.2. The projection to the ordinary part 201

Proposition 1. Let K be a finite extension of Qp(%). If f is a common eigen-

form of all Hecke operators in #4(ro(pa),%;K) normalized so that a(l,f) = 1,

then f is actually a complex common eigenform in a

Proof. We define a K-algebra homomorphism X : Hk(ro(pa),%;K) -» K byf |h = k(h)f. Then a(n,f) = k(T(n)). Since we have

Hk(r0(pa),%:K) = Hk(r0(pa),%:Q(%))(E)K,

we can restrict X to Hk(r0(pa),%:Q0c)). Since Hk(r0(p

a),%:Q(%)) is of finitedimension over Q, X in fact has values in Q. Since

Hk(r0(pa),x:C) = Hk(r0(pa),%:Q(%))®C,

we can extend X to a C-algebra homomorphism of Hk(ro(pa),%:C) into C.

Then by the duality, we can find f in ^ k ( r 0 ( p a ) , % ; C ) such that

a(n,f) = ?L(T(n)) = a(n,f). Thus f = f e fAfk(r0(pa),%;C).

Lemma 1. Let K be a finite extension of Qp and O be its p-adic integer ring.For any commutative Oalgebra A of finite rank over O and for any x e A , thelimit lim x11" exists in A and gives an idempotent of A.

n—»°

Proof. First assume that A is the p-adic integer ring of a finite extension of K.Let p be the maximal ideal of A and write pf for #(A/p). Then we have#((A//)x)=pr"1(pf-l). Therefore for any x e Ax, x P 6 ^ s 1 m o d / . Thisshows that the limit of {xpfh^pf"^} as n ^ o o exists in A and is equal to 1 forx e Ax. Therefore lim xn! = lim x ^ ^ ^ = 1 for x e Ax. When x is in

n—><» n—><x>

p, then obviously, the above limit vanishes. Now we proceed to the general case.If the scalar extension A®0K is semi-simple, then it is a product of finite exten-sions of K and the image of x in each simple factor is contained in the p-adic in-teger ring of the factor. Then applying the above argument, we know the existenceof the limit, which is an idempotent. If the nilpotent radical of A is non-trivial,write x = s + n in A<8>0K with s semi-simple and n nilpotent. (By a theo-rem of Wedderburn, this is always possible.) For sufficiently large f, lim s1^^"1)

n—»°°

exists in the subalgebra 0[s] of A generated over O by s, since O[s] has nonon-trivial nilpotent radical and is of finite rank as Omodule because it is the sur-jective image of A. On the other hand, if rf = 0, then

202 7: A-adic forms, Rankin products and Galois representation

Note that ft ")= pfr!/(pfr-i)!i!. Since pfn! is divisible by (pfr-i)!pfr and

0 < i < j , I ft ]lp< Cp~fr for a constant C independent of r. This shows

that the term 2Ji=l K pp n vanishes after taking the limit, and we know the

existence of the limit e = lim xn! = lim s11', which is an idempotent.n—>oo n—><»

Let K be a finite extension of Qp(x)« We now define the ordinary projector e of

the Hecke algebra Hk(r0(pa),x;O) by e=l im T(p)n!. We see easily that if fn—»oo

is an eigenform of T(p) with eigenvalue a, then

(1) 10 i f I a l p < 1 .

We say a p-adic modular form f is ordinary if f | e = f. There are examples ofordinary forms and non-ordinary forms. For any character % modulo p a

( a > 0 ) , we have cm,x(p) = l+x(P)Pm = 1 and a'm,%(p) = Pm+X(p) = Pm-Thus Gk(%) I e = 0 and Ek(x) | e = Ek(%) if k > 1. Now we define theordinary part of the Hecke algebras and the spaces of modular forms by

d = eHk(r0(pa),%;O), h°k

rd(r0(pa),x;O) = ehk(r0(pa),x;O),

= ^k(r0(pa),x;o) I e, 5^rd(r0(pa),x;o) = 5k(r0(pa),x;o) I e.

By definition, Hkrd(ro(pa),x;^) ^s t n e largest algebra direct summand of

Hk(To(ipa),%;0) on which the image of T(p) is a unit. Again by definition,

<hh\f> = a(l ,f |hh') = <h' , f |h) for h,h' e H k ( r 0 (p a ) ,x ;O) and f ea ) . Thus this pairing induces isomorphisms:

,X;0),0)= m°krd(r0(p

a),X;O) and

One of the fundamental theorems in the theory of ordinary forms is

Theorem 1. Suppose k > 2. Then we have

rankol^rd(r0(pa),X0)-k;O) =

= ranko52ord(r0(p

a),XCB"2;O).


Here % is any character modulo p a primitive or imprimitive. The proof of thisfact will be divided into two steps: (i) the first step is to show that the rank asabove is bounded independently of k, and (ii) the second step is to show that therank is equal for all k. Putting off the second step to the next section, we firstprove the boundedness of the rank by cohomological means. SinceHk(r0(N),%;A) is a residue ring of Hk(Ti(N);A) by (6.3.2b), the boundedness ofthe rank follows from that of H k ( r i (N) ;Z p ) . Since Hk(Fi(N);C) =0xHk(ro(N),x;C) is the C-dual of M&iQi)) = e x ^ k ( r o (N) ,x) , by (6.3.2b),it is sufficient to prove the assertion for sufficiently large N. Thus the bounded-ness follows from

Theorem 2. Let N be a positive integer prime to p. Then the integer

rankZp(h°rd(ri(Npa);Zp)) is bounded independently of k * / k > 2 and a > l .

Proof. Write T for r i (Np a ) . Let L be the intersection of the image L' ofH ^ L f a Z ) ) in H ^ L f o R ) ) with Hj>(I\L(n;R)). Then L is a lattice ofHp(F,L(n;R)), and hk(F;Z) for k = n+2 is by definition a subalgebra ofEndz(L) which is free of finite rank over Z. Let Lp = L®zZ p . Thenhk(r;Zp) = hk(r,Z)<g>zZp is a subalgebra of EndZp(Lp). Thus h£rd(r;Zp) is asubalgebra of Endzp(eLp) for the idempotent e attached to T(p). Thus what wehave to prove is the boundedness of the rank of eLp independent of n. The exactsequence of F-modules

0 -> L(n;Z) ^> L(n;Z) -> L(n;Z/pZ) -» 0yields the cohomology exact sequence

Therefore H1(r,L(n;Z))®Z/pZ can be embedded into H^rjLfaZ/pZ)). Notethat L/pL = Lp/pLp, L/pL injects into L'/pL1, and L'/pL' is a surjective imageof

H1(T>L(n;Z))/pH1(r,L(n;Z)) = H1(r,L(n;Z))(8)zZ/pZ.

Thus it is sufficient to show that the dimension of eH!(r,L(n;Z/pZ)) is boundedindependently of n. We shall show this fact by constructing an embedding of

eH^rjLfaZ/pZ)) into e H ^ Z / p Z ) .

Note that, for P(X,Y) = I ^ a i X ^ Y 1 e L(n;A),

x P(X,Y) = £ a^X-mYf-V andhence P (1,0) = P(l,0).J i=o ^ ^ ^


Let us define maps i : L(n;Z/pZ) —> Z/pZ and j : Z/pZ —» L(n;Z/pZ) by

i(P(X,Y))=P(l,0) and j(x) = xYn. Since Y = L j mod p for any

y e r = ri(Npa), i and j are homomorphisms of F-module. Thus combiningi or j with a 1-cocycle u, we obtain the following two morphisms of cohomol-ogy groups:

I = i* : H^rjLfoZ/pZ)) -» H ^ Z / p Z ) and

j * : H^F, Z/pZ) -> H^F, L(n;Z/pZ)).

We want to show that I is an isomorphism of oH1 (T,L(n;Z/-pZ)) onto^ ) . We consider the exact sequence of F-modules

0 -> Ker(i) -» L(n;Z/pZ) -» Z/pZ -> 0.This yields another exact sequence:

H^r.KerCi)) -» H^r .LfaA)) -> H ^ Z / p Z ) -> H2(r,Ker(i)).

(1 (A t

Note that for a = L. L a leaves Ker(i) stable and hence T(p) acts natu-

rally on Hq(F,Ker(i)). On the other hand, the Z/pZ-module Ker(i) is generated

by monomials X^Y 1 for i > 0 . Thus for a = , the action of a1 is

nilpotenton Ker(i). Since r a r = U£d1ra , the action of T(p) is nilpo-

tent on Hq(F,Ker(i)) for q > 0. This shows the desired assertion.

We shall give another proof of the theorem when a = 1. When a = 1, wemodify j * and construct a map J : H^F, Z/pZ) -» H!(r , L(n;Z/pZ)) so that

J°I = (-l)nT(p) on H^FiCNp), L(n;Z/pZ)). Thus (-lfTip)'1] actually givesthe inverse of I on eH^F^Np), Z/pZ). We consider the double coset F8F for5 E SL2(Z) such that

(0 - n8 = L 0 j mod p and 8 = 1 mod N.

One can always find such an element (see Lemma 6.1.1). Then we have a disjointdecomposition

r s r = Uf1 r55i ** % = (J J).

Similarly we take x e M2(Z) with det(x) = p such that

0 J mod P and T ^ [ o p j m o d N'


We can find such x as follows. By Lemma 6.1.1, we can find a e SL2(Z)

0 -\\such that G = 1 mod p2 and

a s mod N2.

Then x = Q a does the job. Then one can easily verify that x normalizes

F and induces an automorphism of HÎ^Z/pZ) = Hom(F,Z/pZ) which takesu : F -^ Z/pZ to u | [x](y) = uCxyx"1). Note that

because FxSSi = 0 mod Np and det(x88i) = p. Define J to be

[r8noj*°fr3. We compute J°I. Let u : F -> L(n;Z/pZ) be a 1-cocycle. Then if

5iY = Yi55j for y,yi e F, then xSSiy = xyiX^xSSj with xyix"1 e F. Thus

Note that j(i(P(X,Y))) = a0Yn = (-l)nxlP(X,Y). Thus

yiX"1) = (-l)nu I T(p).

Thus, we have J°I = (-l)nT(p), giving another proof of the theorem.

Here we add one more result for our later use in §10.4, which is a special case of[HI, Th.3.2]:

Proposition 2. If k > 2 and p > 5, then e induces

e : rd(SL2(Z);Qp) = 4rd(ro(p);Qp), e : f^rd (SL2(Z);Qp) = ^ r d (T0(p);Qp),where the ordinary part for SL2(Z) is defined with respect to the Hecke operator

T(p) of level 1.

Proof. We start a general argument. We consider anyT = F(N) = {y e SL2(Z) | y = 1 mod N} for N > 3.

Then F is torsion-free. Now we put Fo = FflFoCp). Then we have

I1 = lores: H^F^n jZ /pZ) ) -+ ^(ToM^Z/pZ)) -> H

Since we can show by the strong approximation theorem (Lemma 6.1.1) that

r(o p)r = U o s u <pr(o T ) U r a l o i) (see [Sh'


for a G SL2(Z) with a = mod N, we know thatV ° V)

°Y l l , fP 0>l1J|P) = (alP)((l,0)[0 J) = (alP)(0,0) = 0 (if n>0)

for any homogeneous polynomial P with coefficients in Z/pZ. Since T(p) oflevel N and T(p) of level Np are different, we add the subscript "N" to indi-

i i i fP 0>lcate the level. Then f | TN(P) = f I TNP(P) + f I ol Q X I. By the above argument,

the term corresponding to a I Q A does not affect to the value of i*. Similarly

we consider the trace map as in §6.3:

Tr = [r0l2r] : HHroJL^Z/pZ)) -> HHlM^Then we consider

j ' = Trojôx : H^roJLCO.o^Z/pZ)) -> H^roJLfoZ/pZ)) -»

Then basically by the same computation as in the case of a = 1, we have

J'or = (-l)nTN(p).

Let us now compute I'OJ'. We pick a cocycle v e Z(r0,L(0,co"n;Z/pZ)). Then,noting that T = IIo<u<pro§5uIlro, we see that

r(J'(v))(y) = 2Li(5ulj(v(yu)))+i(j(v(7))) = £i(5u

lj(v(yu))) = v | TN(p)(y),u=0 u=0

where 88uy = yua for an a in {88V I v = 0,...,p-l }U{ I2} for the identitymatrix I2. Here we have used the fact that i°j = 0. By this, we have

(2) H^rd(r,L(n;Z/pZ)) s H^rd(r0,L(0,co-n;Z/pZ)) = H^rd(ro,L(n;Z/pZ)).

Now we consider the cohomology sequence attached to

0 -> L(n;Zp) —^-» L(n;Zp) -> L(n;Z/pZ) -> 0,

which gives rise to, for O = T and To,

(3) 0 -> Hi(O,L(n;Zp))®zZ/pZ -> HÔ.UnjZ/pZ)) -> Hi+1(O,L(n;Zp))[p] -> 0,

where H2(O,L(n;Z))[p] is the kernel of the multiplication by p onH2(<D,L(n;Z)). By Proposition 6.1.1, we know that H2(O,L(n;Z))[p] = 0.Thus we see that

(4) HÔJL ^


Note that H°(O,L(n;Z/pZ)) = L(n;Z/pZ)^ may be non-trivial. We have the mapinduced by the projector e attached to T(p) of level p:

e :

Then (2) shows that e is an isomorphism after reducing modulo p. Then byNakayama's lemma, we know that the map e is surjective. If x is an element in

I (\ NuYL(n;Z/pZ), then x | T(p) = Xo<u 0 because p = 0 in Z/pZ. Thus, we know

(I NuYn ^ c o n s*s t s on^y of terms involving Y.

This shows x | T(p)2 = 0 and hence H°rd(O,L(n;Z/pZ)) = 0 because e =

limT(p)n!. Applying the operator e to the exact sequence (3) for i = 0, we have

H^rd(O,L(n;Zp))[p] = 0.

Thus Hord(0,L(n;Zp)) is torsion-free. Therefore we conclude from (2) and (3)that e induces an isomorphism:

(5) H^rd(r,L(n;Zp)) = Hird(r0,L(n;Zp)).

Note that G = SL2(Z)/r = r o (p ) / r o = SL2(Z/NZ). If N is a prime, then#(SL2(Z/NZ)) = N(N+1)(N-1). Since p > 5, we can always choose N so that#(SL2(Z/NZ)) is prime to p. Then it is well known that

res: H1(SL2(Z),L(n;Zp)) = H°(G>H1(r,L(n;Zp))),

res: H1(r0(p),L(n;Zp)) = H0(GfH1(r0,L(n;Zp))).

This combined with (5) shows that e induces

H1Md(SL2(Z),L(n;Zp)) = H^rd(ro(p),L(n;Zp)).

By this, we know that d im^ r d ( r o (p ) ;Q p ) = dim^r d(SL2(Z);Qp) , whichshows the assertion for f&4- Then the assertion for 5k follows from that for M^.


Corollary 1. For all fields A of characteristic 0, the algebra h£rd(Fo(p);A) is

semi-simple if k > 2 and p > 5.

In fact, the assertion is true even for k = 2. This fact follows from the fact that52(SL2(Z)) = 0 and [M, Th.4.6.13]. In this case, if f e 52(r0(p)) is anormalized eigenform, then f is primitive in the sense of [M, §4.6] and

f | x = ± p ( k - 2 ) / 2 f c f o r x = ^ " ^

Proof. We know that hk(SL2(Z);C) is semi-simple. Thus h°rd(SL2(Z);Qp) is

semi-simple. Thus we can find a basis {fi, ..., fr} of 5£rd(SL2(Z);Qp)

consisting of common eigenforms of all Hecke operators. Let f be one of them.

We write f | T(p) = af. Then | a I p = 1. We take roots a and p of

X2-aX+pk"1 =0 . Then one of a and p, say a , is a p-adic unit, i.e.

I oc | p = 1. Then | p | p = p 1 " k < l . We define f = f(z)-pf(pz). Then it is

easy to check by the formula (5.3.5) that

f I T(n) = a(n,f)f if n is prime to p and f | T(p) = af.

Thus f is an ordinary form. As shown in the proof of Theorem 5.3.2, ifa(q,fi) = a(q,fj) for all primes outside p, then i = j . This implies fi', ..., fr'are linearly independent. Then by Proposition 2, they form a basis of5krd(ro(p),Qp). Therefore h£rd(SL2(Z);Qp) is semi-simple, which shows theassertion.

§7.3. Ordinary A-adic formsIn this section, we study the structure of the space of ordinary A-adic forms fol-lowing the method of Wiles [Wil]. Actually the space of A-adic forms is theA-dual of the p-ordinary Hecke algebra of level p°° defined in [H3] and [H4].Then all the results concerning the structure of the space of ordinary A-adic formsfollow from the structure theorem of the ordinary Hecke algebra proved in [H3]and [H4]. However, we have adopted the method of Wiles, which is more com-pact. We ease (in appearance) a little bit the requirement (A) to be a A-adic formgiven in §7.1: for each character % modulo pap (which may not be primitive), aformal q-expansion F(X;q) = Z°° a(n;F)(X)qn with coefficients in A = O[[X]]

n=0

is called a A-adic modular form F(X) of character % (with values in (?) if thefollowing condition is satisfied: for the generator u = 1+p

(A1) F(uk-l;q)e #4(ro(pap),xco'k;0) for almost all positive k (i.e. all butfinitely many positive k).

7.3. Ordinary A-adic forms 209

When F(uk-l;q) is a cusp (resp. a p-ordinary) form for all sufficiently large k,we say that F is a cusp (resp. an ordinary) form. Let M = M(%;A) (resp.S = S(%;A), M o r d = Mord(%;A), Sord = Sord(x;A)) be the A-module of allA-adic modular (resp. cusp, ordinary modular, ordinary cusp) forms. To intro-duce a Hecke operator on M and S, we consider the characterK : W = 1+pZp -> Ax given by K(US) = K(US)(X) = (1+X)S. It is obviouslya continuous character with respect to the m-adic topology on A, where m is themaximal ideal of A. Note that for integers n prime to p, K((n))(uk-1) =K(us(n))(uk-l) = uks(n) = 0)-k(n)nk, where we write (n) = co(n)-!n = us(n) (s(n) =log((n))/log(u)). Then we define for each A-adic form F e M(%;A) a formalq-expansion F | T(n) by

(1) a(m,F | T(n))(X) = £ b | ( m n ) K«b))(X)%(b)b-1a(mn/b2,F)(X),

where b runs over all common divisors prime to p of m and n. We evaluatethis formal power series F|T(n) at uk-l where F(uk-l;q) is meaningful as amodular form. Then we see that

a(m,F | T(n))(uk-1) = £ b | ( m n ) K((b»(uk-l)%(b)b-1a(mn/b2,F)(uk-l)

= Xb|(m,n) Xa)-k(b)bk"Vmn/b2,F(uk-l)) = a(m,F(uk-l) |T(n)).

This shows that F | T(n)(uk-1) = F(uk-1) | T(n) e ^k(r0(pap),%co-k;0).Therefore, F is again a A-adic form. Thus, the operator T(n) is well defined andso we now have Hecke operators T(n) acting on M and S and their ordinaryparts.

Lemma 1 (Weierstrass preparation theorem). Any power series F(X) in A canbe decomposed into a product of a unit power series U(X), some power of aprime element in O, and a distinguished polynomial P(X) e O[X]. (A polyno-mial P(X) = ao+aiX+---+Xn is called "distinguished" if | a i l p < l for all i.)

Since this fact can be found in any book in commutative ring theory or p-adicnumber theory (for example [Bourl, III], [L, V.2], [Wa, Th.7.3]), we omit theproof. By this lemma, each non-zero power series with coefficients in O has only

finitely many zeros in the disk { x e O | x | p < 1}.

Theorem 1 (A. Wiles). The space of ordinary A-adic modular forms (resp.ordinary A-adic cusp forms) of character % is free of finite rank over A.


Proof. The proof is the same for Mord and Sord. We shall give a proof only forMord. We prove first that Mord is finitely generated and is A-torsion-free. Bydefinition, Mord is a A-submodule of the power series ring A[[q]]. Therefore itis A-torsion-free. We now prove that the rank of any finitely generated freeA-submodule M of Mord is bounded. Let {Fb F2, ••• , Fr} be a basis of Mover A. Since Fi, "-.Fr are linearly independent over A, we can find positiveintegers ni, ..., nr such that D(X) = det(a(nj,Fj)) * 0 in A. By the abovelemma, we can take the weight k so that D(uk-l)*0 and Fi(uk-1) has mean-ing, that is, is an element of f^£rd(ro(paP)>%G)'k;0) for all i. Write £ forFi(uk-1). Then D(uk-1) = det(a(ni,fj)) * 0. Thus the modular forms fi, ..., frspan a free module of rank r in fAf£rd(ro(pap),%co'k;0) whose rank is boundedindependently of the weight k. Thus r is bounded by a positive number indepen-dent of M. This shows that if Fi, • • •, Fr is a maximal set of linearly independentelements in Mord, any element in Mord can be expressed as a linear combinationof the Fi's if one allows coefficients in the quotient field L of A. We thusconsider V = Mord®AL, which is a finite dimensional space over L embeddedin L[[q]]. For each F e Mord, write F = Si xiFi with xi e L. Then xi isthe solution of the linear equations (a(ni,Fj))x = (a(ni,F)) e Ar. ThereforeDxi G A, and thus DMord is contained in AFi + ••• + AFr. Therefore Mord isfinitely generated since A is noetherian. To prove the freeness over A, we notethe following facts:

(i) A is a unique factorization domain;

(ii) A is a compact ring.

The first fact follows easily from Lemma 1 (see [Bourl, VII.3.9]). The secondfact follows from the fact that for the maximal ideal m of A, A/ni1 is always afinite ring and we have topologically A = lim (A/ma) = lim (A/Pa) for any

non-trivial element P in nu Since Mord is finitely generated, we can find k sothat F(uk-1) is meaningful for all F in M o r d . If F(uk-1) = 0, thena(n,F)(uk-l) is divisible by P = Pk = X-(uk-l) for all n. Thus by dividing Fby P, we still have an element of Mord, because (F/Pk) = F(uj-l)/(uj-uk) for allj & k, which is a modular form. Thus

P M o r d = { F G M o r d | F(uk-l) = 0}.

So M o r d / P M o r d can be embedded into ^r d(r0(pap),%0)-k;O). ThusM o r d /PM o r d is O-free of finite rank. Let us take Fi (i=l, — , r ) so thatFimodPMo r d gives an Obasis of Mord/PMord. Note that the Fi's are linearlyindependent over A. In fact, if not, we may suppose that A4F1+ ••• +A,rFr = 0


with at least one of the Xfs not divisible by P. Then reducing modulo P, wehave a non-trivial linear relation between the Fi mod P, which is a contradiction,and hence the Fi's are linearly independent. Consider M = AFi+ ••• +AFr.Then M is a A-free module of rank r and M/PM coincides with Mord/PMord

because if F is an element of Mord, then we can find a finite linear combinationGo of the Fj's such that F-Go is divisible by P. We now apply this argumentto (F-Go)/P and get another linear combination Gi of the Fj's such that(F-Go)/P-Gi is divisible by P. Continuing this process, we can find the Gj'swhich are linear combinations of the Fj's such that F = G0+G1P+ •••H-Gj.iPJ-1 mod Pj. Thus M/PjM = Mord/PjMord. Note that the series G0+G1P+••• H-Gj-iPJ"1 converges in M by identifying M with Ar by the basis Fj's.Thus M = Mord and hence Mord is A-free.

From the above proof, we know for sufficiently large k that Mord/PMord forP = X-(uk-l) is naturally embedded into ^rd(r0(pap),%co"k;0); in particular,we have, for sufficiently large k,

rankA(Mord(%,A)) < rank0(^rd(ro(pap),xco"k;0)).

Now we want to define the idempotent e on M = M(%,A). Take F in M. We

may assume that F(uk-1) is meaningful for every k>a . We consider the sum

a*a,k(A) = I3k

=a ^j(ro(pap),%co"j;A) inside A[[q]] for a subalgebra A of C

or Qp. The space fAfa,k(A) is in fact isomorphic to the direct sum. In fact, over

A which is a subalgebra of C , we see that if Z j = a fj = 0 for

fjG fAfj(r0(pap),xco"j;A), then for any ye Tx(p

ap)9

' j=a = S j = a fj(z>J(Y'Z)J = °-

We can of course choose j[ e r i (p a p) so that det(j(Yi»zy) * 0 (0 < i < k-aand a < j < k ) . This shows that fj = O. For A inside Qp, the space over Ais defined by the scalar extension of the space over QflA; hence, we know theassertion. Thus we have a natural action of T(n) on fAfa,k(A). Since MîO) isO-free of finite rank, the subalgebra R of Endo(â,k(^)) generated by T(p)over O is Ofree of finite rank. Thus we can take the limit ek = lim T(p)n! in

n—>«*

R. This idempotent preserves fWa,j(0) for j < k and induces the projector eon each subspace stable under e. Now we consider the A-submodules

M a , k = { G e M I G(u j- l )e 5Wj(ro(pap).XCO-J;0) for all j in [a,k]},M ' a , k = { G e Ma,k I G(uj-l) = 0 for all j in [a,k]}.


We note that M'a>k may not equal £*kMa,k> where Q^ = Il]La(X-(u:'-l)). By

definition, the map G f-» £k=aG(uJ-l) induces an injection of Ma>k/M'a,k into

Since T(p) preserves M'a>k and Ma,k, the idempotent ek acts on

Ma)k/M'a)k. If a < j <k, we have the natural projection TCkj of Ma>k/M'a>kto MayM'aj satisfying ej°7ik,j = ^k,j°ek- Note that QÂ is the kernel of the

map G i-> Z- aG(uM) on A, and hence Ma?k/M'a>k can be embedded into

(A/Qk)[[q]L The sequence of ideals {^kA} is a decreasing sequence of ideals of

A and f l ^kA = {0}, since any power series has only finitely many zeros inside

the unit disk in Q p . Thus A = lim (A/Qk). Therefore a(n,F|e)(X) =

lim a(n,F | ek) exists in A and a(n,F | e)(uk-l) = a(n,F(uk-l) | e). This shows

that F | e e Mord. Since the projective limit topology of the p-adic topology ofA/Qk coincides with the m-adic topology of A for the maximal ideal m of A,the above proof shows

(2) F | e = KmJF | T(p)n!) under the /rc-adic topology.

Summing up, we have

Proposition 1. There exists a unique projector e on M(%,A) to Mord(%,A)which satisfies F | e(uk-l) = F(uk-1) | e for all F e M(x,A).

In §1, we constructed a A-adic modular form E(%) out of Eisenstein series for

characters % of [i ([i = {£ e Zp | ^P-1 = 1}). We can extend the construction

to characters % modulo pap as follows. Let % : (Z/papZ)x -^ cf be a primi-

tive character, and put %o = % I p.- We define E(%)(X) = E(%o)(£X+(C-l)) for

£ = x(u). Then we see from Proposition 1.1 that E(%)(uk-1) = E(%o)(£uk-1) =

Ek(%G)"k). Thus E(%) is a A-adic form of character % in the sense of (A').

Moreover take a modular form g e fMa(ro(p^p),\|/;0) for a finite order character

\\f : Zpx -^ Ox, and put \\f0 = \\f \ ^. Then multiplying ECxyo-1) to g and

making the variable change X h-» \|/(u)"1u"aX+(\)/(u)'1u"a-l) in gE(%)(X), weobtain g*E(%\j/0-

1)(X) G M (%\J / 0 ,A) such that g*E(%\|/0"1)(uk-1)= gEk-a(%V"lco k) for all k > a, which is an element in #4(ro(pYp)>%a>"k;0) fory = max(a,(3). For any A-algebra A, we define Mord(%;A) = Mord(%;A)®AA.Now we prove

Proposition 2. Let % be either a primitive character modulo pap having val-

ues in Ox or % = id and a = 0. Let g = Ga(\|/) for a character

\\f : (Z /p a pZ) x -> Ox with \j/(-l) = (-l)a. Then the elements of the form


(g*E(x\|f0-1) I e) | T(n) (Vo = v l ^ together with E(%) span Mord(%,L) over

the quotient field L of A. Moreover if k > 2a+2 tfnd %cok w primitive modulo

p a p , ?/*£« ?/*e K-subspace of K[[q]] spanned by the specialized image of

Mord(x,A) at weight k contains the whole space M™d (ro(pap),%co~k;K) and

coincides with f^£rd(ro(pap),%co'k;K) if k is sufficiently large.

Taking a to be 2, we know by Corollary 5.4.1 that any ordinary modular form

can be lifted to a A-adic ordinary form (up to constant multiple) if k > 6 and that

Mord(x,A)/PkMord(%,A)(8)K= ^rd(ro(pap),%co-k;K) if k is large and %Grk

is primitive. Moreover we know the independence of the dimension of

^k r d (ro(PaP)>%co~k;K) from the weight if k is large, as long as %ccrk is primi-

tive. This fact is actually true for all k > 2 as we will see later.

Proof. By Theorem 5.4.1, as long as %cok is primitive, the modular forms of

the type g*E(xw 1 ) I T(n)(uk-1) = g E ^ t y ^ C D ^ ) I T ( n ) ( n = l , 2 , - ) to-

gether with Ek(%co-k) = E(%)(uk-1) and Gk(%Grk) span f*4(F,%co'k;K) for

F = Fo(pap). Hence ordinary forms of the form

(g*E(%\1/0-1) I e) I T(n)(uk-1) = (gEk.a(%\|/-ico-k) | e) | T(n)

together with E(%)(uk-1) span M™d (ro(pap),%co"k;K), because we know that e

annihilates Gk(%ork) because Gk(%ork) | T(p) = pk~1Gk(%Grk). Thus the

subspace of K[[q]] spanned by the specialization of A-adic forms at weight k

contains f^krd(ro(p

ap),XCO'k;K) if k>2a+2. By the proof of Theorem 1, it is

clear that for sufficiently large k, the specialization map from M/PkM (Pk =

X + l - u k e A) for M = Mord(%;A) into ^rd(ro(PaP)5XW"k;K) is injective.

Choosing large k so that %ark is primitive, we then know that {Fi} ie i (in the

proof of Theorem 1) span M o r d ( % ; L ) if { F i ( u k - l ) h span

fAf£rd(ro(pap),%co~k;K) over K. This shows the proposition.

—»By definition, we also know that, for any character e : W/Wp^

(g*E(%W11 e) I T(n)(e(u)uk-1) = (g*E(xw1)(e(u)uk-l) I e) I T(n)= (gEk.a(e%T1co"k)le)|T(n).

Defining formally e^F(X;q) = F(e(u)X+(e(u)-l)) as an element of A[[q]], we

see from the above formula that e*(g*E(%\|/o"1) I e) I T(n) is a A-adic form of char-

acter £% as long as e% is primitive modulo ppmax(a'P). We now show that this

is always the case without assuming the condition on primitivity. The point of the

argument is that for integer P > a, defining Fa?p = Fi(pa)nro(p^), we have


(I 0) (I (A

Jr r | Jr sets-(I ff\

This fact can be shown as follows. Writing r\ for 0 , we know

m r = UiFriSi is disjoint if F = UiOl^rrinnSi is a disjoint union. In our

case, we see easily thatU t ^ and U ^ ^

for the same 8i = ft L This implies that T(p)^"a not only acts on

A:(ro(p^p),%) but also decreases the level up to pap if % *s primitive modulo

pap. In other words, T(p)p-a brings 5k(r0(pPp),%) into 5k(r0(pap),%). Thus

actually £*(g*E(xw!) I T(n) | e) = E*(g*E(%y0-1) I T(n)) | e has level pap,

which is the conductor of e% (i.e. e% is primitive modulo pap). This shows ourclaim. Thus F H> e*F takes ordinary A-adic forms of character % to those of

character £%, because the forms g*E(%\|/o"1) I T(n) | e (n = 1, 2,...) span the totalspace Mord(%;L). Since e* is induced from the ring automorphism of the ring

A, e* is injective. Since (££')* = £*£'* and hence EÊ'1* = E'1^^ = id* = id,

£* has to be a surjective isomorphism. We have £* : Mord(%;A) = Mord(£%;A)

for any finite order character £ : W —> (?. Summing up we have

Theorem 2. For each finite order character £ : W —> O*, we have an isomor-

phism £* : Mord(%;A)= Mord(£%;A) functorial in E (i.e. (££')*=£*£'*;.

Moreover, for almost all positive integers k, F(£(u)uk-1) is an element of

^krd(Fo(pPp),e%co"k;0) if F e Mord(%;A), where p is the minimum exponent

such that E% factors through WAVP . The same assertion also holds for cuspforms.

By this theorem, without losing much generality, we may assume that % is a char-acter modulo p as we did in the definition (A). Hereafter we assume that % is acharacter of (Z/pZ)x. For each character £ of W with values in Qp, we writeO[£] for the subring of Q p generated over O by the values of £. To prove thefollowing theorem, we need a careful analysis of group cohomology attached to thespace of modular forms. Although the following theorem itself is true for allprimes p, the cuspidal theory is empty for p < 7 because a posteriori we findthat Sord(%;A) = 0 for all p < 7. (We should mention that if we considerFo(Npa) in place of Fo(pa) for N prime to p, the theory is not empty even forp = 2 and 3.) Assuming p > 5, the analysis of group cohomology becomes a


lot easier because Fi(p) is torsion-free. For this reason, we throw away the

primes p = 2 and 3 and give the exposition only when p > 5.

T h e o r e m 3 . For all integers k > l and a primitive finite order character e of

W/WpCl and for any modular form f e f5Vfk(r0(pap),e%co"k;0[e]), there exists a

A-adic form F of character % such that F(e(u)uk-1) = f. Moreover if k > 2,

we have the isomorphisms:

M o r d ( x ; A ) / P k , e M o r d ( X ; A ) = f d k

S o r d ( x ; A ) / P k , e S o r d ( x ; A ) = 5

where the map is induced by Fb->F(e(u)uk-l) and Pk,e is the prime ideal gen-

erated by X-(e(u)uk-l).

Proof. We consider the A-adic Eisenstein series E' = XE(id)(X;q). We knowfrom Theorem 3.6.2 that

rV-DCpd-s) I s=0 = r'logiu^-i)which is a p-adic unit. Thus we have E'(0;q) = 2~1/<?g(u)(p -1). We put

E(X;q) = {TUogW^-lÊ'iXiq) e Mord(id,A).

Then E(0;q) = 1 and for any ge ^krd(r0(pap),e%co"k;0[e]), defining

g*E(X;q) = gE(el(u)ukX+(e\u)uk-l)) E M(X;A),

we have (g*E) | e(0) = (g*E(0)) I e = g. This shows the first assertion. ByProposition 2, we get, if k is sufficiently large

(*) {Mord(%;A)/Pk,eMord(x;A)}(8)oK = ^ £

Since the unique Eisenstein series in f^krd(Fo(pap),exco"k;K[e]) is given by

E(%)(e(u)uk-1), we conclude, if k is sufficiently large, that

(**) {SordOc;A)/Pk,ESord(x;A)}®0K s 5™ = j j

This combined with the first assertion shows the second assertion for large k.

Thus we need to show that (*) and (**) hold for k> 2. First we show that

(*) for k> 2. Then (**) follows from (*) by the same argument as above.

Since every classical (ordinary) form lifts to a A-adic ordinary form, the image M

of specialization in O[e][[q]] contains f r£ and M/fAfk

r is O-free. Thus we

only need to prove the following equality of the ranks:

l = rankAMord(x;A).


For this we may extend scalars and may therefore assume that e%co"k has values in

0. Let \j/ : (Z/p^pZ)x —» 0* be a character. We pick a prime element G3 of 0

and put F = 0/G30. We take a normal subgroup A of Fo(p) so that Fo(p)/A

has order prime to p and A is torsion-free. If p > 2, A = F(4) does the job.

Then for any 0fFo(p)] -module M,

H°(ro(ppa),Hq(Anro(ppP),M)) = Hq(F0(ppP),M)

because Tr°res = (Fo(p):A). For the moment, we suppose that p > 2. Then byProposition 6.1.1, H2(Anr0(ppa),M) = 0 = H2(Fo(ppa),M). From the coho-mology exact sequence attached to

(***) 0 -> L(n,\|/;0)—-^->L(n,\|/;0) -> L(n,\|/;F) -> 0,

we get an exact sequence for F = Fo(p^p):

0 -» H1(r,L(n,\|/;0))(8)oF -> H^r jLCn^F) ) -> H2(F,L(n,\|/;0)) = 0.

This shows that dimKeH1(r0(ppp),L(n,\|/;K)) = dimFeH1(r0(pPp),L(n,\|/;F)).

Now we again consider the map i : L(n,\|/;F) —» L(O,\j/con;F) given by

i(P(X,Y)) =P(l,0), which is a homomorphism of Fo(p^p)-modules. Here we

have \j/con in place of \\f because on Xn,

TO(PPP) ^ 7 = fa0 °] modp for ae Zpx

acts by an\}/(a) = \}/con(a) mod p. Exactly in the same manner as in the proof ofTheorem 2.2, we know that i induces an isomorphism

,\|f(Dn;F)).

Note that \|/(u) = 1 mod p for the maximal ideal p of 0 because \\f(u) is ap-power root of unity and there are no p-power roots of unity except 1 in a char-acteristic p situation. Thus if \|/ = e%co'k, we have for T = ro(pap), writingXjx for the restriction of % to ja and noting the fact that % = X\i m °d p,

i# : e H ^ r ^ n ^ F ) ) = eH^F^CO^co^F)) = eH1(r0(p),L(0,Xn(0"2;F)).

Here the last isomorphism follows from the fact that e decreases the level to the

conductor of the character. Now, for any 0-module M, writing M[G3] for the

submodule killed by C3, from the exact sequence (***), we have the following

exact sequence for F = Fo(ppp):


0 -> H°(r,L(n,\|/;O))<S>oF - ^ H°(r,L(n,\j/;F)) -> H^LCnÔ))!®] -> 0.

We see easily by definition, for all j , that

If j > 0, then Xn-jYj | T(p) = 0 mod p. If j = 0, Xn | T(p) mod p does nothave a term involving Xn. Thus Xn [ T(p)2 = 0 mod p. Anyway, we haveeH°(r,L(n,\|/;F)) = 0 for all \|/ and n. Thus eH1(r,L(n,\)/;O)) is O-free.Thus we have from Theorem 6.3.3 that

2dimK Mlri (ro(pap),exco-k;K)--k;

= dimKHj)rd(r0(ppc'),L(n,exco-k;K))

= dimFH^d(ro(pp«),L(n)e%co-k;F))

K) + dimK ^2or

= 2dimKiW!rd(ro(p),xHco-2;K)-l.

This shows that dimK2tfkrd(ro(q),exco~k;K) is independent of k > 2 and e.

This number coincides with the rank of Mord(%;A) because of (*) for large k.

Therefore for any k > 2, we have

dimK^ld(ro(q),exco-k;K) = rankAMord(%;A),

which is what we wanted. This also finishes the proof of Theorem 2.1 whenp > 2.

We now give a sketch of the argument in the case when p = 2. We replaceToCpp") in the above argument by r o (pp a ) = ro(ppa)f]ri(5). Then r o (pp a )is torsion-free, and we know by the same argument that

HL(r'o(paP),L(n,exco-k;F)) s HThis shows first that

and then taking the subspace invariant under ro(pap), we can conclude with thedesired identity:

dimKW"rd(ro(2a+2),e%0)-k;K) = dimK<rd(ro(4),et lo)-2;K) = 1.


Anyway the ordinary part Sord(%;A) = 0 if p < 7 as is clear from the fact that

dim5krd(ro(p)) = dim5krd(SL2(Z)) = O if k < 1 0 (see Proposition 2.2 and

the dimension formula for 5krd(SL2(Z)) in §5.2).

We now define the universal ordinary Hecke algebra Hord(%;A) (resp. hord(%;A))by the subalgebra of EndA(Mord(%,A)) (resp. EndA(Sord(x,A))) generated by allthe T(n)'s over A. For any A-algebra A, we define Hord(%;A) (resp.hord(%;A)) by Hord(%;A)<g>AA (resp. hord(x;A)®AA). Similarly we define

Mord(%;A) = Mord(%;A)®AA, Sord(%;A) = Sord(x;A)<g)AA,M ( x ; A ) = M ( x ; A ) ( g > A A a n d S ( x ; A ) = S ( x ; ^

Then we have the well defined projection map e : M(x;A) -> Mord(x;A).

Theorem 4 (semi-simplicity). Hord(x;A) (resp. hord(x;A)j is reduced; i.e.,

Hord(x;L) (resp. hord(x;L)j for the quotientfield I of A is semi-simple.

Proof. We choose a basis {Fi}i=i,...,r of Mord = Mord(x,A). Then we can

identify EndA(Mord) with the matrix ring Mr( A). Suppose h e Hord(x;A) is

nilpotent. For k > 2 , EndA(Mord)(8)AA/PkA = Endo(Mord/PkMord) for Pk =

X-(uk-l). Note that the image of h in Endo(Mord/PkMord) gives an element of

Hkrd(ro(pap),XO)-k;0) because Mord/PkM

ord(3)oK= <r d(r0(pap),xo)-k;K). By

Corollary 2.1 and Theorem 5.3.2, H^rd(ro(pap),xco'k;0) has no non-trivial

nilpotent elements. Thus the image of h in H£rd(ro(pap),xco'k;0) is trivial.

Thus h is divisible by Pk. Since we have h e f]kPkMr(A) = {0}, where the

intersection is taken over all k > 2, this finishes the proof.

We now define the pairing

( , ) : H o r d ( x ; A ) x M o r d ( x ; A ) - > A by <H,F) = a(l,F I H) e A.

We also define mord(x;A) by {Fe Mord(x;K) | a(n,F) e A if n > 0 } ,where K is the quotient field of A.

Theorem 5 (duality). For any extension A of A, the above pairing induces

(i) HomA(Hord(x;A),A) = mord(x;A) and HomA(mord(x;A), A) = Hord(x;A),(ii) HomA(hord(x;A),A) s Sord(x;A) and HomA(Sord(x;A), A) = hord(x;A).In particular, Hord(x;A) and hord(x;A) are free of finite rank over A.

The freeness of hord(x;A) over A was first proven in [HI] directly without using

A-adic forms for p > 5.


Proof. We can prove the first part of (i) and (ii) in exactly the same manner asTheorem 5.3.1. Since the arguments are the same for (i) and (ii) for the secondpart, we only prove the second part of (ii). Since Sord(%;A) is A-free, we mayassume that A = A. We note that over the integral closure A of A in a finiteextension of L, for each A-torsion-free module M, the double dualM** = (M*)* may not be isomorphic to M, where M* = HOITIA(M,A). Forexample, for the maximal ideal m of A, //*** = A. Writing h (resp. S) forhord(%;A) (resp. Sord(%;A)), we have a natural map h -> h** = HomA(S,A)which is A-free. This map is injective because of the non-degeneracy of thepairing. Thus N = h**/h is a torsion A-module. Since after localizing at anyheight one prime P of A, we get the identity

h**P = HomAp(HomAp(hp,AP),Ap) = h P

because Ap is a discrete valuation ring. Since the Krull dimension of A is 2, Nis killed by a power of m. Thus N is a finite module. Since h** is A-free,S = h***. Thus we have

(*)Homo(h**/Pkh**,O) = S/PkS

On the other hand, from the exact sequence 0 -» h —> h** -» N —> 0, we getanother exact sequence (see Theorem 1.1.2):

Tor^(N,A/PkA) -> h/Pkh -> h**/Pkh** -> N/PkN -> 0.

Since N is finite, the module TorA(N,A/PkA) is finite, and thus we have another

exact sequence0 -> hk(r0(p),%co"k;0) -> h**/Pkh** -> N/PkN -> 0,

because the image of h/Pkh is the subalgebra of the O-free algebra h**/Pkh**generated by the T(n)'s, which is hk(ro(p),%co"k;0). The above exact sequenceyields, by O-duality (see Theorem 1.1.1)

0 -> Homo(h**/Pkh**,O) -* 4rd(r0(p),%0)-k;O) -^ Ext^(N/PkN,O) -> 0.

Then by (*), we know that ExtJ,(N/PkN,O) = 0. Since O is a valuation ring,

ExtJ,(N/PkN,O) = N/PkN = 0 (Corollary 1.1.1). Then Nakayama's lemma

shows that N =0 showing h = h** = HomA(S,A).

By Theorem 5, we know that Hord(%;L) = IIKK for finite extensions K/L. Wefix an algebraic closure L of L and take a finite extension K/L inside L whichcontains all the isomorphic images of the simple components of the Hecke algebra.The elements Fi corresponding to the i-th projection X[ to K inHomA(Hord(%;K),K) = Mord(x;K) give a basis of Mord(%;K) consisting of com-


mon eigenforms of all Hecke operators T(n) whose coefficients at n are given byA,i(T(n))e K. Thus we have

Theorem 6. For a finite extension K in L, Mord(%;K) and Sord(%;K) havebasis consisting of common eigenforms of all Hecke operators. If one normalizessuch a basis of Sord(%;K) so that coefficients of q are equal to 1, then the basiselements are contained in Sor (%;l), where I is the integral closure of A in K.

The last assertion follows from the fact that Hord(%; I) is finite over A and hence

Let us explain a little about the meaning of a common eigenform F e Sord(%; I).The evaluation of power series at e(u)uk-l gives an algebra homomorphismA -> 0, whose kernel is generated by Pk,e = X-(e(u)uk-l) for a finite ordercharacter e : W/WpCt —> Qp. Since I is a A-module of finite type and isintegrally closed, we can find a prime ideal P of I such that PflA = Pk,eA.We identify A/PÂ with 0[e], which is a subring of Q p generated by thevalues of e over 0. Then I/P can be identified with a finite extension of 0.Thus there exists an algebra homomorphism (p from I into Q p extending theevaluation morphism F(X) h-> F(e(u)uk-1). We identify I/P with a finiteextension of 0 by taking such a homomorphism 9 ((p may not be uniquelydetermined if I/P is a non-trivial extension of 0). Anyway, we see that for an

element F = £°° a(n,F)qn e Sord(%;I), taking the reduction F mod P isn=l

equivalent to considering the formal power series cp(F) = Z°° cp(a(n,F))qn

n=le QP[[q]]. We now show that q>(F) e 5k(ro(pap),exco"k;Qp). In fact,

Sord(x;l)®i(l/P) = (Sord(x;A)®Al)(x)|(|/P)= Sord(x;A)(8)A(l(8)l(|/P))= (S0rd(%;A)(8)AA/PkA)®0(p(l)

which is isomorphic to 5krd(ro(pap),£%co~k;(p(l)). In other words, by express-

ing F = £ i ^ i F i with FiE Sord(%;A) and X{ e I, we see that (p(F) =

Si(p(>,i)Fi(e(u)uk-l) G 5krd(r0(pap),e%co"k;(p(l)). If F is a common eigenform

of all Hecke operators, then cp(F) is a common eigenform of all Hecke operators

in 5krd(ro(pap),exco"k;(p(l)). Hence by Proposition 2.1, <p(F) is even a classical

complex common eigenform. We write F(P) for (p(F). Let j^(l) be the set of

all algebra homomorphisms from I into Q p which induce on A the evaluation

F(X) H» F(e(u)uk-1) for some k> 1 and some finite order character

E : W —» Q px . Each point in ^(1) is called an arithmetic point of

7.4. Two variable p-adic Rankin product 221

Spec(l)(Qp). Then {F(P)j for P e #(l) gives a p-adic family of commoneigenforms parametrized by Spec(l)(Qp) = Homo_aig( I, Qp). If I = A, then wemay identify the set of integers > 2 with a subset of A( I) by k t-> Pk = P ^ ,and we get the p-adic eigen-family of modular forms in the sense of §1. Usuallywe do not need to extend scalars to a non-trivial extension I; in particular, if thereis no congruence modulo p between eigenforms in 5k(ro(pap),e%co"k;Qp) for atleast one pair (k,%), it is known that hord(%;A) is isomorphic to a product ofcopies of A. We will return to this question later.

Theorem 7. Each normalized common eigenform of all Hecke operators in^krd(ro(pap),exco"k;Q) for k > l is of the form (p(F) for a normalized com-mon eigenform F in Mord(%;l) for a suitable extension I of A. The same as-sertion also holds for cusp forms.

Proof. Let f be a normalized common eigenform in ^4(ro(pap),e%co"k;Qp) fork > 1. By extending scalars to a finite extension of 0, we may assume that fhas coefficients in 0. We already know from Theorem 3 that f lifts to a A-adicordinary form F. Then there exists an algebra homomorphism XQ of the Heckealgebra H of Mord(x;A)/Pk,£Mord(x;A) (= ^r d(ro(pap),e%co-k;0) if k > 2 )into 0 such that Ao(T(n)) = a(n,f). By definition, we have a natural surjectivealgebra homomorphism Hord(%,A) -> H taking T(n) to T(n). Pulling back Xo

to Hord(%,A) via the above homomorphism, we have an algebra homomorphismô : Hord(%,A) -» 0. Let p be a minimal prime ideal of hord(%,A) contained inKer(X0). Then I1 = Hord(x,A)/p is a finite extension of A. Let I be the inte-gral closure of A in the quotient field K of I'. Then I' is contained in I andA,o factors through the natural projection X: Hord(x,A) —» I. That is, there existsan algebra homomorphism cp of I into Q p such that Xo = (p°X. This showsthat f = cp(F) for the common I -adic eigenform F corresponding to X.

§7.4. Two variable p-adic Rankin productIn this section, we construct the two variable p-adic Rankin product whichinterpolates the values Z)(k-l,f,g) where f and g vary on the families{f = F(uk-l)}k and {g = G(ul-l)}l for two A-adic forms F and G. Here kis the weight of f and F is assumed to be ordinary. Thus this two variableinterpolation is purely non-abelian and does not include the abelian (cyclotomic)variable. We extend this two variable p-adic L-function to a three variable one in§10.4 including a cyclotomic variable. For interpolation with respect to thecyclotomic variable, there is another method due to Panchishkin [Pa, IV].


We start with an abstract argument. Let S be a space of modular forms with theaction of Hecke operators T(n). We assume that S is an A-module of finite typefor a noetherian integral domain A. Let h(S) be the Hecke algebra of S over A,i.e. h(S) is the subalgebra of EndA(S) generated over A by Hecke operatorsT(n) for all positive n. We assume

(51) S is embedded into A[[q]] via the q-expansion f v-> Xn=o a(n»QQn»

(52) S = HomA(h(S),A) via the pairing <f,h> = a(l,f | h);

(53) 1I(S)(8>AK is semi-simple for the quotient field K of A.

By semi-simplicity, we have a non-degenerate pairing ( , ) on D = 1I(S)(8>AK

given by (h,g) = Tro/K(hg). By this, we have a natural isomorphism i : D —> D*given by i(h)(g) = (h,g). Thus we have i"1 : S(K) = D* -> D = S(K)*.Hence we have the dual pairing ( , )A : S(K)xS(K) —» K given by(fJg)A = i"1(f)(g)« By definition, (f,g)A satisfies

(la) (f |h,g)A = (f,glh)A.

We call this pairing (, )A the algebraic Peters son inner product. In particular, iff I h = X(h)f for an algebra homomorphism X : h(S) —» A with a(l,f) = 1(i.e. f is the normalized eigenform), the number

K

is well defined. In fact, by (S3), after extending the scalar field K by a finiteextension if necessary, S(K) has a basis consisting of normalized eigenforms.Then c(f,g) is the coefficient of f when we express g as a linear combination ofnormalized eigenforms.

Now we return to concrete examples. First we consider 5k(ro(pa),%o)- On thisspace, we have a Petersson inner product ( , ) . As seen in (5.3.10b), if wemodify ( , ) to define a new product (,)«, by

for x = L « OJ>

then we see that (f I h,g)oo = (f,g I h)*, and hence we again have, by (5.5.3)

c ( f ' g ) " (f,f)c "

where ( , )c is as in (la) for A = C. Now we suppose that


(PI) f = I ~ ?Lo(T(n))qn and h = I°° (po(T(n))qn for two algebra homomor-n=l n=l

phisms with k > /,

U : h°krd(ro(p

a),xo;Q(xo)) -> Q and (po: hKro(pp),vo;Q(vo)) -» Q;

(P2) %o is either a primitive character modulo p a or the identity charactermodulo p (i.e. a = 1).

By the ordinarity assumption on A., the condition (P2) actually follows from (PI).Then we define

(2) L(s,X

where we have written

n ( 1 - l 3 q O } " 1 and

To make computation easy, we make the following assumption:

(P3) XoW1 is primitive modulo pY with y = max(cc,P).

The condition (P3) ensures that

which is the key in the computation in §5.4. Then we replace g in (lb,c) by

hEk-KYo^Xo) for he fWKro(pp),Vo). Note that f \ x = (pa)(k"2)/2W(X0c)f for

an algebraic constant W(ô) with | W(ô) 1 = 1 . Then if a > p, we have by

the same computation as in §5.4, for A = Q(ô»9o) anc* E = ^

(3) c(f,hE) = WZ*(f,f)A

where Q(ô>9o) is the field generated by the values of ô and (po- Now let Kbe a finite extension of the quotient field L of A and I be the integral closure ofA in K. We consider an I-adic normalized eigenform F e Sord(%o;l). Weconsider the I-algebra homomorphism X : h o r d (X; l ) -> I given byF | T(n) = X(T(n))F. Then for

) = hE(u-/X+(u-/-l);q) with E(X;q) =


we have an element Lp(Xc®(po) e K (for the quotient field K of I) given by

_ , . c ^ , (F,e(h*E))i(4) Lp(?ic<g)9o) = ( F F ) | e K.

Now any element L in I can be considered as a function on

X(l) = HomO-aig(l,QP) = Spec(l)(Qp)

by L(P)=P(L). When I = A, we have X(A) = {x e Q p | | x - l | p < l }

via X H P X with PX(O) = <D(x-l), and L e A gives a p-adic analytic func-

tion on this unit disk. Thus our domain X(\) is a covering space of X(A) via

P H> PI A . Note that Sord(%;K) = KFe(KF)1 for the orthogonal complement

(KF)X of KF under ( , ) | . If P e A(\) with P | A = Pk,e and if e%co'k is

primitive modulo p a p , then 5k(ro(ppa),excQ"k;K[e]) = K[e]f+(K[e]f)± for

f = F(e(u)uk-1) and the orthogonal complement (K[e]f)x of K[e]f under

(» )K[£]- Localizing at P, we see that

Sord(X;lp)/PSord(%;lp) = 5rd(ro(ppa),e%co"k;K[e]) by Theorem 3.3.

In particular, we have (, )K[E] = ( »)l m °d P- This shows that if we write

g = eChEk-KVo^Xp)) and G = e(h*E¥ -ix ) and define Xp = X mod P by

PoX which factors through 5k(ro(pa),e%co"k;0[e]) by Theorem 3.3, we have

mod P -

as long as %0 = %p = e%co'k is primitive modulo p a with a > (3 > 0, and

%o> Vo» ô = ^p and (po satisfy the conditions (Pl-3). We compute the value

Lp(Xc®(po) when p > a. We have

= XP(T(p)) J J ^ if 5 = max(p-a.O).

Since T(p) decreases the exponent of p in the level by one as long as the charac-ter of the modular form is imprimitive, taking 8 to be the difference of the expo-


nent of p in the level of hEk./Ol/o^Xp) and a, we get the last equality in the

above formula. Note that, supposing (5 > a,

p°5Vo(pa)].Thus we see that

= ( h E t K W ^ p ) I [rO(pP)Q p°8]r0(pa)],f I

^ p ) ^ I * I [ro(pa)[PQ

5 °]ro(pp)])rO(PP)-

Since ( P ^ j]rO(pa)[Po

5 j ]n r o (p p ) = ro(pp), we see that

0

This implies, if Xp is primitive modulo p a with a > 0,

p) I T(p)8)~ = p80c-°(hEk./(W1Xp).(fc I ^)(P8z))r0(pP)

Thus we have, in general, if XP is primitive modulo p a (a > 0), under (Pl-3),for 8 = max(p-a,0),

Thus we get

Theorem 1 (one variable interpolation). Let X : hord(%;l) —» I be an

l-algebra homomorphism and (po • hk(ro(p^),>|/o;0) -> Q p be an O-algebra

homomorphism. Then we have a unique p-adic L-function Lp(A,c(8>(po) G K

with the following evaluation property: for P G J4(l) with primitive %p modulo

p a (a > 0) and P I A = Pk,e for integers k > / > 0, assuming (Pl-3) for

Xo = A,P a«J %0 = %p, we have

= pmax(a'P)(k-/){^P(T(p))-19o(T(p))}max(p-a'O)

r(k-/)r(k-i)L(k-i,y<g)(po)

where F e Sord(x;l) is the normalized eigenform attached to X.

We remove the condition (P3) in §10.5. We now want to vary (po along anotherp-adic family. Let M be another finite extension of L and J be the integralclosure of A in M . Let G be another J-adic cusp form, i.e.


G = E°° na(n;G)qne S(\|/,J). We suppose that G is a normalized eigenform ofn—u

all Hecke operators. Henceforth, for each arithmetic point P e ^t(J), we write

k(P) = k and ep = e if PI A = Pk,e- We also write the conductor of e as

pa(P)p. An arithmetic point P e .#(J) is called admissible (relative to G) if

for some p and \|/p = e\|/co"k. If G is ordinary, all arithmetic points are admis-sible relative to G by Theorem 3.3. Then we define the two variable convolutionproduct G*E(%\\fA) as follows. We consider another copy of A identified withO[[Y]]. We regard J as an O[[Y]]-algebra and take the completed tensor productA<§>oJ = 0[[X]]®oJ = J[[X]]. We define

1 for E =

Then, for the A-algebra homomorphism id®Q : A®oJ —» A = O[[X]]

(Q e #(J)), we have

id®Q(G*E) = G(Q)E(eQ(u)-1u-k(Q)X+(eQ(u)"1u-k(Q)-l)) = G(Q)*E.

If one has a formal q-expansion H = Z°° a(n;H)qn e A ® oJ[[q]] and if

id®Q(H) = X I = 1 id(S>Q(a(n;H))qne M(X;A)

for all arithmetic points Q e ^[(J) with k(Q) > a (for a given integer a > 0),then we claim that

(5) H e M(%;A)<§>oJ.

To see this, write H = Z<DOH<D with H $ 6 0[[X,Y]] for a basis {O} of Mover L. Choosing a dual basis {O*} of {O} under the pairing (x,y) =TrM/i_(xy), we can solve H<> = Ti^f[rx]]/L[[X]](**H). This shows that id®Q(H#)e M(%;A) for almost all admissible Q because the denominator of O* has onlyfinitely many roots in X(J). Thus to prove (5), we may assume that J = A.Then we write H = H(X,Y;q)e O[[X,Y]][[q]]. Since Pk is admissible for allsufficiently large k relative to G by definition, we write a for the integer suchthat Pk is admissible if k > a . We put H0(X) = H(X,ua- l ) . ThenH(X,Y)-H0(X) vanishes at ua-l, and thus Hi = (H(X,Y)-H(X,ua-l))/Yi(e 0[[X,Y]]) for Yi = Y-(ua-l) satisfies Hi(X,uk-l) e M(x;A) for allk > a+1. Now we define Y2 = Y-(ua+1-l) and repeat the above process to getH2 = (Hi(X,Y)-H1(X,ua+1-l))/Y2, satisfying H2(X,uk-l) e M(%;A) for allk > a+2. We then define inductively


Then Hj(X,ua+j-l) G M(%;A) and H = X~=o Hn(X,ua+n-l)n]LiYj, which is

convergent under the adic topology of the maximal ideal of 0[[X,Y]]. This shows

H E M(X;A)<8>OJ. Since the projection e : M(%;A)-^ Mord(%;A) extends to

(6) e : M(%;A)<§>oJ -> Mord(x;A)<§>oJ,

we can think of e(G*E(%\|/~1)), which satisfies

id®Q(e(G*E(xV"1))) = e(G(Q)*E0t\|T1)).

Let h(\j/;J) be the J-subalgebra of Endj(S(\j/;J)) generated by Hecke operatorsT(n) for all n and 9 : h(\|/;J) - > J be the J-algebra homomorphism given byG I T(n) = (p(T(n))G. For each admissible point Q e .#(J), Q p for \ j / Q = eQ\|/co-k(Q)

attached to G(Q). Now considering e(G*E%X|ri) G Mo r d(x;l)®oJ and extend-

ing the scalar product ( , ) i on Mord(%;l) to

( , ) I * J : M o r d (x ; l )®oJ x M o r d (x ; l )®oJ -> H 1

(for the denominator H of ( , )i) by J-linearity, we define

Lp(?iC<g>(p) = ^ p y ^

Then, after specializing Lp(?tc®(p) along id(E)Q, we get the function Lp(A,c®(pQ)in Theorem 1. Thus we have, regarding Lp(k

c®§) as a function on X(l)xx(J)

by Lp(?Lc<g>((>)(P,Q) = P®Q(Lp(?ic®(p)),

Theorem 2 (Two variable interpolation). Let X : h o r d (x ; l ) -> I be an

I-algebra homomorphism and G be a normalized eigenform in S(\|/;J). For each

admissible point Q of A(J), we write (pQ : hk(Q)(ro(p^),eQ\(/co'k(Q);0[£]) -> Q p

for the O[e]-algebra homomorphism associated with G(Q). Then we have a

unique ip-adic L-function Lp(?ic®(p) in the quotient field of ld>oJ with the fol-

lowing evaluation property: for P e JZ(l) with primitive %? modulo p a

(a > 0), and admissible Q e ,#(J) with \}/Q modulo p^, assuming (PI-3) for

Xo = Xp, (po = 9Q and %o = Xp* Vo = VQ> we have

r(k(P)-i)r(k(P)-k(Q))L(k(P)-uPc®(Po)

G(Zp-VQ)(-27tV=I)k(p«Q)(47t)k(p)-1(F(P),F(P))ro(pa) * K W K ^ '

where F G Sord(%;l) is the normalized eigenform attached to X.


Note that XE(id)(X) I x=o = ^(l—)/og(u). Therefore, if \|/ = %, we have

e(G*XE(id))(X,X) = 5(l-i)tog(u)e(G).

Noting that, by definition, (F,F)| = 1, we have

Theorem 3 (residue formula). Under the notation of Theorem 2, suppose that% = \|/. Then we have

§7.5. Ordinary Galois representations into GL2(ZP[[X]])Now we shall explain Wiles' method [Wil] of constructing Galois representationsattached to each I -adic common eigenform F. Let K be the quotient field of I.A Galois representation n : Gal(Q/Q) -> GL2(K) is said to be continuous ifthere exists an l-submodule L of K2 such that L is of finite type over I,L®|K = K2, L is stable under n, and as a map n : Gal(Q/Q) -> End|(L) iscontinuous under the m-adic topology on End|(L) for the maximal ideal m ofI. Since L is of finite type over I, there is a surjective homomorphism ofI-modules L. Thus if we write m for the maximal ideal of I, thenL/IT^L is the surjective image of (l/m1)11 which is a finite module. This showsthat End|(L) is a profmite ring and hence compact. Thus it is natural to considercontinuous representations into End|(L) from the absolute Galois group which iscompact under the Krull topology. On the other hand, I is a huge ring of Krull di-mension 2, and thus K cannot be a locally compact field [Bourl, VI.9.3]. Thisimplies that the image of a continuous representation of Gal(Q/Q) into GL2(K)under any topology which makes K a topological field is very small. This is thereason why we take the m-adic topology on End|(L) to define the continuity ofK. This definition of continuity does not depend on the choice of L by theArtin-Rees lemma [Bourl, IH.3.1]. We also say that K is unramified at a rationalprime q if the kernel of 71 contains the inertia group of q (§1.3). We first statethe result:

Theorem 1. Let F be an I-adic normalized eigenform in Sord(%;l) corre-sponding to the A-algebra homomorphism X : hord(%,A) —» I. Let K be thequotient field of I. Then there exists a unique Galois representationn : Gal(Q/Q) -> GL2(K) such that(i) n is continuous and is absolutely irreducible,(ii) 71 is unramified outside p,(Hi) for each rational prime q prime to p,

det(l-7c(Frobq)T) = 1 2

7.5. Ordinary Galois representations into GL2(Zp[[X]]) 229

where Frobq is the Frobenius element at q and K : W = l+pZp —» Ax is the

character given by K(US) = (1+X)S and (x) = ©(x) '^ G W.

This theorem was first proven in [H4], but here we give a different constructionfound by Wiles [Wil]. Before giving a proof of this fact, let us explain a littleabout the reduction of the representation modulo a prime ideal of I (or moduloeach point of 5i(\)). Let P be a prime ideal of I. We sometimes identify P withthe algebra homomorphism P : I —> I/P given by reduction modulo P. For eachelement XG I, we regard X as a function on Spec(l) via X(P) = P(X) = XmodP G I/P. We want to reduce K modulo P; thus, we consider therepresentation of Gal(Q/Q) on L/PL. It should look like a representation n' intoGL2(Kp) (Kp is the quotient field of I/P) such that:

(la) 7i' is unramified outside p;(lb) det(l-7c'(Frobq)T) = l-^(T(q))(P)T+x(q)K«q»q-1(P)T2 for all prime q

outside p.

If PI A = Pk,eA, then A,(T(q))(P) is equal to a(q,F(P)) and

A Galois representation 7i(P) into GL2(Kp) for an algebraic closure Kp of Kpis called a residual representation of % if 7i(P) is continuous under the m-adictopology on Kp, semi-simple and satisfies the conditions (la,b). Since I is ofKrull dimension 2, Kp is always locally compact under ITl-adic topology forP & {0}, and thus the continuity of K modP is clear. Since L may not be freeof rank 2 over I, it is not a priori clear that the residual representation exists for allprime ideal P. In fact it exists:

Corollary 1. For every prime ideal P, the residual representation TE(P) of %exists and is unique up to isomorphisms over Kp.

Proof. We proceed by induction on the height of P. Suppose that P is of height1. Since A is a unique factorization domain [Bourl, VII], the localization Ap ofA at any PflA is a valuation ring. Then, the localization A at P is a finitenormal extension of Ap. Therefore A is also a valuation ring. We take anl-submodule L of K2 of finite type which is stable under n and L®K = K2.Consider V = L® | A, which is stable under n and V®K = K2. Therefore Vis a free A-module of rank 2. Identifying V with A2, we have a Galois repre-sentation n into GL2(A) satisfying the conditions of Theorem 1. Then byreducing n modulo P and taking its semi-simplification, we obtain a residualGalois representation 7i(P). Uniqueness of % follows from the fact thatTr(7c(P)(Frobq)) is given by A,(T(q))(P) and that Frobq is dense in the Galois


group of the maximal unramified extension outside p of Q. Now we replace Aby the normalization of A/PflA and I by the integral closure I' of A/PflA inthe quotient field of I/P. For any prime ideal P' of height 1 in I1, we apply thesame argument to K mod P and get the residual representation(n mod P) mod P . This representation is just the residual representation attachedto a prime ideal in I of height 2, which is the pullback of P' in I. Continuingthis process, we get the claimed result for all prime ideals of I.

Even if the actual representation n is not known to exist, we can consider theresidual representations separately. Thus if an I -adic common eigenform F isgiven, then for each point P e X(l), we call a semi-simple Galois representation7c' into GL/2(Ap) a residual representation modulo P if n' satisfies theconditions (la,b) for P, where Ap is the integral closure of Op in its quotientfield. Our method of proving the theorem is to show that the desired % exists ifthere exist infinitely many distinct primes {P} in X(l) which have the residualrepresentation modulo P. Such a family of infinitely many residual representationsis supplied by the following theorem of Deligne:

Theorem 2 (P. Deligne [D]). Let M be a finite extension of Qp . LetX : hk(ro(N),%;Z[%]) —> M be an algebra homomorphism. Then there exists aunique Galois representation n : Gal(Q/Q) -» GL2(M) such that(i) n is continuous and absolutely irreducible over M,(ii) n is unramified outside Np,(iii) for each prime q outside Np,

bq)det(l-7i(Frobq)X) = 1 - 5i(T(q))X + x(q)qk"1X2.

This result in the case where k = 2 (except the determination of ramified places)is a classical result due to Eichler and Shimura and its proof can be found in [Sh,Th.7.24]. The unramifiedness outside Np was later proven by Igusa in the caseof weight 2. The case k> 2 is treated in Deligne's work [D]. The remainingcase k = 1 is dealt with by Deligne and Serre in [DS]. Note that %(-l) = (-l)k

and thus det(rc(c)) = %(-l)(-l)k~l = -1 for complex conjugation c. For furtherstudy of ramification, see [La] and [Cl]. Then Theorem 1 follows from the abovetheorem of Deligne and the following result of Wiles:

Theorem 3 (Wiles [Wil]). Let F be an l-adic normalized common eigenformand suppose that there exists an infinite set S of distinct points in X(\) such thatfor every P e S, the residual representation n(P) into GL/2(0p) exists, whereOp is the p-adic integer ring of the quotient field of I/P. Then there exists aGalois representation n : Gal(Q/Q) -> GL2(K) satisfying the conditions ofTheorem 1.


In fact, if F is ordinary, we can take A(l) as the set S by the theorem ofDeligne. Even if F is not ordinary, we can take as S the set of admissible pointsin A(l). For ordinary F, we can avoid the use of infinitely many modular formsof different weight, taking as S the set {P e .3(1) | k(P) = k}. Thus for a fixedweight k > 1, if we have Deligne's Galois representation for every commoneigenform of weight k, Theorem 1 follows from Theorem 3. In particular, if wetake k = 2, Theorem 1 follows from the result in [Sh, §7.6], which shows theexistence of Galois representations attached to modular forms of weight 2.

Now we start the proof of Theorem 3. Although Theorem 3 holds even forp = 2, we prove the theorem assuming p > 2 for simplicity. We refer to [Wil]for the proof in the general case. Let us fix P e S and write n for TC(P) andM for Kp. Let A be the p-adic integer ring of M. Thus n has values inGL2(A). Let G be the Galois group of the maximal extension unramified outsidep. Since n is unramified outside p, we may consider % as a representation ofG. We write L for A2 and consider L as a G-module via K. We write c forcomplex conjugation. Since c2 = 1 and det(7t(c)) =-1 , the eigenvalues of 7t(c)are ±1. We decompose L = L+©L_ into the sum of the ±1 eigenspaces of 7t(c).Thus by identifying L with A via the basis of L±, we may assume that

-1 0

f ( ) ()For each a € G, we write ft (a) = ( , ,( J and define a function

x: GxG —» A by X(G,X) = b(a)c(x). Then these functions satisfy the followingproperties:

(2a) as functions on G or G2, a, d and x are continuous,(2b) a(ox) = a(a)a(x)+x(a,x), d(ax) = d(a)d(x)+x(x,a) and

x(ax,py) = a(a)a(y)x(x,p)+a(y)d(i:)x(a,p)+a(a)d(p)x(T,Y)+d(T)d(p)x(a,Y),(2c) a( l) = d(l) = d(c)= 1, a(c) = - l , and

x(a,p) = x(p,x) = 0 if p = 1 or c,(2d) x(a,x)x(p,rt) = x(a,Ti)x(p,x).

The properties (2c) and (2d) follow directly from the definition, and the first half of(2b) can be proven by computing directly the multiplicative formula

b(o)Va(x) bCcfi _ fa(ox) ^

d(a)J[c(x) d(x)J " tc(ax) d(ox)J*

Then, in addition to the two first formulas of (2b), we also have


b(ax) = a(a)b(x)+b(o)d(x) and c(ax) = c(a)a(x)+d(a)c(x).Thus we know that

x(ax,py) = b(ax)c(py) = (a(a)b(x)+b(a)d(x))(c(p)a(y)+d(p)c(Y))= a(a)a(Y)x(x,p)+a(y)d(x)x(a,p)+a(a)d(p)x(x,Y)+d(x)d(p)x(a,y).

For any topological algebra R, we now define a. pseudo-representation of G intoR to be a triple TC' = (a, d, x) consisting of continuous functions on G or G2

satisfying the conditions (2a-d). We define the trace Tr(7c') (resp. thedeterminant det(Tc')) of the pseudo-representation n1 to be a function on G givenby

Tr(7t')(cO = a(a)+d(o) (resp. det(7c')(cj) = a(o)d(a)-x(o,a)).Our proof of Theorem 3 is divided into two parts: the steps are represented by thefollowing two propositions:

Proposition 1. Let TC' = (a, d, x) be a pseudo-representation of G into anintegral domain R with quotient field Q. Then there exists a continuous repre-sentation % : G —» GL2(Q) with the same trace and determinant as K\

Proposition 2. Let a and B be two ideals of I. Let n(a) and n(B) be

pseudo-representations into I/a and MB, respectively. Suppose that n(a) and

K(&) are compatible; that is, there exist functions Tr and det on a dense subset EofG with values in l/of]B such that for all O G Z ,

Tr(7i(a)(o)) s Tr(a) mod a and Tr(n(B)(o)) = Tr(a) mod B,

det(7c(a)(a)) = det(a) mod a and det(7t(£)(a)) = det(a) mod B.

Then there exists a pseudo-representation %{cf\S) of Q into \lcP\B such that

Tr(n(cf]B)(c)) = Tr(a) and dtt(n(c{]B)(o)) = det(a) on E.

First admitting these two propositions, let us prove Theorem 3. Since G is un-

ramified outside p, the set E of Frobenius elements for primes outside p is

dense in G (Chebotarev density theorem: Theorem 1.3.1). We put Tr(Frobq)

= ?i(T(q)) and det(Frobq) = %(q)K((q))q"1. We number each element of S and

write S = {Pi}?\ and K{ for 7i(Pi). We construct out of each residual repre-

sentation 7Ci for P e S, a pseudo-representation n\. Then all the TCVS are

compatible. Then by the above proposition, we can construct a pseudo-representa-

tion Kd into 1/nLPj so that Tr(n\c)) = TT(%*-\O)) mod flrKp\ on E.Both sides of this congruence are continuous functions and hence

Tr(7Cfi(a)) = TrCTc'1"1^)) mod Pifl ••• HPi-i on G.

Note that by definition, if %{ = (ai, di5 xO, ai(a) = 2"1(Tr(7c'i(a))-Tr(7i'i(ac)))

and di(a) = 2"1(Tr(7cfi(a))+Tr(7i:ti(ac))) and xi(a,x) = ai(ax)-ai(a)ai(x) .

Therefore we have

ai(a) = ai. i(a) mod Pifl ••• DPi-i, di(a) = di.i(o) mod Pifl ••• PlPi-i


and xi(a,T) = xi_i(a,x) mod Pifl ••• HPi-i-Thus we can define a pseudo-representation 7c1 into I = Km l/PiPl-'-PlPi by

i

7C'(CJ) = lim 7ca(a).

Then we can construct the representation K out of tf by Proposition 1.

Wenow prove Proposition 1. We divide our argument into two cases:Case 1: there exist p and y e G such that x(p,y)^0, andCase 2: X(G,X) = 0 for all a,x in G.

(a(a) CCase 1. We define n(o) = \(a) d ( J by putting

c(a) = x(p,a) and b(a) = x(a,y)/x(p,y).

Then b(a)c(x) = x(p,x)x(a,y)/x(p,y) = x(a,x) by (2d). Thus we know that the

entry of 7c(a)7c(x) at the upper left comer is equal to, by (2b),

a(a)a(x)+b(a)c(x) = a(a)a(x)+x(o,x) = a(ox).

Similarly the lower right comer of 7i(o)7t(x) is equal to

d(G)d(x)+b(x)c(a) = d(a)d(x)+x(x,a) = d(ox).

We now compute the lower left comer of 7i(a)7c(x), which is given by

c(a)a(x)+d(a)c(T) = x(p,a)a(x)+d(a)x(p,x).

By applying (2b) to (l,p,a,x), we have

c(ax) = x(p,ax) = a(x)x(p,a)+d(a)x(p,x),since x(l,c) = x(l,x) = 0 by (2c). This shows that

c(ax) = c(a)a(x)+d(a)c(x).Similarly by applying (2b) to (a,x,l,y), we have

b(ax)x(p,y) = x(ax,y) = a(a)x(x,y)+d(x)x(a,y) = (a(o)b(x)+d(x)b(a))x(p,y),

which finishes the proof of the formula 7t(o)7i;(x) = 7c(ax). Obviously, by defini-(l 0\

tion, TC(1) = Q and hence TC is the desired representation.

Case 2. In this case, by (2b), we have a(a)a(x) = a(ax) and d(a)d(x) = d(ax)fa(o) 0 ^

for all c,XEi G. Then we simply put n(o) = ^ ., J which does the job.


We now prove Proposition 2. We consider the exact sequence:0 -> l/aPi5-> \la®M6 —2-> \/(a+S) -> 0

ai-> amoda© amod£

a©b h-» a-b mod a+B.

We consider the pseudo-representation n = n(a)®K(6) with values in l/a®l/5.

The function a°Tr(7t) vanishes identically on £. Since this function is continu-ous on G and £ is dense in G, a°Tr(7c) vanishes on G. Thus Tr(rc) hasvalues in l/dT\b. If we write n = (a, d, x), then

a(Q) = 2-1(Tr(7c(a))-Tr(7i(ac))), d(a) = 2"1(Tr(7c(a))+Tr(7c(oc)))and x(o,x) = a(ox)-a(a)a(x).

Thus n itself has values in l/cP\5 and gives the desired pseudo-representation.

§7.6. Examples of A-adic formsIn this section, we briefly discuss some examples of ordinary and non-ordinaryA-adic cusp forms. We will not give detailed proofs but satisfy ourselves withindicating the source where one can find proofs. We start with the lowest weightcusp form of SL2(Z), which is the Ramanujan A function. The function Aspans 5i2(SL2(Z)) and is a normalized eigenform. Then it is known bycomputation that A | T(p) = x(p)A with x(p) e ZpTlZ for 11 < p < 1021.Thus writing as a the unique p-adic unit root of X2-x(p)+p11 = 0 for thoseprimes, we know that f = A(z)-a"1p11A(pz) is a normalized eigenform of levelp and f | T(p) = ocf, i.e. f is ordinary. Then, if | x(p) | p = 1, the fact that5iO2d(r0(p);Qp) = Qpf follows from [M, Th.4.6.17 (2)]. In general, we have

diir^rd(ro(p);Qp) = dimj£rd(SL2(Z);Qp) (Proposition 7.2.2).

Thus h°rd(ro(p);Zp) = Zp via T(n) H> x(n) which is the eigenvalue of T(n)

for A. This implies hord(co12;A)/Pi2hord(co12;A) = Zp. Since hord(co12;A) is a

A-algebra, we have the structural morphism i : A —> hord(co ;A). Then by the

Nakayama lemma, we know that i is surjective. As we have already seen

(Theorem 3.5), hord(co12;A) is A-free, and hence i is an isomorphism. Thus

there exists a unique ordinary A-adic normalized eigenform FA spanning

Sord(co12;A) such that FA(u12-l) = A(z)-a"xpnA(pz) as long as I x(p) | p = 1

holds. Since 12 is the least weight k for which 5k(SL2(Z)) * 0,

hord(x;A) = 0 for p = 2, 3, 5 and 7 for all %.

In the text, we have only discussed A-adic forms of level p°°. If we introduce anauxiliary level N prime to p, there are abundant examples. A formal q-expansionF G A[[q]] is called a A-adic form of level Np°° and of character % for a finite

7.6. Examples of A-adic forms 235

order character % of (Z/NpZ)x , if for almost all positive integers k,F(uk-1) G fA4(ro(Npa),%co"k;0) for some fixed a. Then we define the notionsof A-adic cusp forms, A-adic ordinary forms, etc. in the same manner as in §7.1.The first example of this type is the A-adic cusp forms associated with imaginaryquadratic fields (given by theta series). We thus fix an imaginary quadraticextension M/Q. Then there are abundant arithmetic Hecke characters X such thatM(&)) = oc^1 for a = 1 mod c for some ideal c, where k > 1 is a positiveinteger. The largest ideal c in the integer ring r of M with this property is calledthe conductor of X. Then it is well known (see [M, Th.4.8.2]) that there exists amodular form

(1) h = 5>(*)q"(fl)

where D is the discriminant of M/Q, %(m) = f — ] is the Jacobi symbol and

X(m) = X((m))/rr^~1 for integers m. This form is known to be a cusp form if X

is non-trivial (in particular if k > 1). Then fx, is a normalized eigenform and forprimes /, a(/,f) = 0 if Ir is a prime ideal, a(/,f) = X(C)+X(C) if lr= CC with

I and CC + c = r, a(/,f) = X(C) if Ci) Dc. Here if /"=> c, we agree to puti = 0. Let p be a prime ideal of M given by {xe r\ | x | p < l } for the

p-adic absolute value of Qp. We fix one character X modulo cp for an ideal c

prime to p such that X((a)) = a if a = 1 mod cp. Then we takeK = QP(X) and its p-adic integer ring O. We decompose (7* = WKX|LIK SO thatWK is Zp-free and JJ K is a finite group. We then write the projection map of cf

onto WK as X H (X). First we suppose that pr= pp with two distinct primeideals p and p. Then the subgroup WM of WK topologically generated by(X(a)) for all ideals a prime to p is isomorphic to the additive group Zp and theindex p^ = ( W M : W ) < °°. Here we consider W = l+pZp as a subgroup ofWM by first regarding W as a subgroup of rf and then applying the map: z h->(X(z)). Note that this is in fact the natural inclusion map of W into Qp(?i). Theexponent y = 0 if the class number of M is prime to p. We fix a generator wof WM SO that wpY = u. Then we consider the ring I = O[[Y]] containing the

ring A = O[[X]] with the relation (1+Y)p7= (1+X). Then we consider theseries

(2a) Fx(Y;q) = âMa)(X(a))-2(l+Y)s(a)qN(a) if P^= pp with p*p,

where a runs over all ideals prime to p and s(a) = log((X(a)))/log(w), i.e.

ws(fl)= (X(a)). Then we know that (1+Y)s(fl) | Y=wk-i = wks(fl)= (X(a))k. Noting


that the Hecke character ^k modulo cp given by X^(a) = X(a)(X(a))k'2 has in

fact values in Q, we get

(2b) Fx(wk-l;q) = f k e 5k(r0(DN(c)p),%X(x)2~k;Oy

Here it is easy to compute Xk = % X(O2'k. If we substitute £wk-l (instead of

wk-l) for X in F^(X;q) for a py-th root of unity £, we have

(2c) F^(Cwk-l;q) = f£ k e 5k(r0(DN(c)p),x?lC02-k;O),

where 8 is the finite order character of the ideal class group C1M(1) given by

e(a) = Cs(a)-

Since Pk = X-(uk-l) = II;(Y-(^wk-l)) in I, we know that F^ is an I-adic

form of level DN(c) and of character %X(O2. Since eXy&p) is a p-adic unit and

since the eigenvalue of T(p) for fe k is given by eXk(p), F^ is ordinary. If

one chooses X so that X\ is trivial (this is always possible), then we see that

Thus FJL(W-I) =E(%CO)(U-1) where E(%co)(X;q) is the A-adic Eisenstein seriessuch that E(%co)(uk-l)=Ek(%co1"k) or Ek(x)(z)-Ek(%)(pz) according as CO1"1" isnon-trivial or not. Then, for example, supposing WM = W, we have a strange

A-adic form E(X;q) = ^ ( u - n ^ This implies

rj I T , x a(n,E(%co))E(xco)-a(n,FQF^ a(n,E(%co))-a(n,F^/

E |T(n)= x ^ ^ = x^j) E(Xco)+a(n,FOE.

Note that — l—T?—,—y—2—E A and is non-vanishing at u-1. Thus on

Mord(%co;A)/PiMord(%co;A), T(n) acts non-semi-simply. On the other hand, theaction of Hecke operators T(n) for n prime to Dp on fî(ro(Dp),%co) issemi-simple, because the same argument as in the proof of Theorem 5.3.2obviously works. This shows that inside 0[[q]], the image of specialization ofMord(%co;A) under F H F(u-l) is larger than the space of classical ordinaryforms fA^rd(r0(Dp),%co;0).

Now we assume that p is inert or ramified in M. For simplicity we assume that

p * 2. We have a surjective group homomorphism q>: C1M(^P°°) -> WM given

by cp(x) = (X(x)). On C1M(^P°°)J there is the natural action of complex

conjugation c which leaves the maximal torsion subgroup of C1M(^P°°) stable,

7.6. Examples of A-adic forms 237

and hence c acts on the image WM by (p(x)c = (p(xc). This action may notcoincide with the usual Galois action of c on Q(^). Then WM = Zp

2 and thuswe have two generators wi and W2 of WM. We then may assume that wi = uand W2 = w with wc = w"1 for complex conjugation c. We then write(X(a)) = us(a)wt(fl) and define

(3a) Fx(X,Y;q) = ^X

This is not really a A-adic form but, so to speak, an O[[X,Y]]-adic form. That is,we have, for a pair (£,£') of p-power roots of unity

(3b) F ^ u k - l , i > k - l ) = feXk

where e(a) = ^ ^ ' ^ is a finite order character with conductor pl\ Sincea(p,fx,k) is either 0 or a p-adic non-unit, the family of modular forms {fex,k} willnever be ordinary.

These A-adic forms constructed out of imaginary quadratic fields are called A-adicforms with complex multiplication or of CM type. As is clear from the aboveexamples, there exist l-adic forms having values in a non-trivial extension I ofA.

There are examples of A-adic forms without complex multiplication. We startfrom a primitive finite order character î modulo c of the imaginary quadraticfield or real quadratic field M. We continue to write r for the integer ring of M.Then, even in the case of real M, if X\{oC) = -1 for a e r with the congruencea = 1 mod c and oca = -1 mod c for a non-trivial automorphism a of M, f\

as in (1) is a cusp form in 5i(Fo(DiV(c)), X%) for the discriminant D of M and

the Jacobi symbol %(m) = f — ]([M, Th.4.8.3]). Suppose thatU )

(4a) p is a divisor of DN(c) (resp. D) if M is real (resp. complex);

(4b) There exists a prime factor p of p in r such that f is prime to c,

where a = c when M is imaginary. Then we see that fx\ I T(p) = X(pc)fxi

and hence f^ is ordinary because X(pG) is a root of unity. When p | D, there areno ordinary A-adic forms with complex multiplication. When M is real quadratic,the modular forms associated with real quadratic fields do not exist in weighthigher than 1. However by Theorem 7.3.7, we can lift f?^ to an l-adicnormalized eigenform F which is not of CM type.

We know from [M, Th.4.2.11 and Th.4.2.7] that


(5) dimc(52(r0(4))) = 0

and for odd p, dimc(52(r0(p))) = ^

It is also known that T(p) is invertible on S2(ô(v)) a nd has only eigenvalues ±1([M, Th.4.6.17 (2)]). Thus in this special case, we have

52(r0(p);O) = 5|rd(r0(p);O).Then by Theorem 3.3, we know

Theorem 1. For odd primes p, we have

rankA(hord(co2;A))=rankA(Sord(co2;A)) =

The above dimension formula is due to Dwork when p = 1 mod 12 [K4, p. 140].By the above formula rankA(hord(co2;A)) grows linearly as the prime p grows.This is the only case where we know the exact rank of the space of ordinary A-adicforms in a systematic way. There are several numerical examples computed byMaeda for A-adic forms. We refer to [Md] and [H2] for these examples. Inparticular, in [Md] Maeda gives a criterion in terms of generalized Bernoullinumbers for having non-trivial A-adic forms without complex multiplication whichare congruent to A-adic modular forms with complex multiplication. Such acongruence is very important in studying the Iwasawa theory of imaginaryquadratic fields and CM fields. We refer to [HT1-3] for such applications inIwasawa's theory.

Chapter 8. Functional equations of Hecke L-functions

In this chapter, after giving a brief summary of the notion of adeles ofnumber fields, we prove the functional equations of Hecke L-functions. For fur-ther study of adeles and class field theory, we recommend [Wl] and [N],

§8.1. Adelic interpretation of algebraic number theoryLet us start with the explanation of the adele ring of Q denoted by A. To definethe topological ring A, we consider the module Q/Z and its ring End(Q/Z) ofadditive endomorphisms. Let Q/Z[p°°] be the subgroup of Q/Z consisting ofelements killed by some power of p for a prime p of Z. Then by definition, forany two distinct primes p and q, Q/Z[poo]nQ/Z[qo°] = {0} and hence©pQ/Z[p°°] c Q/Z. Let us show that this inclusion is in fact surjective. For anygiven rational number r, we expand r into the standard p-adic expansion

S°°_ cnpn defined in §1.3 and put [r]p = ZnL.mcnpn (the p-fraction part). Since

r-Mp e ZpHQ, p does not appear in the denominator of r-[r]p. Repeating thisprocess of taking out the p-fraction part from r for all prime factors of thedenominator of r, we know that r-Ep[r]p e Z and hence r = Xp[r]p in Q/Z.Since [r]p e Q/Z[p°°], we know that ©PQ/Z[p°°] = Q/Z. The above processcan be applied to p-adic numbers r e Qp. Then r i—> [r]p plainly induces anisomorphism: Qp/Zp = Q/Z[p°°]. Thus identifying Qp/Zp with Q/Z[p°°], wehave(1) Q/Z s ©PQP/Zp.

This in particular shows that End(Q/Z) = npEnd(Qp/Zp). The multiplication byelements of Z p induces a morphism i : Zp —> End(Qp/Zp), i.e., ip(z)(z')= zz'. We now show that i is a surjective isomorphism. Obviously it is an in-jection. Note that the submodule Qp/Zp[pT] killed by pr is isomorphic to Z/prZvia x i—> prx. Since the endomorphism of Z/prZ is determined by its value at1, we see that End(Qp/Zp[pr]) = Z/prZ via (p h-> p(q>) = pr(p(p~r)- If (pr eEnd(Qp/Zp[p

r]) is the restriction of (p E End(Qp/Zp), then {p((pr)}r is coherentso that p((pr) = p((ps) mod ps if r > s. Then the map p : End(Qp/Zp) -* Zp

given by p((p) = lim p((pr) e Zp gives the inverse of i. This shows thatm

(2a) End(Qp/Zp) = Zp and End(Q/Z) = Z =

For any finite subgroup X of Q/Z, we put

A(X) = {a G End(Q/Z) | aX = 0}.

240 8: Functional equations of Hecke L-functions

Then obviously End(Q/Z)/A(X) = End(X). Declaring {A(X)}X to be a funda-mental system of neighborhoods of 0 in End(Q/Z), we give the structure of atopological ring on End(Q/Z). This is the weakest topology that the quotienttopology of End(Q/Z)/A(X) = End(X) is discrete, and hence it is a topology ofthe projective limit End(Q/Z) = lim End(X). Thus End(Q/Z) is a compact

ring, which is isomorphic to Z = n p Z p as topological ring. We define the finitepart Af of the adele ring A to be the subring of IIpQp generated by Q and Z.Here Q is embedded diagonally into IIpQp by Q ^ a n (•••a,a,a,---) e IlpQp.We give the structure of a locally compact ring on Af so that Z is an open com-pact subgroup of Af and the topology of Af induces the topology on Z. Wedefine the adele ring A to be AfxR and give the product topology on it. Then Ais a locally compact ring. If x e IlpQp satisfies xp e Zp for almost all p(i.e. for all but finitely many p), we can cancel the denominator of x by multiply-ing a rational integer m, i.e., mx e IIpZp. Thus x e Af. It is obvious thatxp G Zp for almost all p if x e Af. Then we know that

(2b) A f = n ' p Q p = ( x e n P Q p i XP e Z P f ° r a i m ° s t a n p}-

Thus Af is the restricted direct product of Qp with respect to Zp. This is thedefinition of Af usually found in the literature. Now we shall see thatAf = Z+Q. For any x e Af, the above definition using the restricted directproduct shows that [x] = Zp[xp]p is actually a finite sum and is a rational frac-tion. Then x-[x] e Z showing Af=Z+Q. In particular, we have

(2c) Af = Z+Q and Af/Z = Q/(ZDQ) = Q/Z.

The multiplicative group Ax is called the idele group of Q. Since Z is a

principal ideal domain, for any x e AfX, we can write xp = up 6 ^ with u E Zpx.

Since primes p with e(p) ^ 0 are finitely many, {x} = TIpp6^ is a positive

rational number and hence x(x)"1 e Zx, which shows

(3a) A x = ZXQXR+X,

where R+x = {x E R | x > 0}. Since Q c A via Q 3 a H (a ,a , . . . , a )G A, A is naturally a Q-algebra. Consider a small open interval Ue = (-8,8)for e > 0 in R. Then O = Z x U e is an open neighborhood of 0 in A. Wesee that QflO = ZflUe = {0} if 0 < 8 < 1. Thus we have found an openneighborhood O of 0 in A such that QflO = {0}. Note that

A/Q = {(ZxR)+Q}/Q = (Z0R) /{ (Z0R)nQ} = ( Z 0 R ) / Z = Zx(R/Z),which is a compact set. Thus

8.1. Adelic interpretation of algebraic number theory 241

(3b) Q is a discrete subring of A and A/Q is compact.

For a number field F (i.e. a finite extension F/Q), we simply put FA = F ® Q A .Then we have the ring embedding F s a i - ) a® 1 G FA and hence FA is anF-algebra. The trace map Trp/Q : F -> Q induces the A-linear mapTrp/Q®id : FA —> A, which is again denoted by Trp/Q and is called the (adelic)trace map. Fixing a basis {coi} of F over Q, we identify F = QtF:™ a s a

vector space, which induces an isomorphism FA = A^F:(^ of A-modules. Thisidentification gives a natural topology on FA, under which FA is a locally com-pact topological ring. Obviously the topology of FA does not depend on thechoice of the basis. In particular,

(3c) F is a discrete subring of FA and FA/F = dxF^/O is compact,

where we put 0 for 0®zZ for the integer ring 0 of F. Since A = AfXR,we have FA = FAfxFoo with FA£ = F®QAf and Foo = F ® Q R . Thus we canwrite x = (xf,Xoo) with the finite part Xf e FAf and the infinite part Xoo e F^.Let I be the set of all field embeddings of F into C. Then Aut(C) acts on Ifrom the right via natural composition. Then the set of archimedean places a isdefined to be the set I/(c), where (c) is the group of order 2 generated by com-plex conjugation c. Each c e a gives rise to a complex absolute value | | aon F by I a | a = | aa | for any representative a e I. Then writing I(R) forthe subset of I consisting of real embeddings and I(C) = I-I(R), i.e., the set ofembeddings whose image is not contained in R. We put a(R) = I(R) anda(C) = I(C)/<c>. Then as seen in (1.1.5b), F^ = R a ( R ) xC a ( C ) . SimilarlyO = UpOp, where p runs over all prime ideals of 0. The unit group FAX ofFA is called the idele group of F, whose elements are called ideles of F. We givethe topology on FAX SO that the system of neighborhoods of 1 is given asfollows. Let 6x(m) be the kernel of the natural map YlpOp

x -> (O/m)x for allideals m of 0. Then a fundamental system of neighborhoods is given by

{0x(m)xU | wan ideal of 0, U a neighborhood of 1 in F ^ } .

Thus FAX is a locally compact group. The elements of FAX are called ideles.

The topology of FAX is stronger than the one induced from the adele ring FA- Infact, the neighborhood of 0 of FA is given by 0(m)xV for the kernel 6{m) ofthe natural map of additive group HpOp —> (O/m) and a neighborhood V of 0in Foe. One sees easily that {l+0(m)xV}nFAX 3 0x(w)x(l+V) but they arenever equal.

Exercise 1. Show that {l+0(m)xV}nFAx^ 0x(m)x(l+V).


In the same manner as in (3a), we see that

(4a) FA fx = {x e UpFp\ xp e Op for almost all p}.

From this, writing xp0p = pe^p\ we know that e(p) = 0 for almost all p. Thusthe ideal xO = Upp

e^ makes sense as a fractional ideal of F. Thus we have anatural homomorphism of groups

(4b) FAfx -> / given by X H XO.

Obviously this map is surjective. By abusing symbols, we write xO for XfO

when x € FAX. We now define the adele norm | x | A by | Xf | Afx I Xoo I «o for

I x I Af = HPI xp I p and I Xoo I co = IToe a(R) I xa I oxHoe a(C) I xo I a2- We normalize

1®P I p = N(p)'1 for the prime element U5p of Op. Then for a e Fx, we see that

a(R) I a a I oxIIacEa<C) I a a | a2 = I l a e i I a° | = |iV(a) I.|a |oo = l l a e

On the other hand, writing the prime factorization of aO as aO = Hpp^p\ we seefrom (1.2.2b) that

I a I Af = TlpN(pei*>yl = TV(aO)"1 = | W(a) I -1.

By this we know the following product formula:

(5) I a | A = 1 for all a e Fx.

Put F ^ = {x e FAX I x I A = 1}- Then by the product formula, we see that

F(i) -x -pxA —-> JP •

Theorem 1. Fx is a discrete subgroup of F ^ and F^ /F x is a compact group,

where the system of neighborhoods of 1 of the quotient group F ^ / F x is given

by the images of the neighborhoods of I in F ^ \

Proof. The set of normalized absolute values of F, i.e.

{I \p I I o I P: prime ideals, o e a} ,

is called the set of places of F. When we do not care much about the difference offinite or infinite places, we just write v to indicate a place of F. When we writep (resp. a), it indicates a finite (resp. archimedean) place. Let a be an idele anddefine

V(a) = ( x e Ap I xv | v < | av I v for all v}.

8.1. Adelic interpretation of algebraic number theory 243

Then we claim that the following assertion is equivalent to Corollary 1.2.1:

(6) for any idele with sufficiently large I a | A, V(a)riFx * 0 .

To see this, let a - aO. Since a = {x e F I x | v < | av | v } , we see

a = V(a)FcoriF. Thus we only need to show that

{ x e a | | x a | < | a a | for all a e a} * {0}.

Since a(V(a)nFx) = V(aa)nFx and I a | A = I oca | A for a e Fx, we may

assume that a is an integral ideal. Then by Corollary 1.2.1, there exists a con-

stant C > 0 independent of a such that if | a<x> I <« = Ilaei I aa | > CN(d), then

there exists 0 ^ a E a (<=> | (Xf I A ^ I &f I A) such that I oca I < I aa | (i.e.

| a a | a < | a o l a for all a). Since N(a) = UfL"1, U | A ^ C if and only if

I aoo I A ^ C I af I A"1 = CiV(a). This shows (6). Now we shall prove the theo-

rem. We already know that F is discrete in FA (3C). Since FAX has a stronger

topology than FA, the induced topology on Fx from FAX is stronger than the

topology induced from FA- Since the discrete topology is the strongest, Fx is

discrete in FAX. Take an idele c such that | c | A ^ C. Then for arbitrary

a E F ^ \ | ca"11 A ^ C. Thus we can find by (6) an element a E Fx such

that a E a'1cV(l) = V(a"1c). That is, aa e cV(l). Since aa is again in

F ^ \ applying the above argument to (aa)"1, we can find p e Fx such that P

E aacV(l) or p(aa)"1 € cV(l). Thus p = p(aa)"xaa e cV(l)cV(l) which is

contained in c2V(l). Since c2V(l) is compact and Fx is discrete, c2V(l)f|Fx is

compact and discrete and is therefore finite. Writing c2V(l)f|Fx = {pi,..., pm},

we see that (aa)"1 e U^Pi^cVQ) for any a e F(^}. Now we put

B =

Then aa G B and (aa)"1 e B. Since B is a compact subset of FA, it is easyto see that B* = {x E B | X"1 G B} is a compact set of FAX. In fact, foreach v, I x | v for x E B* is bounded from above and from below, i.e.3MV > 0 such that | xv | v < Mv and U U ^ M / 1 because x and x"1 e B.These Mv can be taken to be 1 for almost all v. Thus, we see thatFXB* 3 F(^} and B* is a compact set of FA

X. Thus F^}/Fx is compact.

Corollary 1. Let I be the group of all fractional ideals of F and & be the sub-group of principal ideals. Then we have


which is a finite group.

Proof. To any idele a, we have associated an ideal aO = aOflF. This corre-

spondence induces a surjective homomorphism of groups p : FAX —> /. We see

easily that p " 1 ^ ) = F x 0 x F x . This shows the first isomorphism. Since

F A X = F^FX<3XF* by definition, we have the second isomorphism. Since

F x 0 x F x is an open subgroup of FAX, the middle term is a discrete group. Since

F ^ is a compact group, the last term is a compact group. Thus I/(P is a discretecompact group and hence is finite. This gives another proof of the finiteness of theideal classes.

Corollary 2 (Dirichlet-Hasse). Let S be a finite set of places. Let

U ( S ) = { X E F AX | | x | v = l for all v g S} and O(S)X = U(S)OFX. Then

if S contains a, i.e. S contains all infinite places, then O(S)X = JI(F)XZS"1 for

s = #(S), where JI(F) is the group of roots of unity in F, and Z is considered

as an additive group.

Note that 0(Soo)x = 0* and hence the above corollary includes Dirichlet's theo-rem about units as a special case.

Proof. Consider the group homomorphism

0 : F AX s a H> (log I x | v ) v e s e Zs"r x Rr,

where r is the number of infinite places and we have normalized log at each placeso that the above map has values in Z at the finite places. Then Ker(6) = U(0).Thus U(S)/U(0) = Zs"rxRr. Since for the set of infinite places Soo,

F ^ = FAX, we see that

U(S)/F^nU(S) = FAX/F(

A1} = R .

Thus F(^nU(S)/U(0) = Zs-r x R1"1. On the other hand,

F(^nU(S)/O(S)xU(0) = F^nFxU(S)/FxU(0)

is compact but FxU(S)/FxU(0) = C<S)x/cX0)x is discrete. Thus

F^nU(S)/O(S)xU(0) = (R/Z)r4xG

8.2. Hecke characters as continuous idele characters 245

for a finite group G and O(S)X/0(0)X = ZsA. By Lemma 1.2.3, we see that

O(0)x = JLL(F). This shows the result.

§8.2. Hecke characters as continuous idele charactersIn this section, we show that each ideal Hecke character X uniquely induces acontinuous character of FAX/FX such that ^(x) = X(xO) if xc = 1 for the con-ductor c of X. Here xc = (xp)p^c. Let F be a number field and I be the set ofall embeddings of F into Q. Let O be the integer ring of F and m be anon-trivial ideal of O. Write I(m) for the group of all fractional ideals of Fprime to nu Let Fv be the completion of F at v and Ov (resp. n^) be the clo-sure of O (resp. m) in Fv. (We use the symbol /v to indicate the prime ideal ofO corresponding to v and also its closure in Ov if v is a finite place). We de-fine Uv(m) = {a G Ov I a = 1 mod mv}. Thus if mv = pv

e, then

U v W = { a e Ov I a-1 | v < N(p)'e], a closed disk of radius N(p)'e centered

at 1, and \Jw(m) = OyX if pv is prime to nu We define

U(w) = J[ J v finiteUv(m) and Foo

Foo+ = {x = (xv) G F ^ I xv > 0 if v is real},

P(m) = FxH{x G FAX xp G XJv(m) if p I m and Xoo G FOO+}

= {a G Fx | a G Uv(m) if py divides m and a a > 0 if a is real}.

Here the intersection is taken in the idele group FAX. Let T(m) be the subgroupin I(m) consisting of principal ideals spanned by elements a G P(m). Then theray class group Cl(m) modulo m is defined by the following exact sequence:

1 -> <P(m) -> I(m) -> C\(m) -» 1.

We have another exact sequence

1 -> E(w) -> P(w) -» !P(OT) -> 1,

whereE(m) = FxnU(m)Foo+

= {a G Ox I a G Uv(w) if pv divides w and cc° > 0 if a is real}.

Let x = (xv) G FAX be an idele such that xv = 1 if either v is infinite ormv ^ Ov (i.e. pv divides m). Then xv0v = pv

e^ and e(v) = e(xv) = 0 foralmost all v including those v appearing in nu Then we defined a fractionalideal xO in F by xO= IIv/v • This gives a group homomorphism of


FA(W) = ( x e FAX I xv = 1 if either v is infinite or mv * Ow]

into I(m).

Lemma 1. We have the following two exact sequences:(1) 1 -> FxU(m)Foo+ -> FA

X -> Cl(m) -> 1,

(2) 1 -> U(w)nFA(w) -» FA(lit) -» /(w) -» 1.

Proof. We first prove the exactness of the second sequence. The projection mapis given by x h-> xO. If a- IIvPv6^ is an element of I(m), then e(v) = 0 ifpv divides m. Since Oy is a valuation ring, every ideal is principal, and hence

we have p ve ^ = xv0v for some xv G Fv

x. We define xv as above if v isfinite and pv is prime to nu We simply put xv = 1 if either v is infinite or &divides m. Then a=xO and thus the map F A O ) -> I(m) is surjective. IfxO = O, then e(xv) = 0 for all v and hence x e U(O). Since xv = 1 if ei-ther v is infinite or py divides m by the definition of FA(W), we see thatx G U'(w)flF\{m) if xO = O. This shows the exactness of the second se-quence. Since uxO = xO for u e U(m)Foo+, we also know the exactness of

(3) 1 -» U(m)Foo+ -> U(m)FA(m)Foo+ -> /(m) -> 1.

By definition, we have

U(/rc)FA(m)Foo+ = {x G FAX | xp G Uv(»t) if p\ m and Xoo e Foo+},

and hence U(m)FA(/n)Foo+riFx = P(m). Thus we see

because iP(w) = P(m)/E(m) and E(w) is contained in U(m)Feo+. By definition,

F x is dense in Fmx = Tlp\ JF7^. Thus, for each idele x e FAX, there exists

a G Fx such that (a"1x)mG HpOpx. That is, aAx0 is prime to m. Thus we

can find y e U(WI)FA(WI)FOO+ such that a'lx0 = yO. That is,

oc^xy'1 G U(0)Foo+. Since

XJ(O)fU(m) = n ^ | m(Owx/\Jy(m)) = n ^ | m(Ojmv)

x = (O/m)x,

we can find ye O such that y = a ^ x y " 1 mod U(m), and hence

x G yayU(m). Therefore FAX = FxU(m)FA(/n)Foo+. Since

we see that

8.2. Hecke characters as continuous idele characters 247

FAx/FxU(i»)F«H. S

= U(ifi)FA(w)F^{FxnFA(w)U(w)Fo.+}U(w)F«+s Cl(ift).

This shows the exactness of the first sequence.

Let X* be a Hecke character modulo m; that is, X* is a character of I(m) such

that A*((oc)) = oc if a e P(m), where § = X ^ a G Z[I] and a5 = UGa°^ for

a e F x .

Theorem 1 (Weil). There exists a unique continuous character X : FAX/FXU(7TC)

-> Cx such that X(x) = X*(xO) if x G FA(*K) 0«d Mx~) = *~^ = nax"aâ if

Xoo e Foo+, w/zere x a = xv and x o c = xv for o = Gv and complexconjugation c.

Proof. By the exact sequence (3), we know that

Thus we can define a character A,: \J(m)FA(m) -> Cx by X,(x) = A*(xO) for x e

UO)FA(/tt). We extend this character X to U(w)FA(w)Foo+ by X(x) =

A,*(xfO)x~ where x" = ITax"aâ and Xf is the projection of x to U

Here note that xf0 = x0. Thus for a e P(i») = U(m)FA(m)Foo+nFx,

A,(ax) = A,*(axfo)(ax)"^ = A.*((a))A*(xfo)(x'5x"* = ^*(xfo)x"^ = X(x).

Now we extend this character to FAX = FxU(7rx)FA(/n)Foo+ by X(jx) = X(x) if

y e F x and x E U(/n)FA(w)Foo+. This is well defined. Indeed, if yx = 8y

for x,y G U(w)FA(m)Foo+ and y,8 G FX, then

F B yl8 = y-1x G U(m)FA(m)Foc+ and yS"1 G P(W).

Then X(yx) = X(x) = ^ ( y ^ y ) = X(y) because X(ax) = X,(x) if a G P(O T ) .

This shows the existence of X. The uniqueness is obvious because

Exercise 1. Show the continuity of X as above. (A sequence xn e FAX con-

verges to x if and only if for any ideal a of O and e > 0, there exists a positive

number M such that if n > M and m > M , then xn-xm G U(a)Foo+ and

I xn-xm I v < £ for all infinite v.)


Exercise 2. Show that if t, = 0, then X* is a character of Cl(/n) and X is

just the pullback of X* : CIO) —> Cx by the isomorphism

FAx/FxU(w)Foo+ = Cl(m) given in Lemma 1.

Exercise 3. Let X be as in the theorem.(i) Show that ^(x) is contained in a finite extension K of F for all

x E FA(m).(ii) Enlarging K if necessary, we suppose that K contains all conjugates of Fover Q. Fix one finite place v of K and write p (resp. | | v) for its residualcharacteristic (resp. the v-adic absolute value on K). Then show that every placew of F over p is given in such a way that I x | w = | xa | v for some d e l .Write this place as w(a).(iii) If w = w(o), then a : F —> K extends to a : Fw -> Kv by continuity.

Define for x e F px = ( F ® Q Q p ) x = n w | P F W

X , x^ = n a x a ^ a where

x a = (xw(a)) a . Then show that there is a unique continuous character

Xp : FAx/FxU(w)Foo+ -> Kv such that Xp(x) = X*(xO) if x e FA(/np) and

^p(xp) = X(xp)Xp"^ for all xp e Fpx.

Now for a given character X : FAx/FxU(/n)Foo+ -» Cx such that (Xoo) = Xoo" ,

we can recover a Hecke character X* by putting X*(xO) = X(x) forx e FA(m). In fact, if xO = yO for x and y in FA(m), then xy"1 e \J(m)

and hence ^(xy'1) = 1, that is X*(xO) = X*(yO). If a e P(m), thenX*((a)) = X(am) for the projection am of a to FA(w). On the other hand, theprojection a m of a to IIvlmFv is contained in \J(m) becauseU(/rc)FA(7rc)Foo+nFx = P(w). Thus ^ ( a j = 1. On the other hand, 9L((XOO) =a"^. Since X is trivial on Fx, we see 1 = X(a) = Xia^Xia^Xia^) =

X*((a))a'^ and hence X*((a)) = ofi if a e P(m). Thus the correspondenceX <-> X* is bijective for a given infinity type ^ and for a given defining ideal m;that is, we have

Corollary 1. The correspondence of Hecke characters: X<r*X* is bijective

for a given ideal m and the infinity type \. (The same statement is true for thep-adic version of Hecke character Xp given in Exercise 3 (iii).)

Hereafter we identify ideal characters and idele characters and study continuouscharacters X : FA

x/FxU(w) -> Cx with A,(Xoo) = x ^ for % e Z[I] insteadof ideal characters.

Exercise 4. Let m and n be two ideals. Show that XJ(m)\J(n) = U(m+n).

8.3. Self-duality of local fields 249

For a given Hecke character X as above, if X is trivial on U(w) and XJ(n), thenby Exercise 4, X is trivial on U(wri-n). Thus there exists a largest ideal c suchthat X is trivial on U(c). This ideal c is called the conductor of X.

We now study additive characters of FA- We start with the simplest case of A.Pick a prime p. Considering the p-fraction part [z]p of z e Zp, we define thestandard additive character

(4) ep : Q p -> T = {z e C | | z | = 1} by ep(z) = exp(-27C V^lMp).

For any finite idele x e Af, we define ef(x) = npep(xp) = exp(-27i V~-IZp[xp]p).This is well defined additive character since for almost all p xp e Zp and henceep(xp) = 1. We define eoo(Xoo) = exp(2rc V-T Xoo). Thus we have an additive char-acter e : A —> C given by e(x) = ef(xf)e(Xoo). If a is a rational number, then

a - X p [ a ] p e ZflQ = Z.

Thereforee(a) = exp(-27c VzlEP[a]p)exp(27cV::Ta) = exp(2^z^[-[(a-Tp[a]v)) = 1.

Namely e is trivial on Q and hence is a non-trivial additive character of A/Q.

Theorem 2. For any number field F, there exists a non-trivial additive charactere = eF : FA/F -> T.

Proof. We have explicitly constructed e = eQ when F = Q. For a generalnumber field F, we simply define ep by e?(x) = eQ(Trp/Q(x)). Since the tracemap sends elements in F onto Q, ep is a non-trivial additive character of FA/F.

§8.3. Self-duality of local fieldsWe fix a place v of F and consider the completion K = Fv. Let

T = { z e C | z | = l } , which is a multiplicative group. We want to show that

the group of continuous additive characters Homconti(K,T) of K is canonicallyisomorphic to K. We start proving this first assuming K = R. If a : R -» Tis a continuous homomorphism, we consider its kernel L = Ker(cc), which is aclosed subgroup of R. Suppose that it has an accumulation point x. Then x it-self belongs to L, and we can pick y e L arbitrarily near to 0 but not equal to0. In other words, z = y-x is arbitrarily near to 0 but not equal to 0. Then Zzis a subgroup of L whose adjacent elements have distance I z |. Since we canmake | z | —> 0 keeping z in L, we know L is dense in R. Since L isclosed, L = R and a is the zero map. Thus for non-trivial a, L is a discretesubgroup of R. Let z be the element in L-{0} with the minimum distance to 0.Then by the euclidean algorithm, for each x e L, we can find q e Z so that


x e [qz,(q+l)z). Then x-qz e L has absolute value smaller than z, andhence by the minimality of the distance between z and 0, we know that x = qz.Namely L = Zz. Since the homomorphism (p :xh ) exp(27iV-Iz"1x) has thesame kernel as a, there is a unique automorphism P : T = T such that a =p°q>. First we fix the lifting i : T-{-l} -> ( - £ , £ ) by i(exp(2rc V ^ x ) ) = xfor x with I x | < j . Since p is an automorphism, p sends a small neighbor-hood U of 0 onto another neighborhood U' of 0. We take a still smallerneighborhood W in U so that W+W c U. Thus pf(x) = i(p(exp(27i-vcTx)))is a well defined additive map on Wo = i(W), which is a small neighborhood of0 in R. Since any additive map on WoDQ is induced by a linear map x h-> bxfor some b e R , the continuity of p' tells us that pf coincides with x h-> bxfor some b e R in a small neighborhood Wo. Then by the linearity of p, wesee that p(exp(27cV~Ix)) = exp(27tV--lbx), which is an automorphism of T,and hence b = ± 1 . This shows that a(x) = exp(±2rc V-lz^x) . We haveshown the surjectivity of the natural map R B b H eb G Homconti(R,T)given by eb(x) = exp(2rc V-l bx). The injectivity of this map is obvious. Thuswe have

Homcont(R,T) = R via eb <-> b.

By this, HomCOnt(R2,T) = R2 and hence Homcont(C,T) = C. We can check thatthis isomorphism is induced by eb <-> b for eb given by eb(x) =exp(2n;V~lTrc/R(bx)). We now extend this result to any finite place v. For that,we recall the duality theory of locally compact groups. Let G be a locally compactabelian group. We consider its dual group G* = Homcont(G,T) and on it we putthe topology of uniform convergence on every compact subset of G. Thus asequence of characters a n : G -> T is convergent to a character a if a n

converges to a uniformly on any compact subset X of G. Let e : K -» Tbe the additive character defined in the previous section, i.e.

e(x) = exp(-27C-v/-T[TrK/Q (x)]) if the residual characteristic of v is pand

e(x) = exp(27cV-T[TrK/R(x)]) if v is infinite,

where [x] is the p-fractional part of x. Then we can define a pairing( , ) : K x K -> T by (x,y) = e(xy). Then we want to prove

Proposition 1. The above pairing induces HomcOnt(K,T) = K.

Before proving the proposition, we prove a preliminary lemma:

8.3. Self-duality of local fields 251

Lemma 1. Identify T with R/Z and let Ik be the image of the interval

(-—,—) in T for positive integers k. If a homomorphism a of a group G

into T satisfies l\ 3 oc(G), then a = 0.

There are two consequences of this lemma:

(i) If G has a system of neighborhoods of the identity consisting of (open) sub-groups H, then any continuous homomorphism a : G —» T becomes trivial ona sufficiently small open subgroup H.

(ii) If G is compact, then G* is discrete. In fact, if a continuous character ocn

of G converges to a in G*, then the convergence is uniform on G (because Gitself is compact) and oc-an has values in Ii on G for sufficiently large n.This means (Xn = a and thus G* is discrete.

We add one more remark:(iii) If G is discrete, then G* is compact, because G* is easily seen to be theclosed subspace of compact space TG, which is the set of all functions of Ghaving values in T.

Proof (Pontryagin). We claim that if jt e l\ for all j = 1,2,3,-.., k, thent e Ik. We prove this by induction on k. If k = 1, the assertion is triviallytrue. Suppose that the assertion is true for k-1 and try to prove the case of k.

Choose a representative x of t in (-«,, n , -ZTT——). If x is already in

(-—,—), there is nothing to prove. Thus we may suppose that

| x | e [g£, 30^1))- Then | kx | e [ - , - ^ - ^ - ) . This interval is inside [0,1)

and hence kt e Ii means that kx e (- -, -) and hence x e (-—,—), which3 J 3K JK

means t e Ik. Now let us prove the lemma. If oc(G) is contained in Ii, thena (kg) = ka(g) E Ii for any positive integer k and any g e G. ThusIk D a(G) for all k, which means that oc(G) = 0 because D iA^ lO}-

Proof of Proposition 1. We may suppose that v is a finite place over a rationalprime p. First assume that K = Qp. If <() e Homcont(Qp,T), then for a suffi-ciently small neighborhood V of 0 in Qp, Ii z> (|)(V) by continuity of <|). Wecan take the subgroup prZp as V. Thus Ii z> (|)(prZp) and hence by the lemma,0(prZp) = 0. Since for any x e Qp, we can find a sufficiently large exponent ksuch that pkx e prZp, we see pk(|)(x) = <|)(pkx) = 0. Thus identifying T withR/Z via xh-»exp(27tV-Tx), the value of continuous character is contained in theimage of fractions whose denominator is a p-power, i.e. Tp = Qp/Zp z> <|)(QP)

252 8: Functional equations of Hecke L-f unctions

(see (1.1)). Thus what we need to show is Homzp(Qp,Tp) = Qp under the pair-

ing (x,y) = xy mod Zp, because any continuous homomorphism <|): Qp —»Tp is

Zp-linear. Thus Q p is sent into Homzp(Qp,Tp) via X H (()X for

$x(y) = (xy mod Zp). Since Tp is divisible (i.e. for any y G Tp, we can

find x G Tp such that px = y), applying the functor Homzp(*,Tp), we have

the following commutative diagram:

Q P

4a ip0 -> HomZp(Tp,Tp) -> HomZp(QP,Tp)

where both rows are exact ((1.1.1a)), a takes ze Zp to multiplication by z onTp = Qp/Zp, p(x) = (|)x and y takes t G Tp to the unique homomorphism <[)such that <|)(1) = t. The surjectivity of 8 can be proven as follows. If<|> : Zp -» Tp is a Zp-linear map, then we can extend it to p"rZp by putting(|)(p"r) = xr for xr in Tp such that prxr = $(1). Thus <|> is extensible to p"rZp

for any r and hence extensible to Qp. By definition, y is an isomorphism. Thuswe only need to show that a is an isomorphism in order to show that p is anisomorphism. If oc(x) = 0, then multiplication of x on p"rZp/Zp = Zp/prZp iszero and hence x is divisible by pr for arbitrary r. This shows that x = 0 andhence a is injective. If ty : Tp -> Tp is a Zp-linear map, then $ induces amap (t>r: p' rZp/Zp -> p"rZp/Zp. Thus <|>n G End(Z/prZ) = Z/prZ. Since <|>r

induces (t>s if r > s , as an element of Z/prZ, <|>r = <|>s mod ps. Picking an in-teger xn such that xn = <|)nmodpn for each n, the sequence {xn} converges tox G Zp because | xr-xs | p < p"s if r > s. By definition oc(x) = 0. Thus ais surjective and hence the assertion follows when K = Qp. By this we knowHomZp(Qp

r,Tp) = Qpr via the pairing (x,y) = ZJ=1XJVJ mod Zp. Now we

treat the general case. Identify K with Qpr by choosing a basis {vi,...,vr} over

Qp. We can also identify the Qp-dual vector space K* of K with K via thepairing (x,y) = TrK/Qp(xy). Thus by choosing the dual basis Vj* of K (i.e.(vk,Vj*) = 8kj), we can identify K with Qp

r. Then by the above argument, weknow that Homcont(K,Tp) = K via the pairing (x,y) = exp(27cV-lTrK/Qp(xy)).This shows the proposition.

Theorem 1. Define a pairing ( , ) : FA X FA -» T by (x,y) = e(xy) for

the adelic standard additive character e : FA/F —> T. Then this pairing induces

an isomorphism FA = Homcont(FA,T).

8.4. Haar measures and the Poisson summation formula 253

Proof. Since FA = FAfXFoo and Foo is a product of finitely many copies of R

and C, we only need to prove the self-duality for FAf. Since n(5 for positive

integers n gives a system of neighborhoods of 0 in FAf, for any continuous

homomorphism <|> : FAf -» T = R/Z, we can find n so that (|)(nd) = 0 by

using Lemma 1. Thus we know that Q/Z z> (|>(FAf). We first assume that

F = Q. As seen in (1.1) and (1.3b), e f : Af/Zf = 0p(Qp/Zp) = Q/Z. Using

the commutative diagram analogous to the one in the proof of Proposition 1,

0 > Z > A f > 0 p T p > 0

l a i (3 4 70 -> Hom(Q/Z, Q/Z) -> Hom(Af, Q/Z) -> Hom(Z, Q/Z) -> 0,

8

we know that Hom(Af, Q/Z) = Af. This shows the assertion for Q. Now wecan take a basis vj of O over Z; then F = Q d and O=Zd (d = [F:Q]) viathis basis. Then we see that FAf = Afd. The character e takes, by definition,FAf into Q/Z (in fact e = eQ©Tr for the trace map Tr : FAf —> Af). Thus weknow from the same argument as in the proof of Proposition 1 thatHom(FAf,Q/Z) = FAf via the pairing induced from e. This finishes the proof.

By Lemma 1 and the above proof of the theorem, for each non-trivial continuouscharacter <|> : Fv —> T (resp. <|) : FA -> T), there is a fractional ideal pv~

5 inFv (resp. -ft"1 in F) maximal among the ideals a with (|>(a) = l. The inverseof this ideal is called the different of <(>. If e is the standard character, then the in-verse of the different of e is the dual lattice of O under the pairing(x,y) = Trp/Q(xy) and hence is the different inverse &l in the classical sense. If<|)(x) = e(ax) for all x e FA, then the different for (J) is given by aft.

Since the pairing ( , ) on FA is continuous under the topology on FA, the iso-morphism of Theorem 1 is in fact an isomorphism of topological groups.

Exercise 1. Show that the isomorphism of Proposition 1 is continuous if

§8.4. Haar measures and the Poisson summation formulaTo prove the functional equation for Hecke L-f unctions of number fields, we needthe theory of Fourier transform on a locally compact abelian group G, which weexplain now. We assume the following conditions.

(la) There is a continuous pairing ( , ) : G x G — > T = { z e C | | z | = l }


which induces an isomorphism G = Homcont(G,T) (on the right-hand side, weput the topology of uniform convergence on all compact subsets of G and thisisomorphism has to be an isomorphism of topological groups).

(lb) G has a cocompact discrete subgroup F. Here the word "cocompact" meansthat G/T is compact.

(lc) The orthogonal complement F* = {x* G G\ (X*,X) = 1 for all x G F} isagain a cocompact discrete subgroup of G.

In fact, the third condition is superfluous and it is known that it follows from(la,b) (see the remark after Lemma 3.1). In our application, G will be a vectorspace over R or the adele ring FA. Let us check the conditions (la,b,c) for areal vector space V = Rr. Let S : VxV -* R be any non-degenerate innerproduct and L be a lattice of V. Then L is a discrete subgroup of V and bychoosing a basis {vi,,..,vr} of L over Z, we can identify L with Zr. Sincethe Vj's form a basis of V over R, we can identify V with Rr via this basis.Then V/L = (R/Z)r is compact. Now we define ( , ) : V x V - 4 S by(x,y) = exp(2jcV=lS(x,y)). Then

L* = {x* G V | (x*,x) = 1 for all x G L}= {x* G V | S ( X * , X ) G Z for all XG L} .

Then L* is the dual lattice generated by the dual basis Vi* such thatS(vi*,Vj) = 8y. Thus L* is again a lattice and is discrete and cocompact. WhenG = FA, we can take F as a discrete subgroup of FA-

On the group G, there exists a Haar measure dji with values in R, which sat-isfies the following conditions:

(2a) JLL is defined on a complete additive class containing all compact subsets of G(e.g. a union of countably many compact subsets is measurable);

(2b) 0 < |Li(K) < +oo for all compact subsets K o / G (|n(K) > 0 if the inte-rior of K is not empty),

ji(U) = SUPUDK compactM-(U) for all open sets U and\x(X) = Infuz,x, u openM-(U) for all measurable subsets X;

(2c) ja.(x+X) = [i(X) for all measurable subsets X and x G G.

Under the conditions (2a,b,c), the Haar measure is unique up to a constantmultiple. The uniqueness is intuitively obvious because if K is a disjoint union ofK' and x+K', then |i(K') = |i(K)/2. In this way, if one fixes a compact neigh-borhood of 1 with positive measure, by subdividing it, the measures of allcompact subsets are uniquely determined. The condition (2c) implies the equality

8.4. Haar measures and the Poisson summation formula 255

(3) Jef (x+y)dji(x) = fGf(x)d*i(x)

for any integrable function f with respect to JLL. When G is a real vector space,the Haar measure on V is a constant multiple of the Lebesgue measure induced bythe identification G = Rr.

Now we fix the Haar measure (I on G and consider the Hilbert space L2(G/T)

of /^-functions on G/F. In fact, by taking the fundamental domain K of F in

G so that K is compact and the complement of K is open, the measure |i in-

duces a measure on K, which gives a Haar measure on G/T. By multiplying by a

suitable constant, we may assume that Jo/rdM-W = *• The Li space Z^G/F)

is the space of functions f: G/F -> C which are square integrable; i.e,

JG/T ' x ) ' 2^M-(X) < +o°- Thus w e c a n define the positive definite hermitian inner

product on Li(G[T) by

For y* e F*, we consider the character \\f = \|/y* : G —» T given by

\j/T*(x) = (y*,x). Then for y e T, \|f(x+y) = (y*,x)(y*,y) = \\r(x) because

(Y*,Y) = 1. Thus \|/ is a continuous character of G/F. Since G/F is compact

and \|/ is continuous, we know that \}/ e L2(G/F). Moreover

= L

Since G = Homcont(G, T), if y*,8* e F* and y* * 8*, then for some

y e G, Vy*(y) * V8*(y)« On the other hand, we consider an operator

Ty : L2(G/F) -> L2(G/F) given by Tyf(x) = f(y+x). Then we see from (3)

that

<Tyf,g> = J G / r f ( y + ^ ) g ( x ) ^ ( x ) = JG / rfWg(x-y)d|l(x) = <f,T.yg>.

This implies1 = <Ty\|/Y*,\|/5*) = \|/5*(y1

and hence (xj/^,^*) = 0 if y* * 8*. Moreover it is known (see [P]) that

(4) \|/yn for y* e F* gives an orthonormal basis of L2(G/F).

Thus, any f e L2(G/F) is expanded into the /^-convergent series

f = Xy*er* C(Y*)VT* for C(Y*)


Exercise 1. By applying the functor Hom(*, T) to the exact sequence of

abelian groups 0-»M—» N -» L -> 0, we have naturally another sequence

(5) 0 -> L -> N -> M -> 0,

where M denotes Hom(M, T) for any abelian group M. Show that the se-quence (5) is exact. For the surjectivity of the last arrow: For each a e M,consider the set A of pairs (X,£) consisting of subgroup X of N and a ho-momorphism E, : X -> T such that ^ | r = ex. Put an order (X,£) > (X',£') onA when X D X 1 and £ I x1 = £'• Then by Zorn's lemma, there exists a maximalelement (X,^) in A. Show that X = N and the image of Z, in M is a.

Now we define the Fourier transform for any integrable function f on G by

(6) J(f)(y) = Jr(0(y) = JGf(x)(x,y)dn(x).

Since | (x,y) 1 = 1 , the above integral is always convergent if f is integrable.

Theorem 1 (the Poisson summation formula). Suppose that

(i) f is a continuous function on G integrable with respect to \i,

(ii) ZYGrf(x+y) and Zy*Gr* jF(f)(x+y*) are both absolutely and locally uni-

formly convergent. Then we have

£ y s r f(x+y) = Y

Inparticular, ^yeT f(y) =

Proof. The function O(x) = Z 7 e r f(x+y) is invariant under translation by the

elements of T. Thus we may consider <& as a function on G/T. Since G/T is

compact, O is square integrable. Thus we can expand

C(Y*)\J/Y*(X) for c(y*) =

Let K be the fundamental domain of G/T we have already chosen. Then

,> = JG/r(-y*,x)O(x)dn(x) = JG / r ( -Y*,x)Iy e r

= E v e r JG/r(-Y*'x)f(x+Y)d|4(x) = JUYK+7(-Y*,x)f(x)dn(x)

= JG(-Y*,x)f(x)d^(x) =Thus we have

8.5. Adelic Haar measures 257

Since the convergence is locally uniform, the above identity is not only in theL2-space but the actual identity of the two functions valid everywhere.

§8.5. Adelic Haar measuresNow we want to construct explicitly the Haar measure on FA and FAX. When vis infinite, Fv is C or R and thus we have the Lebesgue measure dx on Fv.The Haar measure on the multiplicative group Fv

x in this case is given byI x I-1dx when v is real and djiv

x(z) = |x2+y |"*dxdy (writing z = x+V~Ty)when v is complex. Now we treat the case where v is finite. We only need tomake explicit the integration of locally constant functions on Fv for finite v underthe Haar measure. A function f on a topological group G is called locally con-stant if for any x e G , there exists an open neighborhood V of x such that fis constant on V. Thus for a given x e V, the set ( ye G | f(y) = f(x)} isan open set. In particular, any locally constant function is continuous. To knowthe volume of open compact subsets of the Haar measure on CV, we may assumethat JI(OV) = 1. Since {jp^}j=i,2,... gives a system of neighborhoods of 0 andOy = Uaa+pv-" is a disjoint union of open subsets, where a runs over arepresentative set for CK/pyK Thus we must have

This shows that for any generator G5V of

(1) j

Of course this formula is valid for all a e Fv and i e Z (not necessarily posi-

tive). If f: Fv —» C is a locally constant function, then as already remarked,

the set f^fCx)) = {y e Fv | f(y) = f(x)} is an open set for a given x. Thus

we can write, for f(x) = c, f-1(c) = UaCa+jv1^) as a disjoint union. Then

formally

and jiv(fx(c)) = 2 > ^

If the above sum is absolutely convergent, then f is called integrable. Inparticular, if f is compactly supported, that is, the closure of the set( x e Fv I f(x)*0} (which is called the support of f) is compact, then f isintegrable. We then have the property (4.2a-c) for this Haar measure. Similarlywe can define the multiplicative Haar measure d|Lix by

25 8 8: Functional equations of Hecke L-functions

(2) \i - I ) ' 1 if j > 0

for a e Ovx. Since a+/v* = a(l+;v*) and the subgroups {l+pv

J}j>o give asystem of neighborhoods, the same argument as in the additive case shows the in-variance property of |ivX-

Exercise 1. (i) Show that ( x e Fv | f (x )*0} is closed if f is compactlysupported and v is finite.(ii) Show that for any locally constant function f whose support is in Fv

x,f * i f I I 1\v xf(x)djiv (x) = N(jtfy)(N(pv)-l) Jp f(x) I x | v d|iv(x).

J Jrv • rv

(Reduce the problem to the formula Jivâ+pv1) = (l-Nipy)'1)'11 a I ~ M-vfa+^v1)-)

(iii) Show that for any compactly supported locally constant function f on Fv

and a * 0, JFvf(ax)d|iv(x) = I a | v"1JFvf(x)djiv(x) (use (ii)).

We now define the Haar measure on FA. For each fractional ideal a, we write &

for the closure of a in FAf, which coincides with Ylv fmite v- First, on FA^ we

define the additive measure |if and the multiplicative measure |Hfx by

|U.(a+a) = N(a)~l for each fractional ideal a in F,

îx(aU(a)) = (U(0):U(a))4 = WQ/a)*)'1 for each ideal a in a

Since {U(a)}a (resp. {&}a) gives a system of neighborhoods on AfX (resp.

Af), we can verify that |Lif and |ifX are well defined. Let [U (resp. |ieoX) be theadditive (resp. multiplicative) Haar measure on F«> induced by the Lebesgue mea-sure as already defined. Then if a function is of the form 0(x) = <|)f(xf)<t>oo(Xoo)with a locally constant function % on FAf and an integrable function L, on F«»,

then the integration under the Haar measure \x = |if<8>}ioo (resp. JLLX = |iXf0jiXoo)can be computed by

= JFAf<))f(xf)d|if(xf)xJFeo(|)oo(xee)djieo(x<>o).JFAf<Exercise 2. Show that (if = ®v fmiteM-v. First show that the problem is re-duced to proving \if{a) = Tlv finiteM'v(flv) for all integral ideals a and then showthat the infinite product of the right-hand side is in fact a finite product and givesthe volume of the left-hand side.

8.5. Adelic Haar measures 259

Examples of integrals: Orthogonality relations (Lemma 2.3.1): Let % a nd A,be continuous characters (with values in T) on G = Ov

x or Ov. Let JJ, be aHaar measure on G. Then

Proof. Since %^ is continuous, it becomes trivial on a sufficiently small sub-group H of finite index by Lemma 3.1. Thus we see

(3)

By Lemma 2.3.1, we see that

2 , X E G / H X M X ) - | ( G : H ) . f x = r i >

Then the assertion follows from the fact that |i(G) = |i(H)(G:H).

Gauss sum. Let % be a continuous character of Fvx and $ be a non-trivial addi-

tive character of Fv. Let G5~r0v = fty1 be the different inverse of <|> for a gen-

erator 03 of pv (i.e. G5"rOv is the maximal fractional ideal in Fv such that

(j)(G3"r0v) = 1) and G3f0v be the conductor of % (i.e. G5fOv is the maximal ideal

such that %(l+GJfOv) = l if % is non-trivial on 0 / and m = 0 if % is trivial

on OvX). When f > 0, we consider the integral

(4a) I G3r | J G 3 O v

This is a Gauss sum. In fact, by the variable change: x i-» 03 x, we see

I C5r | vx(B3r+f)J05.r.fOvXx<Kx)d|a(x) = %(03r+f) | nf |

= I ®-f I vJOvxX(x)$(03-f-rx)d^(x) = | C3-f | V X X mod ajf

= I °5"f I v l x mOd rofX(x)JOv(|>(C5-r-f(x+03fy))dn(03f

y) = x mod ^

which is visibly a Gauss sum. We only integrate %<|) on G5"f"rOvX but the same

integral on G5mOvX vanishes if m ^ -f-r. In fact, we have

(4b) J03mOvX%(|)(x)d|i(x) = 0 if m*-r-f and f > 0 .

Proof. We see that


First suppose that m < -r-f. Then the above integral is equal to

| C3 | v £ x mod S 5 f X ( v

We then compute

J C 3 f o v * ( G 3 m ( x + y ) ) d ^ ( y ) = I G5f I v<]>(G5mx)JOv<t>(G3m+fy)d|J.(y).

Since m+f < -r, y h-> <|)(G5m+fy) is a non-trivial additive character. Hence by(3), we see the right-hand side of the above integral vanishes. Thus we have thevanishing of (4b). Now we suppose that m > -r-f. Then

= X x mod raf-iJx+05f-iOvX(x(l+03f-1y)Wn3mx(l+05f-1y))d^(x(l+05f-1y))

= Zxmod®M I xG5M I v$(03mx)%(x)Jn5f.1OvX(l+ti3f-1y)dH(y).

Since Jn5f.ic,vX(l+O3f'1y)d|i(y) = l^(G5fa)Iyel+rof-iOv/1+03f-iOvX(y)

= n(G3fOv)J1+05f.iOvX(y)d^x(y) = 0 by (3),

the vanishing of the integral we wanted follows.

Integrals at °°. Write I for the set of all embeddings of F into C. We consider

the inner product S : FxF -> Q given by Trp/Q(xy). This inner product ex-

tends to V = Foo as a real bilinear form. We identify Foo with Ra ( R )xCa ( C ) .

Then for each a e F, the image of a in V is given by (a a ) a € a(C)Ua(R> We

consider the function \|/y(v) = exp(-7tZa € lya I v a I 2) for yo € R+ with

ya = ya c for complex conjugation c, where for a e a(C), we agree to write

va c for va. Then we see that for any polynomial P(v), \|/p(v) = P(v)\j/(v) is

integrableon V and we have

Lemma 1. Let du be the usual Lebesgue measure on V. Then

(5a) JFeoVy(u)e(TrF/Q(uv))du = 2"W(y)-1/2\i/y-i(v),

where e(x) = exp(2TcV-lx), t = #(Z(C)) and N(y) = UGeiyo. Moreover let0 < r| e Z[I] such that T]a=0 or 1 for a e a(R) and r\oV[cc=0 fora E a(C). Then we have

8.6. Functional equations of Hecke L-functions 261

(5b) JFoouVy(u)e(TrF/Q(uv))du = 2-W(y)-1/2yî^}v^>

where [r]} = Zael'Ha e Z.

Proof. Note that Trp/Q(uv) = Zaea(R)UaVa+Z<y6a(C)(uava+uava). Thus theproblem is reduced to the computation of

(5c) exp(-7cyu )e(uv)du and exp(-27iyuu)e(uv+u v)du.J— oo J—ooJ—oo

The first integral is equal to -y/y exp(-7ix2/y), and the second follows from thefirst immediately. We obtain the last assertion via the differentiation by

j - \ = n ae 11 g^" I of the first formula (5a).

Exercise 3. Give a detailed explanation of the computation of the integral (5b,c).

§8.6. Functional equations of Hecke L-functionsIn this section, we prove the functional equation for Hecke L-functions via themethod of Tate-Iwasawa. We basically follow the treatment in [W1,VIL5]. Wefirst deal with the Fourier transform of standard functions on Fv or FA. For afinite place v, a function f on Fv is called standard if it is locally constant andcompactly supported. For a infinite place v, a function f is called a standardfunction if

f(x) = cxAexp(-7Cx2) for a constant c and A = 0 or 1 when Fv = R,f (x) = cxA xBexp(-27i I x I ) for a constant c and

integers A > 0 and B > 0 with AB = 0 when Fv = C.

A function f on FA is called standard if f(x) = Ilvfv(xv) f°r a standard func-

tion fv on Fv for all v and fv is a characteristic function of (\ for almost all

v. Let G be either FA or Fv. Let e : G —> T be the standard additive charac-

ter. We define a pairing ( , ) : GxG -» T by (x,y) = e(xy). Then we al-

ready know that under this pairing, we have G = Homcont(G,T). Let \i be the

additive Haar measure defined in the previous section. We modify JI as follows.

Define \i\ by 2JJ,V if v e a ( C ) , jn'v = |Liv for v e a ( R ) and

|TV = I G3r | V 1 / 2 J I V for finite places v, where we write the local different

T% = G3r0v f°r a prime element C3 of Ov. Then we define p,1 = ®|Ufv as the

Haar measure on FA. We will see later IFA/F^H' = 1. We now define the

Fourier transform of standard functions by


When G = Fv for a finite place v, then f is compactly supported and hence in-tegrable. Thus f(f) is a well defined function on Fv because | (x,y) 1 = 1 .When v is infinite, any standard function is integrable and thus the Fourier trans-form is again well defined. When G = FA, then by definition of standard func-tions, f(x) = foo(xoo)ff(xf) and ff is compactly supported and f<x> is integrable.Then we see

JGf(x)(x,y)dî'(x) =

and each integral of the right-hand side is well defined and hence the Fourier trans-form is a well defined function on FA- We also know that if G = FA andf = Ilvfv is a standard function, then

Thus the computation of Fourier transform of standard functions is reduced to localcomputation on Fv for each place v. As seen in §5, we already know that

(1) tffv) = JRxAexp(-7ix2)(x,y)d^'(x) = V^VexpC-Tiy2) if Fv = R,

T(U) = J c x A xBexp(-27C | x | 2)(x,y)d^(x)

= V=IA+ByByAexp(-27r | y 12) if Fv = C.

Thus we compute the Fourier transform on Fv for finite places v. Let K be agenerator of the maximal ideal of C\ and G3rOv = $v be the different of e. Firstwe take the characteristic function O = Ov of Oy. Then we claim

(2) n&)(y)= lG3r|v1/2O(G5ry).

Let us prove this. Note that the additive character y h-> e(xy) is trivial on a, ifand only if x e G3~r0v. Thus the orthogonality relation tells us that

J F O(x)(x,y)dji(x)=J0 e(xy)d|i(x) =«J^v J U v [0 if x ^ 03 Ov,

which shows (2), because |i 'v = I G5r I v1/2M-v Since r = 0 for almost all v,

^•(Ov) = Ov for almost all v. If f is a standard function on FA, then f = nvfv

and fv = O v for almost all v, and we know from (1) and (2) that> = FIv7(U) is again a standard function.


Now we take a continuous multiplicative character X of Fvx and assume that X

is non-trivial on OvX. Thus if G3f Q, is the conductor of X, then f > 0 (G3f CV isby definition the maximal ideal such that (l+G3fOv) = 1). Let O^ be the locallyconstant function defined as follows:

Then defining K = KV = | G5f | V"1/2JO x^"1(x)e(G5"r"fx)d|i(x), we have

(3) * * 0 ( y ) = 1®f+r I v1/2X(©^f)Kv<l>x.i(©

r+1y).

In fact, by the computation of the Gauss sum, we already know that

JFvOx(x)(x,y)d|i(x) =

0 if yOvx * G3"r"fOv

x,

|05f+r|v1/2?l(03r+fy)Kv if yOvx = G5-r"fOvX,

which shows (3). By (3), we know that (G5r+f)Kv is determined independentlyof the choice of G3 since the Fourier transform has nothing to do with the choiceof G5. This number is called the local factor (or local root number) of X and it isnot so difficult to show that | KV I = 1 by using the Fourier inversion formula(see Exercise 2.3.5, Corollary 1, and the proof of Theorem 1 below).

Let O be a standard function and O' be its Fourier transform. Consider another

function Oz(x) = O(zx) for z e FAX, which is again a standard function. Then

by definition,

= JFAO(zx)e(xy)d^'(x) = = | z |

This formula will be useful. Let X : FAx/FxU(c) -> Cx be a Hecke character of

conductor c and of infinity type £, = Za£a<3. We may assume that X has values

in T by dividing A, by a suitable power of the norm character cos(x) = |x|^ (in

fact, if Xoo = l,then |x|^ = N(xO)'s and L(s,A,cot) =

Exercise 1. Show the existence of cos with the above property, i.e. show that

there exists S G C such that I ^(x) | = IXIA*


Let Xv be the restriction of X to Fvx. If Fv = R and a : F -> R is the cor-

responding field embedding, then Xw(x) = xâ | x | '^° = x"Av | x | Av for

A v = 0 or 1 according to the parity of £ a . If Fv = C , then

Xv(x) = x-A vx"B v |x |A v + B v for integers Av > 0, Bv > 0 and A v B v =0, be-

cause X has values in T.

Exercise 2. Show that if % : Cx —> Cx is a continuous character, then thereexist integers A > 0, B > 0 and AB = 0 and a complex number s e Csuch that %(x) = x"Ax"B | x |2 s .

Take an idele b such that bO- cb for the absolute different d of F and

boo = 1. Then for each v, we may assume that bv = 03 in the above compu-

tation. We then define a standard function attached to X by Ox = IW>x,v as fol-

lows. If Xv is trivial on 0VX (this is the case for almost all v), we put

®XV = 3>v (the characteristic function of Ov) and if Xv is non-trivial on OyX,

O^v is as in (3) and if v is real,

O?tv(x) = xAvexp(-7ix2),and if v is complex, then

= xAvxBvexp(-27i | x | 2 ) .

Thus by (1), (2) and (3), we have

(4)

where K = nvKv and KV is as in (3) if Xv is non-trivial on Ovx, Kv = 1 if

Xv is trivial on Ovx and KV = V-TAv for v real and KV = V-TAv+Bv for v

complex.

Now we define for any standard function O the zeta integral:

(5) Z(s,A.;*)

By definition, Z(s,A,;O) = IIVJF x^>v(xv)^v(xv)|xv|^d(iXv(xv). Now we compute

this integral locally and show that it is essentially the Hecke L-function L(s,A,).

We start computing the integral for finite places v. First suppose that Ov is the

characteristic function of CV and Xv is trivial on OyX. Then

( 6 ) J F v | | ^ ^=0 J t s j O v | vJxd^ x

v = (l-X*(pw)N(Pvy*y\


which is the Euler factor of L(s,X) at p,. For other finite places v, Supp(Ov) is

contained in G3"nOv for a sufficiently large n. Thus taking the characteristic func-

tion % of this set, we have, for a sufficiently large constant M,

| vs | < MJC(X) I x | v

a for a = Re(s). Then

(7a) | JFvXOv(xv)?iv(xv)|xv|svd^x

v(x)

If Ov = O^v in (3), we can compute JF xOv(xv)Xv(xv) I xv I vsd|iv(x) explicitly.JF x

In fact, we see that

(7b) JFvXOxv(xv)Xv(xv)|xv|^d^xv(x) = JOyXd|ix

v(x) = 1

because O^v = ^v"1 on CVX and 0 outside. Anyway we have

(8a) |JFAfXO(x)?i(x)|x|sAd^fX(x)| <M^F(c) for a = Re(s),

which is convergent if a > 1. We conclude that

(8b)

Now we compute the integral at infinite places. Since the standard function Ov(x)is equal to either xAexp(-?cx2) or xAxBexp(-27C I x | 2 ) according as Fv = R orC, <X>V decreases exponentially if | x: | —> +<*>. Thus if Re(s) is sufficiently

large, the integral

J ) |xv|'djixv(x)

converges absolutely if Re(s) is sufficiently large. Thus Z(s,^;O) is a well de-fined analytic function of s if Re(s) is sufficiently large. Now we compute

x^)v(xv)A,v(xv) I xv I vsdjj,Xv(x) for Ov = O^v. First assume that Fv = R and

x) = x"A | x | A . Then Ov = xAexp(-7Cx2) and we have

(9a) JFvXOv(xv)?iv(xv)|xv|svdîx

v(x) = J ^ exp(-7cx2) | x | ^ ^

J exp(-7cy)y(s+A/2)-1dy = 7i'(s+A)/2r((s+A)/2) =

When Fv = C and Xw(x) = xATB | x |A+B,

266 8: Functional equations of Hecke L-f unctions

(9b) JFvx*v(xv)^v(xv) |xv|svdnx

v(x)

= JF v Xexp(-2* I x | 2 ) | x 2+ y 2 1 - ( ^ ^

= 2-1(27i)1-(s+(A+B)/2)r(s+(A+B)/2) =

We put Gj^(s) = IIv infiniteGxv(s). Then we have

(9c)

Thus finally we see that

(10) Z(sXOx) = 2-lGxJs)L(s,X) if Re(s) > 1.

Until now we have only used the fact that X is a continuous character ofFA X /U(C). Hereafter we suppose that X(FX) = 1 and prove the functionalequation. Let O be a standard function on FA and O1 be its Fourier transform.Since FA/F is known to be compact and F is a discrete subgroup of FA (13C),

we can apply the Poisson summation formula in §4 to this situation. We firstreview the formula. Let F 1 = {x e F A I (x,F) = 1}. Then F 1 =Homcont(FA/F,T). Since FA/F is compact, F 1 is a discrete subgroup of FA byidentifying FA with Homcont(FA,T) (Lemma 3.1 and Theorem 3.1). Since thestandard character e is trivial on F, (!]£) = e(r|£) = 1 for § and r\ in F.Thus F is contained in F1. Thus F^/F is a discrete subgroup of FA/F. SinceFA/F is compact, F^/F must be discrete and compact. Thus F^/F is a finitegroup. Let x E F X - F and suppose N X G F (such an integer N > 0 alwaysexists because F^/F is finite). Since F is a field of characteristic 0, we can findy e F such that Ny = Nx. Thus x-y is killed by N in FA- Since FA istorsion-free, x = y, a contradiction. Thus F 1 = F. Let G be a locally compactabelian group. We suppose that there is a pairing ( , ) : GxG —» T underwhich G = Homcont(G,T). Let F be a discrete cocompact subgroup andF* = {x e G I (x,F) = 1}. Then F* is again a discrete cocompact subgroup.Let |J, be a Haar measure on G. Let K be a fundamental domain of G/T andnormalize [i so that jKd|J, = 1. Then the Poisson summation formula reads

if both sides are absolutely convergent and the Fourier transform F(f) with respectto JI and ( , ) is well defined (i.e. f is continuous and integrable on G). Letus apply this formula for f = O and G = FA, F = F* = F. Thus we need to


show that Jp^dM-' = 1. By (1.3c), we have a canonical isomorphism

Thus JFA/F^M- ~ JFoo/o M-00* -êt {a)i»-*»»c°d} (d=[F:Q]) be a basis of O over

Z. We identify Foo = Ca(C)xRa(R) and C with R2 taking the basis (1,V-1).

Thus COJ can be considered as a vector in F^ = (R 2 ) a ( C ) xR a ( R ) whose

component at each a e a is given as follows: For c e a ( C ) ,

(Re(c0ja),Im(c0ja)) and for a e a(R), cof. Then

~ = I det(coi,...,cod) | = 2"11D | 1 /

This shows that |i' on FA is the right choice, i.e.,

Theorem 1. Let <$> be a standard function on FA and <!>' te /& Fourier trans-form. Then the function s h-> Z(s,X,;O) defined by (5) when the integral is con-vergent can be continued analytically as a meromorphic function on the wholecomplex plane. It satisfies the following functional equation:

If one specializes O in the theorem to Ox, we derive from (4) and (10) thefollowing result:

Corollary 1. L(s,X) can be continued to a meromorphic function on the wholes-plane and satisfies the following functional equation:

)( ID | W(where c is the conductor of X, D is the absolute discriminant of F, b is the

idelefixedin(3)and K is the root number for X.

Before proving the theorem, we start with the following lemma in [Wl, VII.5].

We can decompose FAX/FX = ( F £ } / F X ) X R + via X H (xf(Xco|x^1/[F:Q]), |x|A).

We know that (F^/Fx) is a compact group (Theorem 1.1).

Lemma 1 ([Wl, VII.5]). Let Fi be a measurable function on R+ with0 < Fi < 1. Moreover suppose that there exists an interval [to,ti] in R+ suchthat Fi(x) = l if x < to and F i (x )=0 if x > t i . Then the integral

f(s) = J Fi(x)xs"1dx is absolutely convergent for Re(s) > 0. The function

f(s) can be continued analytically in the whole s-plane as a meromorphic function.Moreover f(s)-s"1 is an entire function. If Fi(x)+Fi(x"1) = 1 for all x e R+,then f(s)+f(-s) = 0.


Proof (A. Weil). First take as Fi the function $ such that <|)(x) = 0 if x > 1l sAdx = [xs/s]J = s"1 ifand <t>(x) = 1 if x < 1. Then f(s) = j l xsAdx = [xs/s]J = s"

Re(s) > 0 and the assertion is obvious. For general Fi, we see that

fCs) "1 = J~ (F^Xx^dx.

Since F -<() is a bounded measurable function with compact support on R+, theabove integral is (locally) uniformly and absolutely convergent for all s, whichgives an entire function of s. Note that (|)(x)+(|)(x"1) = 1. If F1(x)+F1(x"1) = 1,then F2 = Fi-<() satisfies F2(x-1) = -F2(x). Thus replacing x by x"1 in theabove integral, we see f(-s)-(-s)"1 = -f(s)+s"1. This shows that f(-s) = -f(s).

Proof of the theorem. We already know that the integral

Z(sA;«) = JFAXO(x)^(x)|x|sAdjlx(x)

is absolutely convergent if Re(s) > 1. Let Fi be a continuous function as in thelemma and define Fo by F0+F1 = 1. Thus 0 < Fj < 1 and Fo(x) = 0 ifx < to and Fi(x) = 0 if x > ti. Take arbitrary B > 1. Then, for anya € R with a < B, we see that xaF0(x) < to

a"BxB. We define

ZfafcQ) = JFAxO(x)?i(x)|x|sAFj(|x|A)d|ix(x).

Then writing a = Re(s), we have

J F A X I * (x) I |x|° Fo( I x I A)d|ix(x) < toa-BJF Ax I ®(x) I |x£dji*(x).

Thus this integral is absolutely convergent for all s since B > 1. ThusZo(s,A,;0) is an entire function of s. Now replacing X by X'1, s by l-s, Oby O' and Fo by the function x h-» F^x"1), by the same argument as above,we have an entire function of s defined on the whole s-plane,

Z'od-s,*-1;*1) = JFAxO'(x)V1(x)|x|1A-sFi(|x|;1)dîx(x).

Let <£z(x) = O(zx) for z e FAX. Then the Fourier transform of Oz is given

by I z I A ' ^ ' Z - I a s a^rea(iy remarked. We now write, choosing a fundamental

domain X of FAX/FX, FA

X = U^F-{0}^X . Then


A ^ Mx) |x|sAFj( |x|A)d|ix(x).

Similarly, we have

By the Poisson summation formula, we see that

Thus we see that

x | AsFi(|x|A)djlx(x).

By the variable change x H» X"1, the first integral equals Z'oQ-s,^"1;©') andthus

) I x | A-1-*(0)}X(x)|x|8AFi( I x | A)d|ix(x).

Now we use the fact that FAX/FX = KxR+ for a compact group K = F(

A}/FX.

Write X = Xfa as a product of characters of K and R+. Since log : R+ = R

and Homcont(R,T) = R, ^ 0 0 = I x | A1 for some t e V-1R. Then we have

Thus we may assume that t = 0. Then

ZiCsÔJ-Z'od-s.V1;®1) = JKWjixJR+{*l(O)-*(O)x}x8-2Fi(x)dx.

Since K is compact, c(^) = jKX,d|LLx is a constant and vanishes if X * 1 on

K. Anyway, by Lemma 1,

for a meromorphic function f(s) satisfying the following conditions: (i) f(s)-s"1

is entire and (ii) f(s)+f(-s) = 0. Since Z(s,X;O) = Z0(s,X;O)+Zi(s,^;©), the


above fact shows the analytic continuation of Z(s,A,;O) because Zis entire. Moreover we have

Z(s,?i;O) = Z0(s,?t;O)

Since f is an odd function, we know the functional equation

because of the Fourier inversion formula !F.!F(O)(x) = O(-x). We have notproven the inversion formula yet but for the special function Oa, it is obvious by(3).

We now give a sketch of a proof of the inversion formula when v is finite (a simi-lar argument works also for R or C). Let f be a standard function and^n(x) = Oo(7Cnx) be the characteristic function of G5~nOv. Then we compute forG = Fv

We already know that ^(On)(x) = \K\ v"nOo(Ttr~nx) by (2). Instead of the above

integral, we compute

JGJGcX)n(y)f(x)(x,y)dja(x)(y,z)djl(y) = JGJGOn(y)(y,x+z)d^(y)f(x)d^(x)

= JGJ(On)(x+z)f(x)d^l(x) = I G3 | v-nJGd>0(7Cr-nx)f(x-z)d^(x)

By taking the limit as n —> °o, we see that

f(-z) I 031 v"r = f(-z)JGO0(G3rx)dn(x) = | G51 ^

because the difference of (I and JLL1 is I G5r I v1/2- This shows the result.

Exercise 3. Show that ZêF^K^) is absolutely convergent when F = Q.

When X is the trivial character, we see that <E>x(x) = Of(xf)Ooo(Xeo) for the char-

acteristic function Of of 6 and Ooo(Xoo) = exp(-7cZaGil XooG I 2) . Thus

0(0) = 1 and by (4), J(O;0(0) = | D | -1/2Ox(0) = | D | "1/2. Thus we have

O'(0) = | D | ~1/2. By the proof of the theorem, we see that

'(0) (K =


We now compute JKXd|Lix. By Corollary 1.1, we see that

for K'

where h(F) is the class number of F. It is then plain that K1 = X/E forX = {x e F x | N(x) = 1} where E is the subgroup of (f consisting of to-tally positive units. We then consider the logarithm map in (1.2.7)/ : F x ^ Ra. We decompose the Haar measure on C so that dxdy = rdrdGfor r = I z 12 for z e C and the Haar measure d0 on T. Then we see easilythat

vol(/(X)//(E)) = J/(X)//(E)dr = | R | = R. ,

where R = d e t ^ e ^ i j ) for a basis {ei,...,es} (s = #(a)-l) of the torsion-free part of E. Note that the kernel Y of / can be written as({±l}a(R)xTa(C))/M-(F) for the group |i(F) of roots of unity in F. Then we have

JYd0 = 2r(27c)Vw for w = #M<(F) (r = #(a(R)) and t =

f ^ x 2r(27i)tR0Oh(F) A u .Thus JK?id|ix= —————, and we obtain

Corollary 2 (Residue formula). We have

r ^ 2r(27i)tRooh(F)Ress=1CF(s)= | D | 1 / 2 .

Chapter 9. Adelic Eisenstein series and Rankin products

In this chapter, first we shall give an adelic interpretation of modular formsand then we will compute the Fourier expansion of adelic Eisenstein series forGL(2) over Q. After this computation, we will know that the Eisenstein serieshas analytic continuation with respect to the variable s. Using this fact, we willshow the analytic continuability of the Rankin product and its functional equations.

§9.1. Modular forms on GL2(FA)Hereafter, we assume that F is a totally real field. We start from the definition ofmodular forms on the group GL2(FA), which consists of invertible 2x2 matriceswith coefficients in FA. We use the notation of Chapter 5. In particular, Hdenotes the upper half complex plane. We put, identifying F<x> with R1 for thesetofrealembeddings I of F,

G ^ = {x G GI^CFoo) = GL2CR)1 I det(xa) > 0 for all o e I}.

Then we can let the group Ge*>+ act on Z= ?{ via linear fractional transforma-

o+ = {x G G^+ | x(i) = i} for i = (^, . . . , ^p[) e Z.

tions x(z) = (xo(zo))O€i, where ^ dJ(z) = a ^ j . We put

Exercise 1. Show that for a e GL2(R) with det(a) > 0, a(i) = i

(i = V—1) if and only if there exist t € R x and 0 G R such thatfcosG -sinB^

a = V e cose} EsPecially C ^ = RXSO2(R).

afa b^For a = A and z G #", we put j(a,z) = (cz+d). Then we see easily that

j(ap,z) = j(a,p(z))j(p,z). Thus if a (0 = p(/) = /, then j(ap,z) = j(a,0j(P,0

and the map ah^ j ( a , 0 is a group homomorphism of RXSO2(R) into CX. In

;A fcosB sinB^ v

fact, j(a,0 = tezo if a = t^s]nQ C Q S 6 J. We define jk(x,z) = UoK^cô) for^ Jeach positive integer k, x e Geo+ and ze Z, and | j(x,z) | s = Flo I K^c^zc) Is

for s G C. Then the map x h-> jk(x,i) is a group homomorphism of CL+ into

cx.

We consider the open compact subgroup

S = G L 2 ( 6 ) = npGL2(O;,)

of GL2(FA) and its subgroup of finite index, for each integral ideal m

9.1. Modular forms on GL2(FA) 273

For a given character % : Cl(m) = ¥\X/Fx\](m)¥oo+ -^ T, a continuous functionf : GL2(FA) —» C is called an adelic modular form (in a weak sense) of weightk, of character % and of level m if it satisfies the following condition:

(Ml) f(ocxu) = XmOOfWjkOw)"1 for all u e S(m)Coo+ and oce GL2(F),

fa b\

where %m( I) = IIV | m%v(dv)- Modular forms play the role of Hecke charac-

ters for the group GL(2) in place of GL(1). In fact, any Hecke character

X : FAX = GLI (FA) -» Cx for a general number field F satisfies the condition

A,(ocxu) = 9i(x)uoo" for a e Fx = GLi(F) and u e U(/rc)Foo+ for its infinity

type £, which is an analog of (Ml).

We can define, from the modular form f and a given element t e GL2(FAf) as

above, a function ft: Z —» C as follows: for z = x+iy e Z (x, y € Foo),(y x

we pick one element Uoo G Goo+ so that uî) = z, for example, Uoo = L

satisfies the requirement. This in particular shows that Goo+ = B00+C00+, where

We put ft(z) = f(tUoo)jk(Uoo,i). This definition of ft(z) does not depend on thechoice of Uoo. In fact, if ueo(/) = u'oo(0> then uOo"1u'Oo(0 = / and hencec = Uooû'eo G Ceo+. That is, UooC = u'oo and

f(tu'oo)jk(u'oo,i) = f(tUooC)jk(UooC,i) = f(tuOo)jk(c,i)"1jk(c,i)jk(Uoo,i) = ft(z).

Now put r t = GL2(F)nt"1S(N)tGoo+. Then Tt is a subgroup of GL2(F), and ift = 1,

(a. b^ 1r i = {a = I d j 6 GL2(O) \ ce m and det(a) e Foo+}.

In particular, if F = Q, thenfa b

F i = T0(m) = {y = ^ J G SL2(Z) | c G m}.

We now show that ft satisfies ft(y(z)) = %*(d)ft(z)jk(y,z) for y e Tt if t is offa Oj

the form , where %* is the ideal character associated to the idele character

%. Thus ft is a modular form on Z for the discrete subgroup Ft in the classicalsense: if y G Ft, then

274 9: Adelic Eisenstein series and Rankin products

ft(y(z)) =

since YooUoo(i) = Y(z). The adele matrix Yoo has non-trivial components only at in-finity and t is concentrated on finite places. Thus Ttt'Vf^t = YYf~!t = Y~t = tY~.Moreover t^yfhe S(m) and we know that

ft(Y(z)) = f(tYooUoo)jk(Y-Uoo,i) = f(Ytt" V ^

= Xm(d)-1f(tuoo)jk(Yoo,z)jk(Uoo,i) = Xm

Note that 1 = %(d) = %m(d)%*((d)) for the ideal character %* associated to the

idele character %. Thus XmCd)"1 = %*(d). Similar computation shows that for

more general a e GL2(F) with det(a) »0 (the symbol "»" means that

det(oc) is totally positive, i.e. det(a)a > 0 for all a e I), we have

(la) ft(a(z))jk(a,z)-1 = fa r i t (z).

Thus to one modular form f on GL2(FA), we can attach a system of classicalmodular forms {ft}. This system is basically parametrized by the double cosetspace:

GL2(F)\GL2(FA)/S(/n)Goo+ = GL2(F)\GL2(FAf)/S(/rc)

which is an analog of the class group Cl(m) = Fx\FAx/U(fft)Foo+. In fact, it is

known that if {ai}i=i h is a representative set of Cl(l) in FAX, then

(lb) GL2(FA) = UiL1GL2(F) ISin^G^ (approximation theorem).

Thus each modular form on GL2(FA) corresponds to a set of h classical modularforms on Z. We now impose the following holomorphy condition on f:

(M2) ft is a holomorphic function on Z for all t e GL2(FAf).

Let us give a proof of (lb): Consider Lf= 6 2 and L = O 2 as lattices in thecolumn vector space V = F2. For each x e GL2(FAf), xL = xLfflV is a lat-tice of V. We first show that there is a vector 0 * y e xL such that xL/Qy istorsion-free. Take one non-trivial vector y € xL. The places v such that theimage y(v) of y in xL//vxL is zero are finitely many (note that y(v) = 0 if andonly if xL/Oy has non-trivial py-torsion). Let Z be the set of such places. Wechoose z e xL so that the image z(v) in xL/pvxL is non-zero for everyv e Z. We may assume that z and y are linearly independent over F. Then theimages of z and y in xL/pvxL are linearly independent for almost all places v.Let Y be the finite set of places v where the images of z and y in xL//vxL arenot linearly independent. Thus we can write z(v) = ?c(v)y(v) for v € Y with


X(v) e Ov/py. Since 0v//v has at least two elements, we can find be O suchthat (b modpy) * X(v) for all v e Y (by Chinese remainder theorem). Thenput t = z-by. Then if v e Y, then t(v) = t mod pvxL * 0 by the choice ofb. If v is in Z, then t(v) = z(v) ^ 0 by our choice of z. If v is in neither Znor Y, then z and y are linearly independent in xL/p^xL and hence t(v) * 0.Thus xL/Ot is without torsion. Thus we may assume that xL/Oy is torsion-freefrom the first. Since xL/Oy can be embedded into F, it is isomorphic to an ideala of O, which is protective because O is a Dedekind domain ([Bourl, VII.4.10]).Thus xL = at0py for an ideal a and t e xL. Take a e FAf such that aO = a,and write a = pai with p in F and for some i (we can choose the parity of Parbitrarily at infinite places by changing i if necessary). Then, writing the matrix

for t = [tl 1 and y =\h)

We may further assume that det(oc) » 0 by choosing a suitable parity of p. Thusfa- 0^ ,

x = ocI ju for u e S since S = {xe GL2(FAf) | xLf = Lf}. This

shows (1) for m=O. For general m, we can easily approximate u as above

by y e b^SbflGL^F) for b = k j I so that yu e S(m). This is a special

case of the strong approximation theorem but it is not so hard to verify it in thiscase. When F = Q, we have already proven the strong approximation theorem asLemma 6.1.1 and its proof applies to the general case.

Now we define the Fourier expansion of modular forms. Since any modular formf on GL2(FA) has a prearranged move under the left translation by GL2(F) andthe right translation by S(m)Coo+, by (Ml), it is determined by the value on thesubgroup

B(FA)+ = ( b e B(FA) | det(boo) » 0},where for any Q-algebra A, we put

B(A) = j f j j l l a e (A(g>QF)x and b e A(g>QF

(fy x\\Let f be a modular form as in (1). We write simply f(y,x) = f 0 1 I . Now

(I u\for u e FA, we consider a unipotent matrix oc(u) = . Then

a(u)y i ] = o X1UJ* I n P a r t i c u l a r ' i f ^ G F, then x(^) e GL2(F) and

thus f(y,x+£) = f <x(u) = f(y,x). Thus f(y,x) is translation invariant

under F in the variable x. Thus for a fixed y, we can consider x h-» f(y,x) as a


continuous function on FA/F (which is of C°°-class in x«o G Foo). The groupFA/F is a compact additive group. Thus we can expand f(y,x) into an adelicFourier expansion on FA/F (see §8.4); namely,

where <|> runs over all additive characters <|) G Homcont(FA/F,T) and

c(y,<()) = JFA/Ff(y,x)(|)(-x)d|i(x).

Here the additive Haar measure \1 is normalized so that JI(FA/F) = 1. We al-ready know that Homcont(FA/F,T) = F so that each character of FA/F is givenby X H e(J;x) for the standard character e : FA/F —» T. Thus we can writethis expansion as

^ for x e FA.

Let us verify several properties of this expansion. Since f is invariant under

right multiplication by the matrix a = L with u G U, we see that

ffy x i ^f(uy,x) = f a = f(y,x). This shows that c(uy,£) = c(y,£). Thus

(2) c(y,^) depends only on the ideal yO and yoo.

Similarly f(T|y, rjx) = f(y,x) for T[ e Fx, and thus

(3) cCny,$) = c(y,^Ti) for ^ E F, 0 * r| G F.

Let ft = dO (de FAfx) be the different of F/Q. Thus d'1 O is the maximal

additive subgroup in Af of the form x 6 so that ef(x 6 ) = 1. Now for

u e 6 , oc(u) = G S(w), we know that f(y,x+uy) = f(y,x).

Therefore c(y,^) = c(y,^)e(ûy) for all u e 6 . This implies that ifc(y,^) ^ 0 , ^y G d"1 6 ; in other words,

(4) £dyO is an integral ideal if c(y,£) * 0.

fa (ANow we compute the function ft : Z -» C for t = L with a fixed

a G Af. Write a = aO for the corresponding ideal. Then for z = Xoo+VoJ e Z,

taking as Ueo, we see that


(5a) ft(z) = f(ay«,x«) = X ^ e

Especially, if ft is a holomorphic function, then -r^- = -—+ *\— kills each

term and hence -—c(ay <*>,!;) = -2rci;ac( ay «>,!;). Thus c(ayoo,£) is a constantWe

multiple of exp(-2rcTr(S;yoo)). We write this constant as c(^da); thus

c(ayoo,£) = c(^da)exp(-2jcTr(^yoo)). We now show that c(^da) = 0 unless

£a > 0 for all a when F ^ Q . For that we pick one £ * 0 which is not to-

tally positive. We then define [£] = {a e 11 ^ a < 0}, which is not empty. Fix

one element a € [£] and take a totally positive unit e such that e° > 1 for a

and 6T < 1 for all % ^ o. We can always find such a unit (see the proof of

Dirichlet's unit theorem, Theorem 1.2.3). Then

c(^da) = JFA/Ff(l,x)\|/(-^x)d|i(x)exp(27cTrF/Q©)

JF A /F I f(l,x) I d^(x)exp(27iTrF/Q(ên)).

Note that £ a e o n -> -©© as n - » ©o and JjV*1-» 0 as n -»o©. Thus we

know that exp(27cTrp/Q(ên)) —> 0 as n —> ©o. This in particular shows that

c(^da) = 0 unless £ a > 0 for all a e l . When F = Q, this argument does

not work and we need to impose the following condition:

(M3) ft has Fourier expansion of the following form:

) f°ral1 tG

where n runs over a lattice of Q. As for the constant term ao = ao(ft), if we(y A

restrict t to be (ye FAfx, x e FA), it is a function of y, and hence we

fa 0°)write ao(y). Since f is invariant under left multiplication by I 0 I (a e Fx)

fu 0\and under right multiplication by L for u e O x, the function y h-» ao(y)

factors through FAfx/Fx 6 x, which is the absolute class group. Thus we can

define new functions n h-> a(n;f) by a(n;f) = c(^da) if n = ^da for \ e F+

and ni->ao(rt;f) by ao(n;f) = ao(yd"!) if rt=yO. We denote by

for the space of functions satisfying the conditions (Ml-3). Then f e

has adelic Fourier expansion of the following type:


(5b) f(y,x) =

where ni-» a(n;f) e C is a function of fractional ideals vanishing outside inte-gral ideals and n h-> ao(n;f) factors through the ideal class group of F. Thespace of cusp forms Sk(w,%) is the subspace of Mk(m,%) consisting of func-tions f e Mk(m,%) satisfying the following cuspidal condition for the standardHaar measure dji on FA/F:

(S) JFA/Ff(jo i]xJd|i(u) = 0 for all x e GL2(FA).

This just means the vanishing of the constant term of ft for all t. Sinceft I a = fotrit for a e GL2(F) with det(a) e F+, this simply implies that ftis a cusp form for all t.

When F = Q, we know from (lb) that GL2(A) = GL2(Q)S(m)Goo+ since(\ (A

Ax = QXU(1)R+X (8.1.3a). We already know that ft for t = L A gives a

homomorphic modular form in #4(ro(ffO>%*) since Cl(m) = (Z/m)x and thecharacter %* can be considered as a Dirichlet character. Conversely, startingfrom <|) e ^k(ro(^),%*) and writing each X E GL2(A) as au withas G L 2 ( Q ) and u € S(m)Goo+, we define f : G L 2 ( A ) - ^ C byf(x) = Xm(u)<l)(z)jk(Uoo,i)"1 for z = Uoo(i). We need to show that f is a well de-fined element in Mk(m,%). If x = au = a V , then

GL2(Q) 3 y = a ' ^ a = u'u"1 e S(m)Goo+

and hence ye GL2(Q)nS(w)Goo+= Fo(m). Then we see that

Thus f(x) is well defined independently of the choice of a and u. It is obviousfrom (5.1.la,b) that f e Mk(m,%) and fi = 0. Thus

(6) Mk(m,%) = fA4(r0(m),%*) and 5 k (« ,x )s4(ToW,X*) via f H> fi.

In this sense, we can take Mk(m,%) as a generalization of the spacefor general totally real field F.

We have, for t = L A and fe


(7) ft(z) = Xo«5Ea-U-i câ)txp(2n^TT(^z)) for c(a) e C.

In particular, when F = Q and a = 1, the above expansion has the usual form:

X°° c(n)exp(27cV-Inz). In general,

c(y£) = c ( y U ) = c(^dyO)exp(-27iTr(^y)).

Thus we may regard the Fourier coefficient c(a) as a function of integral idealsa n c(a). The L-function of f is then defined by L(s,f) = £ac(a)N(a)"s. Thisfunction is known to have an analytic continuation and a functional equation similarto Hecke L-functions. These L-functions are first investigated in detail by Heckeand have now become very important tools in number theory. Of course, whenF = Q, this L-function is nothing but the one studied in §5.5. In this book,however, we do not go into details of the theory of such L-functions. We list [JL],[G] and [W3] as standard textbooks for this subject. Instead, we introduce nowthe Eisenstein series as an example of these modular forms and try to give someaccount of how to compute the Fourier coefficients of Eisenstein series whenF = Q in the following section. Let T = GL2(F)rif1U(m)tGoo+ for the abovet. Thus

r = { r d ] e GL2(F) I a, d e O, c e am, d e a'1, ad-bc e E } ,

where E is the group of all totally positive units in (?. Let

Then j(y,z)k = (N(d)/\ N(d)\ )k if y = f* Jl e r . . . Thus we see

j(Y5,z)k=j(y,8(z))kj(5,z)k = (iV(d)/|iV(d)|)kj(5,z)k for y e r_ . Now we put

for a character % : (O/m)x -> T such that %(e) = (N(e)/ \ N(e) I )k

(8) Er(z,s,%) = ysX7ere o\r%(8)J(5 ' z)"k I J(5>z) I "2s>

ffSL b\\where %\\ A = %(d). Then this series is absolutely convergent if

Re(s) > l-(k/2) (see §§2.5 and 2.8, where we expressed Eisenstein series interms of Shintani zeta functions) and Er(y(z)) = %(y)"1Er(z)j(y,z)k for y e Tby definition. We now interpret this definition in adelic language. In the adeliccase, the object corresponding to the congruence subgroup T is GL2(F), and thatof Too is Boo = 0*B(F), where


B(F) = {(* J] I a e Fx and b e F | .

Let % : Cl(m) = FAx/FxU(/rt)Foo+ -> T be a character and suppose that

%(Xoo) = (N(Xoo)/ |N(Xoo) I )k. We define the following functions on GL2(FA):

(9) ri(x) = < M y ' A i f x = (o l j a u â G F A > < a n d U G s ( w ) c ~ + ) >10 otherwise

and

%#ffa b l l = jnvUXv(dv) if (* J ) e B(FA)S(m)Goo+,^ C d ^ 10 otherwise.

These functions are well defined. To see this, let Lf = 6 2 (column vectors).(y b>i a v\

Then, if au = a'u' as above, then looking at the second row of(y b>\ fy b>y\

a L uLf = a' u'Lf which only depends on a and a' because

uLf = u'Lf, we know a d = a' 6 . Thus I af I A = I a'f I A- By taking the de-terminants of both sides, we then conclude that I yf I A = I y'f I A- O n the otherhand, Im(Xoo(i)) = yoo = y'oo, which shows that I y I A = I yf I A- Similarly %# iswell defined. By the product formula |£ I A = 1 for t, G F X (8.1.5), we knowthat Ti(yx)=r|(x) if y e B^. We also see that %#(yx)j(YXoo,i)"k = %#(x)j(Xoo,i)"k

for all y e B^.

Exercise 2. Suppose that %(xoo) = (Af(Xoo)/1 A (Xoo) | )k . Then show thatX#(yx)j(yxeo,i)-

k = x#(x)j(xoe,i)-k for all ye B^.

We define

E*(x,s,%) = IYGBeo\GL2(F)X#(yx)rl(/Yx)sj-k(yxoo,i).

Suppose that xf = t = L A. Then yt e B(FA)S(m)Goo+ if and only if

a f ^ *X (Yt)Ti(yt) * 0. Write y = , then if % r|(yx) ^ 0, there exists s e S(m)

\° aJ° a

1 = sClwith s = . . Thus c e a1 = sClOt d e O and ca+dO= O. This shows that

we can find e e O and f e a such that cf-de = 1. Namely 8 = J e I \

One can easily show that this correspondence


BooMylyte B(FA)S(m)Goo+} s y H-> 5 e TooXT

is bijective and thus each term of the summation of Ep and E* can be naturallyidentified. Moreover by the above computation, writing (yt)f = bfSf as above forb e B(Ap), we see that Tj((yt)f) = | det(bfSf) I A = I det(yf) I A I a I A- On the otherhand, TI((YX)OO) = Im(yxoo(i)) = |j(y,Xoo(i)) |"2|det(yoo) I ATI(XOO). Thus

(10a) ri(yt) = I j(y,xo.(i)) I "2 I det(Y-) I AIm(xoo(i)) | det(yf) | A | a | A= |a|AIm(x0O(i))lj(Y,x0O(i))|-2.

Thus we know that

(10b) E*(txoe,s,%)j(xoo,i)k = | a | A

sEr(xo.(i),s JC*'1) ,

because %*((d))%(dm) = %(d) = 1 for the ideal character %* associated with %.This shows the convergence and we now know E* corresponds to Ep naturally.

By definition, we have E*(yx,s,%) = E*(x,s,%) for y e GL2(F) and

E*(ux,s,%) = %(u)E*(x,s,%) for u e 6 X . Thus for z e FAfx, the modular

form %-1(z) | z | AkE*(zx,s,%) depends only on the class of z modulo 6 XFX.

Note that FAfx/ 6 XFX = FA

X/ O xFxFooX which is isomorphic to the absolute

ideal class group Cl of F. We now define

(11) Ek(x,s,%) = Ek,m(x,s,%) =Lm(k+2s,%-1)£zGC1%-1(z) | z | AkE*(zx,s,%)

and Gk(x,s,%) = Gk,N(x,s,%) = %(det(x))Ek(xxf,s,%-1),

(0 -I)where % = n \ with a finite idele m such that mO = m and

^m(s,%"1) = Hn%l{n)N(n)~s in which the sum is taken over n prime to m. Note

that for z1 e FAX, we have

S"* 'V" ( v \ -7 A T**('77*T[ C f\f\ — ^V 7 M771 ^ I7P7 A P * ^ 7 Y <2 V^

= X(z1)lz'|A-kXzeCiX"1(z)E*(zx)s;%).

Thus Ek satisfies (Ml) for %k : z \-> %{z) \ z | A"k. Using the fact that

a b ^ _ , ( d -c)lmc dj i -mb a

we conclude easily that Gk also satisfies (Ml) for %k. We now state the principalresult:


Theorem 1 (Hecke-Shimura). Let % be a finite order Hecke character modulo

m. Define a function on the set of fractional ideals by am>5C(a) = Hg^J&ftNiGf1

and a'm^Ca) = EO a%(a/£W(£)m if a is integral and otherwise om>x(a)

= o'm.xCfl) = 0. Here we understand %(a) = 0 if a and m have a non-trivialiV(Xoo)k

common factor. Let k be a positive integer such that %(xoo)=-j ^nr- Thenl A ^ ( ) r

Ek,m(x,s,%) and Gk,OT(x,s,%) c#« £e continued to meromorphic functions in s $0that there exists a non-zero entire function f w s such that f(s)Gk(x,s,%) and

f(s)Ek(x,s,%) are entire. Moreover they are finite at s = 0, and except when

k = 2, F = Q, and % is trivial, we have

and if m* O,

l l,0,

C anti C are non-zero constants depending on k and m. In the excep-tional case when F = Q, % is trivial and k = 2, G2,m(x,0,id) is non-holomorphic.

Here e(/ eoyoo)e( Xoo) = exp(2jc/Tr(^z)) for z = x«o+iyoo and * = V-T. Wewill prove this theorem for F = Q in the following two sections. A proof for thegeneral fields F can be found in [Sh9] and [H8, §6].

§9.2. Fourier expansion of Eisenstein seriesIn this section, we assume that F = Q and compute the Fourier coefficients ofEisenstein series. I follow Shimura [Sh2, 7, 9] in this computation, which can begeneralized to bigger groups (e.g. symplectic and unitary groups as was done in[Sh9]). We exploit the fact that our group is GL(2) to simplify the computation inmany places. We restate the definition of the Eisenstein series when F = Q. LetN be a positive integer. We change the notation here and the ideal NZ plays therole of m in the previous section. We first prove Theorem 1.1 when N > 1.We will later remove this assumption to include the special case of N = 1. LetX : (Z/NZ)X = C1(N) -> Cx be a Dirichlet character with %(-l) = (-l)k (0 < kG Z). We define two functions TI,%# : GL2(A) -> C by

9.2. Fourier expansion of Eisenstein series 283

ii (y b>i

I y I A if x = I Q J a u (u e

) otherwise,

# f f a D l l = I n , M X P ( d p ) i f f! : i e B ( A ) S ( N ) G « + ,xfc ) {^V JJ 10 otherwise.

Then the function x h-> %#T|(x)sj(Xoo,i)"k is invariant under left multiplication by

elements in B = {±1 x 11 a e Qx, b e Q}. Then

E*(x,s,%) =

where GL2(Q)+ = {a e GL2(Q) I det(a) > 0} and B+ = BflGL2(Q)+. Put(0 - n

e = L 0 I. We compute the Fourier expansion of E(x,s) = E*(x£f ,s,%) and

later relate it to the Eisenstein series Gk(x,s,%) in the theorem. Note thatE(yx,s) = E(x,s) for all y e GL2(Q), and hence E(x,s) has a Fourier expan-sion. We define its Fourier coefficients for £ e F by

b(5,w,s) = JA/QE(a(x)w,s)e(-^x)d|Li(x),

where w e AXB(A), JLL is the additive Haar measure on A/Q such that(\ x]

|J,(A/Q) = 1 and oc(x) = I . We observe that

a(x)w

Suppose that w e AXB(A)+. Then the non-triviality of

%#(ya(x)wef"1)r|(Ya(x)w£f-1)s

means that ya(x)we{1 e B(A)S(N)Goo+ and ya(x)w e B(A)S(N)Goo+6f.fa b | fa b\

Since u = 0 . mod N for all u e S(N), if we write y = I , I, then

cÔ. Thus Y = [ c 2 d e t ( Y ) c ^JceaCc-M). That is, ye B^.QxeU for

U = {a(x) | XG Q). Thus B+\B+QxeU = Q+eU. Hence we see that


b($,w,s)

= J Xx#(eya(* + 8)w£f 1)Ti(eya(x+5)wef-1)sj(yoo(a(x+8)w)e«,0"ke(^x)d|i.Q6

We now prove a lemma, which shows that the above summation with respect to y

e Q is in fact reduced to the summation over a fractional ideal of Q.

(y 01 (y x]Lemma 1. Write w = a(x)a 0 = a n for a G A . Then we have

(ewe'^f e B(Af)S(N) (i.e. %#(ewef'1) * 0) if and only if

X G a 4 N Z , a e y"xZ and axZ+ayZ = Z.

1 ( a 0>lProof. By computation, we see that (ewe )f = • From this,

a 0\ . . . . .e B(A)S(N) implies ax G NZ, ay G Z and axZ+ayZ = Z

(<=> axZ+ayZ = Z). On the other hand, if axe NZ, ay G Z and

axZ+ayZ = Z, then we can find t,s G Z such that axt+ays = 1 and

u = f S 1 1 G S(N) andl -ax ayj

(1) (ewe'V1 = fY

which shows the converse.

(y 0>iApplying the above lemma to 7a(x+8) (instead of w itself) for a given yG y-1Z, we see that

%#(eYa(x+8)wef"1) * 0 <=> y(x+8) e NZ and y(x+8)Z+yyZ = Z.

Let O = O 7 : Af -> C be the following function depending on y and y:

O(x) = np^)p(Xp) and <Dp is the characteristic function of NZp if yypZp = Zp

and O p is a characteristic function of Z px if pZp z> yypZp. Thus if

NZ+yyZ = Z (i.e. yyZ is prime to N), then %#(eyx(x+8)wef'1) ^ 0 if and

only if O(y(x+8)) = 1. Thus we see that

w,s) = JA/Q

J%#(eycc(x)we71)O(yx)r|(eya(x)wef"1)sj(yoo(a(x)w)oo,0"ke(^x)dja(x),


where y runs over y"1Zf|R+ such that yyZ+NZ = Z. By (1), we can compute

%#(eya(x)wef"1)ri(eYa(x)wef"1) explicitly:

Lemma 2. Suppose that O(yx) = 1 and yxZ+yyZ = Z for y e Q+ and

[y x\and z = Xoo+V-lyoo e 9{

rj(£ya(x)wef-1) = I yf

2y I A I z I "2 and %#(eya(x)wef"1) = n

Therefore, for %N(yy) = rip|NXP(YPyp)

\s) = XJ 0<Y€y"1ZA

2s'k I yf I AsjAO(yx)z-k I yoo I s I z I -2se(-^x)dja(x),

because %(Y^NyN^)%N(yy) = %(y) and %(y^NVN)) = %(yyZ) as the ideal

character (where y(N)ITp | NyP = y).

Now we compute the following two integrals:

. 1 i r

and

Then of course, b($,(j j],s) = bK§,(j i],s)b-(§,^ *j,s). We first

compute the finite part of the integral. First suppose that yypZp = Zj(<=> yZp = yp^Zp). Then

"1jNZpep(-^r1x)d|ip(x) = I y4N | pJZpep(^y-1Nx)d|ip(x)

" y p N | p if £ y ! N e Z p (i.e. \ e N ' V p ' ^ p ) ,0 o therwise .

Next suppose that pZp 3 yypZp (<=> yv'l7uv 3 yp^Zp) (in particular, p is

prime to N). Then we have


if x h-» epC^y^x) is non-trivial on pZp, i.e. ttf1 £ p^Zp. Thus we may as-sume that § G yp^Zp. Then

Thus we know that if £ e y^N^Z, then

J.s) = x(y) Xx

= Z(y) I yf I A - ^ N " 1 X%"' (^Z) I fyy)f I0<7ey~1Z

To compute this sum, we introduce the Mobius function \i on the set of idealswith values in {±1,0}. Let X : Z -» C be any Dirichlet character and considerthe L-function

Then write L(SjX)"1 in the form of a Dirichlet series Z°° |i.(n)X(n)n"s. Thenn=l

and hence |i(l) = 1, |j,(p) = -1 for a prime p and |x(piP2 Pr) = (~l)r fordistinct primes pj and |i(n) = 0 if n has a square factor. For any positiveinteger m, by the above definition, we see that

In particular, if we specialize s = 0, then we see thatrl if m = 1,

2.o<d|m^d)={o if m > 1.

Thus for positive integers m and n,

We want to interchange the two summations: for each divisor d of m, the r withd I (r,m) is just the multiples of d; thus such an r runs over the set

{d, 2d, ..., (m/d)d).Thus


, exp(-27ujdn/m)

exp(-2jc/jn/d) by d i-» m/d.

Note that by the orthogonality relation of characters of Z/dZv^d , „ .. /1<v fO if n is not divisible by d,> exp(-27izjn/d) = i .

J - 1 Id if d I n .This shows that

From this fact, we conclude that

= X(y) I Yf I A " 8 - ^ ^ - 1 X X " 1 CyyZ) I (7V)f I A2 s + kI r e ( Z / y y Z )xexp(-27ci^Ny/7y)0<Y€y"!Z

= x(y) I yf I A - ^ N - ^ ^ z x k Y y Z ) I eyy)f I A2s+kIo<d

= x(y)lyfiA"8"k+1N-1XnS=1x'

We again want to interchange the two summations as above. For each divisor0 < d of £N/y, n runs through all positive multiples md of d. Thus

Zn=15C (")n 5>(n/d)d =

= L(2s+k)x"1)-1Xo<d |

= JL(2s+k,x-1)-1Xo<dUN/YX"1(d)d-2s-k+1 if

U(2s+k>x'1)'1L(2s+k-l,x"1) if % = 0.

We now know that

(2) 1

{ k - ifU(2s+k-l,x-1) if \ = 0.

Now we compute y"sbo.(^,[0 A,s) = J_°° (x + /y)"k Ix+/y I "2sexp(-2ra^x)dx.

Consider the function C,(z;a,$) = j°°e'îx+l^x^dx for ze C with

Re(z) > 0 and ct,p e C. Here for ze Cx za = | z! a e ' a 9 writing


z = I z I e*e with -K < 0 < %. Then one knows that this integral is convergentif Re((3) > 0 by a standard estimate. The divergence of the integral whenRe(s) < 0 is caused by the singularity at 0 of the integrand. Thus by convertingthe integral into a contour integral on

for sufficiently small+ o° e > 0, we can avoid the

singularity at 0 and havean integral expression for (e2mP-l)£(z;a,P), which is convergent for all p e C:

(e27Ifp-l)C(z;a,p) = z*\^zfi.+zh)*'l$'lGl&l (see §2.2).

Thus the function (e27uP-l)£(z;a,p) is holomorphic on the whole space

H ' x C x C , where H' = {z e C | Re(z) > 0}. By using the well known

formula: T(p)r(l-p) = 2ni(en®-tn®y\ we know that

co(z;a,p) = ZPr(p)"1C(z;a,p) = ^ - r

is also holomorphic on H' x C2. When P = 0, by computing the residue of(l+z'hy^h^t'1 at t = 0 (see the computation in Exercise 2.2.1), we know thatco(z;a,O) = 1. On the other hand, when a = 1,

Jpâ+z-HrHP-V^t = jmt*'ltldt = (e27l/p-l)r(p) (see (2.2.2)).

Thus we again obtain co(z;l,p) = l by rflJ)r(l-p) = 27ci(em'p-e"w*)-1. Actuallywe will prove the following functional equation in Lemma 3.1 (see [Sh6]):

co(z;l-p,l-a) = co(z;a,p);and the above evaluation follows if one knows either co(z;l,p) = l orco(z;a,O) = 1.

Lemma 3. We have

rrk(27c)k+T(k+s)-1(2y)-s^k+s"1e-27C^co(47T^y;k+s,s) if § > 0,

= jrk(27i)T(s)-1(2y)-s-kUls-1e-27c|^lyco(4TcUly;s,k+s) if § < 0,

Lrk(27c)k+2T(k+s)-1r(s)-1r(k+2s-l)(47cy)1-k-2s if § = 0.

We will give a proof of this lemma later. By admitting this lemma, we now writedown, for Tc(s) = (27c)"T(s),


(3a) L(2s+k,x)%(y)E*(^

N~s~k v^.. . i-s-k+i , „ , Jtiz+(nx| ( n ) ^ ( n y z ) e ( ^ iJtiz

y)flA 2 s k + i , x ( y ) (n = 1

+(21 y I A ) - k ^ I M f f ^-2s-k+i,z(nyZ)e(- ^n=1

where as>%(yZ) = ZdZz.yZX(d)ds and z = Xoo+/yoo. Let x = I Q I. Then

we see from x P l f ) = J t P W ' 1 = (-l)k, E*(zx) = %(z) | z | A'kE*(x) and

that

(3b) Gk([o ij,s,x) = (-DkNkL(2s+k,x)%(y)E*((-N-1)f^ jjxf.s.x"1)

NXfYyoo N?

AJ r(k+s)r(s)oo

+rc(s+k)-1 Xl(ny)f \l'k+l a^.k+ia^y2)^1 1 2^1 1*)^©^^n=l

+rc(s)-1(21 y I A ) ' k a.2s.k+i,x(nyz)e(-nz-(nx)f)co(47inyeo;s,k+s)}.

The explicit Fourier expansion (3a,b) shows the analytic continuability of Ek andGk because we already know the meromorphy of L(s,%) and by an estimatesimilar to (5.1.4a,b), we easily know the convergence of the Fourier expansion forall s e C as long as the constant term is finite. Writing mZ (0 < m e Z) fornyZ, we see that

|(ny)f|1

Ak-sa.2s.k+1,x(nyZ) | s=0 = mk-1

m/dh*d

Ik 1 XNoting that

{r(2s) /r(s)} | s = 0 = 2-1 and Ress=0L(2s+l,%) = ^^Yl^d-ip-1),

we now know from the above formula and (3b) that for 0 < k e Z,

C = V^Pck- l ) ! " 1 ^ )* and for U Z = I Q *


(4a) Gk(uz,0,%) = c | ^ % ^ - 8 ^

and for C = V ^

(4b) Gk(u2,l-k,X) = CJLL(12

k>X) + f>k_u(n)e(nz)|,I n=l J

where 8k,j is the Kronecker symbol and 8x,id = 1 or 0 according as % = id

or not, cp(L) = #((Z/LZ)X) and LL(s,%) = {np|L(l-X(p)p"s)}^(s,X).

Now we assume that N = 1 and hence % = id. The only difference in

computation from the case when N> 1 is that for w e AXB(A)+, the

non-triviality of %#(ya(x)wef"1)r|(Ya(x)wef"1)s does not necessarily mean that

(a b\ fc"2det(y) c"!a^ ,c * 0 for Y = AV Since we have y = rt ^ cex(c d) if

\c aJ \ 0 1 yc ^ 0, we have

GL2(Q)+ = B+Q^B^-QÛ (disjoint).

Thus we have an extra factor coming from B+Qx in the computation of

5J

Note that, by the formula IAJQ&\L = 1 (see §8.6),

J A / Q 1 ! (uz)s\[/(~^x)diLL = I y I A

sJA/Q\|/(-^x)dji = 8?>01 y I As.

Thus we have, for Gs(n) = as,id(n)

(5a) GuCu.s.id) = C(k+2s) I y I AS

+ ^ { 2 K ( 2 I y I ^ nr(k+s)F(s)

00

+rc(s+k)-1^|(ny)f|^s~ka.2s.k+i(nyZ)e(nz)e((nx)f)co(47tny»;k+s)s)

n=l

+rc(s)"1(21 y I A)-k£|(ny)ffA~sa.2s.k+1(nyZ)e(-nz)e(-(nx)f)co(4jcny00;s>k+s)}(

n=l

and by the functional equation for the Riemann zeta function, we have forC = V=lk(k-l)!-1(27c)k and C = 4-ik2kn

(5b) C-1Gk,i(x,0,id) = 2-1C(l-k)+8u(2jc I y I AY^Î ^k-


This finishes the proof of the theorem. We note here a byproduct of ourcomputation:

Theorem 1. Let the notation be as above. Then Fc(s+k)Gk,N(x,s,%) is entire

if Re(s) > - 1 — and %Nk is non-trivial. When % is trivial and k = 0, it is

2 —kentire if Re(s) > —-—. The singularities of this function are at most simple

poles.

We now prove Lemma 3. Following the argument given in [Sh6], we show that

(6) b (^y ;a ,p ) = f (x4-/y)-a(x~/y)-pexp(-2^x)dx

,p) if £ > 0.

The case where £ < 0 can be dealt with similarly. Then the lemma is the special

case where a = k+s and p = s. We have

Here note that a = y-/x e Hf (i.e. Re(a) > 0). Recall the formula in Exercise2.4.2:

Jo~ e ^ V ^ d u = a"T(s) if a e H' and Re(s) > 0.

Thus (y-/x)-a = r(a)-1J0°Oexp(-(y-/x)u)ua-1du if Re(a) > 0. Using this

formula, we obtain

(7) b(^,y;a,p) = /p

exp(-uy)ua-1

Now by putting f(x) = (y+/x)"p, we consider the inner integral

P exp(zx(u-2^))(y+/x)"pdx= f°° exp(27i/xH:^)(y+/x)-pdx ^ ( ) (J-°° J-°° 2TC 2K

Here jT(f) denotes the Fourier transform of f. On the other hand, we also knowthat


Now define a function g : R —> C by

fexp(-yu)up'1 if u > 0,g(u) = i

[0 if u < 0.Then, by (*), we see that

f(x) = (y+ix)-P = r ( p ) ' 1 f exp(-27c/x^-)g(u)du = (p) J (g ) (^ )J° 2K 2%

Thus J(f)(x) = 27cr(p)"1 JJ(g)(27tx) = 27tr(P)-1g(-27tx) by the Fourier in-version formula. Thus we obtain

f- ixC-2,*), , B J j2Icr(p)-1exp(-y(u-2^))(u-2^)P-1 if u > 2 ^ ,I e v ^J (y+zx)"pdx = s

[0 otherwise.We plug this in (3) and then obtain

b(^,y;a,p) = /P-(X(27r)r(a)-1r(p)-1 Jêxp(-2yv)(v+^)a-1(v-7U^)P-1dv.

By the simple variable change (V-TC )/2TC^ h-> t, we obtain the desired formula.

§9.3. Functional equation for Eisenstein seriesHereafter we always assume F = Q. Assuming that % is primitive modulo N(allowing the case % = id and N = 1), we now look at the explicit Fourierexpansion of Gk(x,s,%) given in (2.3b) to know whether there is a functionalequation for Eisenstein series Ek and Gk. We know from (1.9) that

s,%) for z = xee(0,

where T = T0(N) and Er(z,s,%) = ysS7€reeNr%*-1(5)j(8,z)-k I j(8,z) | "2s. Note

f a b>|that roo\T = R/{±l} via L N d I h-> (cN,d) for

R = {(cN,d) e Z2 | cNZ+dZ = Z } .

Thus, regarding %* as a Dirichlet character on (Z/NZ)X,

Er(z,s,%) = 2-1ysX(cN,d)6R^*"1(d)(cNz+d)"k I c N z + d I "2s*

Note that RxN = {(mN,n) I (m,n) e Z2} via (cN,d)xt h-> (tcN.td). Thus

E'k,N(z,s,%) = ysX(mNtnM0>0)%*"1(n)(mNz+n)-k | mNz+n | "2s

t=i-1= 2L(2s+k,%-1)Er(z,s,%) = 2Ek,N(Xco,s,%)j(xoo,0

k.

9.3. Functional equation of Eisenstein series 293

Let x = L . n I. Then %l = -N'lz and we have

Efk,N(z,s,%) I kx = N ^ E ' N ^ S O C X N Z ) * = 2Nk-1Ek,N(Teoxee,s,x)j(xee,0k

= 2Nk-1Ek,N(Txooxf-1,s,%)j(xeo,0

k = 2Nk-1Ek,N(XooXf-1,s,%)j(Xoo,0k

= 2Nk-1x(-Nf-1)Ek,N(xOoXf,s,%)j(xee,0

k = 2N-1(-l)kGk,N(Xoo,s,%)J(Xoo,0k.

Now we have, for Tc(s) = (2TC)'T(S),

(1) rc(s+k)rc(s)Gk(^ *},s,x) = 8%,idrc(s+k)rc(s)C(k+2s) | y | As

+*k(2N)"s{(2 | y | A)1-k-Tc(k+2s-l)L(2s+k-l,x)

+rc(s) 2J(ny)f |A a-2s-k+i,x(nyZ)e(nz+(nx)f)co(47cnyoo;k+s,s)n=l

+Fc(s+k)(21 y I A)"k ^|(ny)f |A s(T.2s-k+i,%(nyZ)e(-n z-(nx)f)co(47inyoo;s:

n=l

The idea is to compute the Fourier expansion of Ek directly and relate it toGk(x,l-k-s,%). Here we assume that % is a primitive character modulo N allow-ing the identity character when N = l . We know, for z = x«o(0,

I mNz+n | "2s.

Since we have shown in the previous section how to compute the adelic Fourierexpansion of Gk, here we shall give a computation in a classical way:

Ek(x-,s,)c) = 2-1ysX(mN,nM0,0)ZJk'1(n)(mNz+n)"k I m N z + n I "*

(mz + - - + n ) ' k '** be(Z/NZ)x m=l n=-oo ^

= ysL(k+2s,x*-1)+N"k"2s X5C*"1 (b) £ S(mz+^;k+s,s),be(Z/NZ)x m=l

where S(z;a,p) = Sm€z(z+m)"a(z+m)"p (a,p e C). For each z e C a n d s e

C, we define zs = | z | s e t s 9 writing z = | z | e/e with -rc < G < n. We

compute the Fourier transform of cp(y;a,P;x) = (x+/y)"a(x-iy)'P. Note that

S(z;a,P) = 9(y;a,P;x+m) for z = x+/y. We have by the Poisson summation

formula (Theorem 8.4.1)

S(z;oc,p) = ]TnGZe(nx)B(y,n;a,p)


where B(y,£;oc,p) = £ e(-£x)(p(y;oc,p;x)dx.

Exercise 1. To justify the use of the Poisson summation formula, show:(1) If Re(oc+p) 2. 1, 9(y;a,(3;x) is continuous and bounded as a function of xand is integrable; (ii) XneZ I B(y,n+x;a,(3) | is convergent locally uniformly inx. (You may use Lemma 2.3 and Lemma 2 below.)

By Lemma 2.3, we have, for z = x+/y e M

(2) Ek((jJ *],s,%) = ysL(k+2s,%*-1)

+N-k-2s £ % *-i ( b ) £ £ e ( m n x + ^)B(my,n;k+s,s)b€(Z/NZ)x m=ln€Z W

= ysL(k+2s,%*"1)+N-k-2s/'k(27i;)s

x{(27c)k+T(k+s)-1r(s)-1r(k+2s-l)(47cy)1-k-s ^%*~l (b) ^m1"1""2 8

be(Z/NZ)x m=lu °° k+s-1

X 1 ' f I 2n>0b€(Z/NZ)x m=l m

n>0be(Z/NZ)x m=l m

= y*L(k + 2s , X -V2- s N-^f*5 x i d { r (2 s + k - l )L(2s + k - l , X ) 2 SXl r c (k+s)r c (s )

Here we have used Lemma 2.3.2 to obtain the last equality. Now we replace s in(1) by 1-k-s and compute for z = x+iy s 9{

rc(l-s)rc(l-k-s)Gk([j i],l-k-s,z) = 8z,idrc(l-s)rc(l-k-s)C(2-k-2s) | y | As

+ikNs+k-12k+s-1{(2y)Tc(l-k-2s)L(2s+k-l,%)

+ r c ( 1-k-s) ]T ~=1 n"s ak+2s-i,z(n)e(nz)co(4jcny; 1 -s, 1 -k-s)

+rc(l-s)(2y)-kX~=1n"s"kok+2S-i,z(nyZ)e(-nz)o)(45my«;l-k-s,l-s)}.

We will prove the functional equation co(z;l-p,l-oc) = co(z;a,P) later. Then,

noting the facts that F(s+1) = sF(s) and %(-l) = (-l)k, we know that

rk(2N)1-s-krc(l-s)Gk,N(w)l-k-s)x)-2sNk+2sJkrc(s+k)G(x-1)-1Ek,N(w,s)x)= c(s)ys+d(s)y1-k-s


for suitable meromorphic functions c(s) and d(s) of s. Since we know that Gk

and Ek are both modular forms of weight k and of character %. Then we see for

c d 1 e ro(N) that

c(s)ys | cz+d | -2s-k+d(s)y1-k-s I cz+d 12s+k"2

= (c(s)ys+d(s)y1-k-s) I ky = X(Y)(c(s)ys+d(s)y1-k-s).

By choosing y and Y = [ c. d. I e Fo(N) suitably, we can make

- x(y) \d z+df» o f o r a t a l o s t a l l ,

This implies c(s) = d(s) = 0 for all s. Since GL2(A) = GL2(Q)S(N)Goo+, theabove identity between Ek and Gk holds all over GL2(A). That is, we have

Theorem 1. Suppose that % is a primitive character modulo N (we allow thetrivial character when N = 1). Then we have

Exercise 2. Show the functional equation for L(s,%) using Theorem 1.

We need to prove

Lemma 1. We have co(z;l-(},l-a) = co(z;a,p).

Proof. By definition, £(z;a,p) = J°°e~zx (x+ iy^x^dx , which converges ab-

solutely for arbitrary a e C if Re(p) > 0 and Re(z) > 0. We compute

irx^dx = JVZX JV^V-Mux^dx,

because r(oc)(x+l)-a = Jo°°e"u(x+1)ua-1du if Re(a) > 0 (Exercise 2.4.2).

From Fubini's theorem, we see that

-Mxdu = r(p)

= r(p)za-P JVû^cu+iyPdu = r(p)za-PC(z;i-p,a).

This implies the functional equation of co because co(z;a,p) =Now we prove the following estimate for our later use:


Lemma 2. For any given compact subset T in C2, we can find two positivereal numbers A and B such that if (oc,p) e T and y e R+, then

|co(y;oc,p)|

Proof. We have C(y;cc,p+1) = J ^ e ' ^ C x + l f ' ^ d x . Using the formulas

e"yX = ^(-y"le"yX)' ^ { ( x + i r ' x P ) = (

we perform the integration by parts and obtain

If Re(|5)>0, [-yV^Cx+iy^xPlÔ and thus we have

PC(y;a,p) = ypC(y;a,p+l)+(l-a)pC(y;a-l,p+l).

Interpreting this using co, we get

(3) co(y;a,p) = coCyja.p+^+Cl-^y^coCysa-l.p+l).

Iterating this formula n times, we have

(4) co(y;a,p) = £ £ = ( ) (£)(l-cc)(2-a) (k-a)y-kco(y;a-k,p+n)

Now we choose a positive integer n so that Re(a-l)<n for all (oc,p) e T.

Then for x e R+, | (l+x)""11 < (l+x)n = Zk=0(J)xk. Thus if Re(p) > 0

for all (cc,P) e T, we have

|co(y;a,P)| <

S I T(p) |

= ILoLo ©r(k+Re(P))y-k.

From this the assertion of the lemma is clear. When there exists (a,p) e T suchthat Re(p) < 0, we choose a sufficiently large integer n so that Re(P+n) > 0for all (a,P) e T. Then by (4), the proof in this case is reduced to the case al-ready treated.


From the computation of the Fourier expansion we have done, we can at least de-termine the form of the Fourier expansion of Ek at various cusps. That is, we seethat

where E'N(z,s;(a,b)) = ysI(m,nKa,b) modN, (m,n)ô.O)(mz+n)"k | mz+n | "2s for(a,b)e (Z/NZ)2. We also see easily that for y e SL2(Z)

E'k,N(z,s;(a,b)) I kY = E'k,N(z,s;(a,b)Y).Thus the Fourier expansion of E'k,N(z,s;(a,b)) at the Cusp Y(°°) is given by theFourier expansion of E'k,N(z,s;(a,b)Y)- Thus we only need to computeE'k,N(z,s;(a,b)) for general (a,b). First writing

E'k,N(z)s;(a,b)) = ^ m o d N> m

and then applying the Poisson summation formula to S(—TJ—;k+s,s), we have

E'k,N(z,s;(a,b)) = c(s)ys+d(s)y1-k-s+ ^=1alVN(s)e(nz/N)co(47cny/N;k+s,s)

where c(s) and d(s) are meromorphic functions of s and an/N and bn/N areentire functions of s. Moreover, as long as s stays in a compact subsetI an/N(s) I ^ A'nA and | bn/N(s) I < B'nB for suitable positive real numbers A,A\ B, B1. This shows

Lemma 3. For any given compact subset T in R and y e SL2(Z), there ex-

ist positive numbers A and B such that if %*id

Ek(z,s,%) I kY I ^ A(l+y"B) as y -» °° as long as x = Re(z) G T1-k

and if % = id, the above estimate holds if T c R-{ -j-}.

Exercise 3. Compute explicitly the Fourier expansion at ©o of E'k,N(z,s;(a,b)).

We now make the following definition. For a C°°-function f: #"-» C satis-

fying flkY = f for all Y e T for a congruence subgroup T of SL2(Z),

(5a) f is called slowly increasing if for any a e SL2(Z), there exist positive

numbers A and B such that |flkOc(z)| <A(l+y~B) as y —> °°;

(5b) f is called rapidly decreasing if for any B G R and a e SL2(Z), there

exists a positive constant A such that |flkOc(z)| <A(l+yB) as y —> °°.


Thus if f is slowly increasing, then for every a e SL2(Z), flkCX has polyno-mial growth in y as y -> oo. If f|k(x(z) for every a e SL2(Z) decreasesexponentially with respect to y as y —» «>, f is rapidly decreasing. If g and fare both of weight k for F and f (resp. g) is rapidly decreasing (resp. slowlyincreasing), then <D(z) = gf(z)Im(z)k is F-in variant and rapidly decreasing atevery cusp. Since every small neighborhood D = {z e r«NH I Im(z) > M} of thecusp i°<> is isomorphic to an open disk with variable q (q = exp(27uz)) andsince such neighborhoods are transformed by an element a e SL2(Z) toneighborhoods of a given cusp, | O(z) | is a bounded function on Y = PsH.Since Y has finite volume under y dxdy, the inner product (f,g) is welldefined. Let us record this fact:

(6) Iff and g are C°°-class modular forms of weight k with respect to Fand if f is rapidly decreasing and g is slowly increasing, then the integraldefining the Petersson inner product (f,g) is absolutely convergent.

§9.4. Analytic continuation of Rankin productsWe start by introducing a slightly more general space of modular forms. Let % be

a finite order Hecke character modulo N and %' be a character of (Z/NZ)X. We

define Mk(N;%,%') to be the space of functions f satisfying (M2-3) in §1 and the

following replacement of (Ml): for u € S(N)Ceo+ and a e GL2(F),

(M'l) f(axu) = Xf(u)xN(u)f(x)jk(ueo,i)"1,

((* bY\ iwhere %N(u) is as in (Ml) in §1 and %'N J l = %'((a-1d)N) for

a h\J <= S(N)Coo+. If f G Mk(N;%,%') satisfies the cuspidal condition (S) in

C Qy

§1, f is called a cusp form and we write Sk(N;%,%') for the subspace consistingof cusp forms in Mk(N;%,%'). For each arithmetic Hecke character0 : AX/QXU(M)R+ -» C x of finite order and f e Mk(N;x,%'), we define anew function

f<g>0 : GL2(A) ~> C by f®9(x) = 9(det(x))f(x).

Then it is plain that f<8)0 G Mk([M,N];ze2,%feM"1), where [M,N] is the least

common multiple of N and M and 0 M is the restriction of 0 to

ZMX = np|MZpX. Note that any character %' can be lifted uniquely to a finite

order Hecke character 0 of AX/QXU(N)R+ so that %' = 0N- Then the associa-

tion f h-» f®0 induces a map 0 0 : Mk(N;%,%') -» Mk(N;%02) whose in-

verse is given by 00"1. Thus we have

9.4. Analytic continuation of Rankin products 299

(1) ®0 : Mk(N;x,0N) s Mk(N;%02).

One can of course define the space M ^ ^ x ' ) for GL(2) over a totally realfield for a character %': (O/m)x —» Cx. However in this case, %' may not beliftable to an arithmetic Hecke character 0, and even if the lifting is possible, 0may not be uniquely determined. Although we can construct a theory similar to theone presented here in the general case (see [H8]), this difficulty certainly addsmore technicality in the treatment. This is the one of the reasons why we assumedthat F = Q here.

Now we look into the Fourier expansion of f e Mk(N;%,%!). We take the finiteorder Hecke character 0 such that 0N = %'. Then f<8>0 has the followingFourier expansion:

f ® 0 ( (o l

Thus applying 00"1, we find for y e Ax+ = AfxR+ that

Here we have used the fact that 0((£y)f) = 0(§y) = 0(y) = 0(yf) for y e A\and \ e Q+. Thus let us put

(2) a(y;f) = ^(vfMyZjf®©) and ao(y;f) = e ^

Then the function of ideles y i-> a(y;f) no longer factors through ideals but reallydepends on the finite part of the idele y and satisfies

(3a) a(uy;f) = 0N-1(uN)a(y;f<8)0) for u e U(l) = Z x .

This shows the restriction of a(*;f) to ideles in

( A f ( N ) ) x = ( x e A fx | x p = l if p l N }

factors through the ideals prime to N (thus f e Mk(N;%,%!) with non-trivialNf is a GL(2)-analog of Hecke characters with non-trivial conductor). As for theconstant term, we have

(3b) ao(auy;f) = Z!"1(u)a(y;f) for u e U(l) = Z x and a e Q+.


Thus the function y h-> ao(y;f) factors through AfX/U(N) = C1(N) instead of

CL(1) for f®0. Now there is another operation. Let 0 be a primitive Dirichlet

character of (Z/CZ)X (2 < C e Z). Then choose a representative set R in

Zcx = n p | c Z px modulo CZC and define for f e Mk(N;%,x'),

(4a) f|e(x) = C1G(e)XreRe"1(r)f(xa(r/C)) for a(r/C) = (J ' ^ j .

Then for u = I Q J E S([C,N]), ua(r/C) = a(ab"VC)u and thus

f te(xu) = C^GCO) Xe^Wfixuair/Q) = C^GCG) XreR reR

reR

Note that S([C2,N]) is generated by the u's as above, <x(v) for v e Z c (Zc =IIplcZp) and w e S(l) with w = l m o d C 2 . Since wa(r/C) = a(r/C)w' withw1 G S(N) for the w's as above, we know that

(4b) f | 6 e Mk([C2,N];%,%'0).

We compute the Fourier expansion of f 10:

^fleffj fll =ao(y;f)L0"1(r)+ S ^ y .

£vrHere note that by the definition of e (8.2.4), e(^=r) = exp(-27n[£yr/C]) for the

fractional part fcyr/C] of ^yr/C in Q c x = n p | c Q Px . Hence E ^ R ^

= 6(-^y)G(e-1) = e(^y)C"1G(0) (see (4.2.6a)) if £yZ is prime to CZ and

otherwise it vanishes. This shows a(y;f | 0) = G(yc)a(y;f) and ao(y;f |0) = O.

Thus we have

(4c)

where we consider 0 is supported on ZQX extended 0 outside Zcx.

Let % be a finite order character of Ax/U(N)Qoo+ = (Z/NZ)X. We write %* forthe associated Dirichlet character modulo N. When confusion is unlikely, wewrite % for %*. We pick a Z[%]-algebra homomorphism


X : hk(To(N),x;Z[x]) -» C

and consider its normalized eigenform f = iT_ X(T(n))q" e 5k(ro(N),%*). We

may write the corresponding cusp form f e Sk(N,x) as

f ( (o i)) = X ^ Q +a ( ^ y z ; f ) e ( ^ x ) e ( i ^ ) for a(nZ;f) = X(T(n)).

Similarly we take another normalized eigenform g e M/(J,\|/) associated with aZ[\|/]-algebra homomorphism (p : h/(ro(J),\}/;Z[\|/]) —> C. Then we define theL-functionof X®(p<8>CG for an arithmetic Hecke character co (of Q) as in (7.4.2):

(5) L(s,A,®q>®G>) = L(2s+2-k-/,%\|/CG2) ]T ~=1 ^(T(n))(p(T(n))co*((n))n"s

= L(2s+2-k-/,%Vco2)Xr=1 a(n;f)a(n;g)co*((n))n-s,

which is an Euler product of degree 4 (see the proof of Lemma 5.4.2). Now tak-ing the product *F(x) = (f(x)jk(xeo,0)((g I G))(x)j/(Xoo,0) as a function onGL2(A), we consider the integral

(6) Z(s,f,g,co) = JAX+/Q J A / Q T | J j *)jd^(x)co(y) | y | Asd|ix(y),

where d|J. and d|Lix are the additive and multiplicative Haar measures discussed in§8.5. Disregarding the convergence of the integral, it is formally well defined.We will look into the convergence later and for the moment concentrate on comput-ing it formally in terms of Dirichlet series. Let C be the conductor of CO andwrite O for the characteristic function of ZQX in Qcx. Then we see that

a(y;f)ca(y;g I co)co(y) = O(yc)a(y;f)ca(y;g)co(y(c)),

where a(y;f)° = a(y;f) and y ^ = yyc"1 for y e Ax+. Then by the orthogo-

nality relation (8.4.4):

we have, for Ax+ = AfxxR+ (R+ = {x s R | x > 0}) and Q+ = QflR+,

Z(s,f,g,co) =


= co(y))

J A K / O

= JA x <&(yc)a(y;f)ca(y;g)co(y) I y I Ase(2iy^)d^x(y)

* (yc)a(yf;f)ca(yf;g)co(yf) I yf I Asd|a.fx(yf)JR+e(2 iy »)y Jd| i x (y) .

Thus we compute each integral, for U = U(l)

<&(yc)a(yf;f)ca(yf;g)co(yf) I yf | Asdnf

x(yf)

= £ co * (n) A,c(T(n))9(T(n))n-s = L(2s+2-k-/,%" Vco V£(s,kc<S)(p<S>co)n=l

and

JR+e(2fyeo)yoosd^x(y) = J ^ e ^ y - M y = (4ic)-r(s).

This shows

(7) L(2s+2-k-/,%-Vco2)Z(s,f,g,co) = (47t)-T(s)L(s,A,c®cp<g>CG).

As already seen in §5.4, the series Z°° co*(n)A,c(T(n))(p(T(n))n~s actually con-n=l

k+/ kverges absolutely either if Re(s) > I+-9- o r if Re(s) > /+ j according as g isa cusp form or g is not a cusp form. The integration over A/Q needs no justifi-cation because A/Q is compact and the series are uniformly convergent. As forthe integration over Ax/Qx, replacing each term in the summation by its absolutevalue, we can perform the interchange of summation and integration because weare dealing with series with positive terms. The resulting series with positive termsafter the interchange of the integrals converges if Re(s) is sufficiently large. Thenwe apply the dominated convergence theorem of Lebesgue quoted in §2.4 to per-form the same interchange without taking absolute value. Thus the identity (7) isvalid if Re(s) is sufficiently large.

Now we compute the integral (6) differently. Recall that

¥ ( x ) = (f(x)jk(xM,0)((g|co)(x)ji(x-,0).

We consider a measure dv = I y I A'1d|ix(y)®d)J.(x) on

B(A)+ = U*Q jj e B(A) I y~ > Ok


(y x\ (y iC\ II r

Since L J a ( u ) = a(yu)l Q A and d|i(yx) = | y | Adî(y) (because JxZpdUp

= I x | p (8.5.1)), b'*d|iB(b) = d|iB(b'b) = d|iB(b), i.e. d|iB is left invariant.Then we see, for B(Q)+ = B(A)+HB(Q), that

Z(s,f,g,co) = JB(Q)+NB(A)+^(b)co(det(b))Ti(b)s+1d^B,

where ri(x) is as in (1.9...). Let GL2(A)+ = | X G GL2(A) | det(Xoo) > 0} andrecall that C«>+ = {x e GL2(R) I det(x) > 0 and x(0 = /} . Now we choose asuitable Haar measure JUL on GL2(A)+ depending on SQL) as follows. Let Sbe any open compact subgroup of GL2(Af). Let djioo be the Haar measure on thecompact group Coo+ZZoo (for the center Z«o of Goo+) with volume 1 (note thatC00+/Z00 = T because C00+ = S02(R)Zoo). Then we define a measure |Lis onB(A)+SCoo+ such that

= JB(A)

for all functions cp on B(A)+UCoo+, where d|Uo is the tensor product measure onSCoo+/Zoo of the Haar measure on the compact group S with volume 1 and themeasure d|ioo on C00+/Z00. We take the measure on GL2(Q)+\GL2(A).f/SCoo+ in-duced by this measure. In fact, by taking a fundamental domain 7 of GL2(Q)+in B(A)+SCoo+/SCo^, we define

This is possible because of GL2(A)+ = GL2(Q)+.B(A)+-S-Coo+ (Lib). The mea-sure we have constructed depends on the choice of S in the following sense. Ifone takes an open compact subgroup S1 of S, then for any right S-invariantfunction f,

= [S:S']JG L 2 ( Q ) A G ( A ) + / S C o e +f (x)d^s(x) ,

because d|io f°r S1 is d|J,o for S multiplied by the index [S;S']. LetE(S) = QxnSCoo+ (which is either {±1} or {1}) and (p be a function onB(Q)+\GL2+(A)/SCoo+ such that cp is supported by B(A)+SCoo+ = B(Af)SGoo+.Then we have

We know from (1) and (4b) that, for % = X^CD" 2 ,

xF(xu)co(det(xu)) = ^#(u)xF(x)co(det(x)) for u e S(L)


where L = [C2,N,J] is the least common multiple of C2, N and J. Thus^#(x)xF(x)co(det(x)) is left B(Q)+-invariant and right S(L)Co+-invariant. Notethat ^#(b)vF(b)co(det(b)) = xF(b)co(det(b)) for b e B(A)+ ) and writeY = GL2(Q)+\GL2(A)+/SOo+. Applying (8a) to S = SQL) and E(S)={±1},we have, writing djis as d|i.L and £, = XXJ/W2 ,

We now compute

T(TX) = (f(Yx)jk(Yx-Of(x)c((g | co-1)(x))jk(yx00,O

cj/(YXoo,0 = f(x)c((g |

(yThen we see from (1.11) and (1.10a) that for w = 0

(8b) L(2s+2-k-/,%-Vco2)Z(s,f,g,co)

= L(2s+2-k-/,x'1\j/co2)JYfc(g I co-1)(w)co(det(w))

I co-1)(w)co(det(w))L(2s4-2-k-/,x-Vco2)E*(w,s-k+l,^) | y | Akd^L(w)

I co-1)(w)co(det(w))Ek./,L(w,s-k+l,O | y | AkdîL(w).

Here we claim that

(9) Y = GL2(Q)+\GL2(A)+/S(L)Coo+ = T0(L)W via Xeo h-> Xoo(/).

In fact, if Xoo = yx'ooU with u e Coo+, then ye S(L)GOo+nGL2(Q) = T0(L).Therefore the above map is well defined and surjective. The injectivity followsfrom the fact that GL2(A)+= GL2(Q)+S(L)Goo+ (Lib). Thus the fundamentaldomain J we have taken in order to define djiL can be thought of as afundamental domain of Fo(L)V# Then our measure on Y is the familiar one,

y"2dxdy, by the definition of d|J.B. Noting the fact r\ ~ , ] = y«>» we know

that(10) L(2s+2-k-/,%-Vco2)Z(s,f,g,co) = 2"x((g I co)EI

k./,L(s-k+l,%\)/-1co-2),f)ro(L),

where ( , )r is the Petersson inner product defined in §5.3 and g | co

G f^(ro(J),\}/co2) (resp. f e 5k(Fo(N),%)) is the modular (resp. cusp) form

corresponding to (g | co)<8>co (resp. f). By Lemma 3.3, E(z,s) =


E'k-/,L(s-k+l,%\i/"1co"2) has only polynomial growth as y - > s e P!(Q) (i-e.E(z,s) is slowly increasing). The same fact is true for g | co (see Section 5.3).Since f is rapidly decreasing (see (5.3.8a)), f (g | co)E(z,s) is also rapidlydecreasing, and hence ((g I co)Ef]C./fL(s),f) converges absolutely for any s. Thusthe function assigning ((g | co)E'k_/,L(s),f) to s is a meromorphic function of swell defined for all s. In particular, when k * I, it is an entire function. Thisshows the analytic continuability of L(2s+2-k-/,%~1\|/a>2)Z(s,f,g,CQ). Thus wehave

Theorem 1 (residue formula). Let X : hk(Fo(N),%;Z[%]) -» C and (p :

h/(ro(J),\|/;Z[\|/]) -> C be algebra homomorphisms with k > / and let co be a

primitive Dirichlet character. Then L(s,A,c®(p®co) can be continued to a mero-k + /

morphic function on the whole complex s-plane and is entire if Re(s) > ——

and if either %\(d2 Nk~l is non-trivial or X(T(p)) * co(p)(p(T(p)) for at least

one prime p outside NJC. If X = (p and if % is primitive modulo N, the

function L(s,Xc<8)X) has a simple pole at s = k whose residue is given by

Ress=kL(s,?ic<sa) = 22k-17ik+1(k-l)!-1N-1np|N(l-p"1)(f,f)r0(N>where f e 5k(Fo(N),%) whose Fourier expansion is given by

Proof. The first assertion for non-trivial x V ^ 2 ^ " ' follows from Theorem 2.1and the argument given above. Note that

Ress=iGo,N(x,s,id) = Ress=iE0,N(x,s,id) = 7cnp|N(l-p"1)-

From this, we get, with a non-zero constant c,

Ress=kL(s, c<S>(p<8)co) = c(g I co,f)ro(L).

We know from (5.3.10) that T(p)* = %(p)'1T(p), although we only proved thisfact when N and J are powers of a prime in §5 (see [M, §4.5] for the generalcase). Thus

co(p)(p(T(p))(g | co,f) = (g | co | T(p),f) = (g | co,f | T(p)*) = %(p)Xc(T(p))(g | co,f).

Since ?ic(T(p)) = x(p)"^(T(p)) (see (5.4.1) and [M, (4.6.17)]), if co(p)cp(T(p)) ^>.(T(p)), (g | co,f) = 0. When X = cp, we can easily compute c using the aboveresidue formula of Eisenstein series and conclude with the residue formula in thetheorem.

Returning to the integral expression (8b), the integrand


I co-1)(w)co(det(w))Ek./,L(w,s-k+l^) | y | Ak

is the restriction of Lye{±i}B(Q)+^L2(Q)+xJ/(7x)co(det(7x))^#(7x)ri(7x)s+1 to B(A),

which is left invariant under GL2(Q) and right invariant under S(L)Coo+. Notethat the same fact is true for

l°(g I co-1)(x)co(det(x)) | det(x) | AkEk_,,L(x,s-k+U).

This shows that

(11) L(2s+2-k-/,x-Vco2)Z(s,f,g,co)

co"1)(x)co(det(x)) | det(x) |

§9.5. Functional equations for Rankin productsIn this section, we prove the functional equation for Rankin products which alsoestablishes the holomorphy of the L-function L(s,A,c®(p) if X * (p. We followthe treatment given in [H5, 1.9]. We start with algebra homomorphismsX : hk(r0(N),%;Z[%]) -» C and cp : h/(ro(J),\|/;Z[\|/]) -» C and suppose:

(la) k > /,

(lb) % and \|/ are primitive modulo N and J, respectively,(lc) Z 'V is primitive modulo L,

where L is the least common multiple of N and J. The first assumption (la) isharmless, but the other two conditions impose a real restriction. To remove thisassumption, the simplest way is the use of harmonic analysis on GL2(A) [J],although one can do that in classical way adding a large amount of technicality.The reason for this difficulty is that, without the conditions (lb,c), the L-functionL(s, c<8>cp) lacks some of the Euler factors (whose exact form can be predictedusing Galois representations attached to modular forms discussed in §7.5; see [Dl,(1.2.1)]) at places p dividing L, and thus we cannot expect a good functionalequation without supplementing missing Euler factors. In [J, IV], all the Eulerfactors are defined in terms of admissible representations and are computedexplicitly when the attached local representations are subquotients of an inducedrepresentation of a character of a Borel subgroup. Then the functional equation isproven for any automorphic L-functions of GL(2)xGL(2) in [J,§19] includingL(s,A,c<8>(p) treated here. The Euler factor for super-cuspidal local representationsis recently computed in [GJ] (see also [Sch] and [H6]). Here we do not intend tobe selfcontained. In fact, we shall use the semi-simplicity of hk(Fo(N),%;Q(%))(for example, [M, Th.4.6.13]), when % is primitive modulo N, which is provenin the text as Theorem 5.3.2 when N is a prime power. Anyway, the proof in the

9.5. Functional equation of Rankin products 307

general case is basically the same as in the case of p-power level and is a goodexercise after studying the proof in the special case.

Let f (resp. g) be the normalized eigenform corresponding to X and (p. Wewrite f and g for the corresponding classical modular forms fi and gi,respectively. We start with the integral expression

2-Tc(s)L(s,A,c<8>(p) = JY(f°g)(x)Ek_a(x,s-k+l£) I det(x) | AkdHL(x),

where Y = GL2(Q)+\GL2(A)+/S(L)Coo+ = T0(L)W and % = j ^ " 1 . Applying(y ^

the functional equation for Eisenstein series (Theorem 3.1): for w = 0

r c ( s - /+ l )E k . a (w,s -k+U)

(-l)G(^1)rc(k-s)^(det(w))Ek.,(wxL,/-s^-1),

we have

(2a) 2-Tc(s-/+l)rc(s)L(s,Xc®(p) , ^ | det(x)

)rc(k-s)

xJYf g(xee)^(det(x))Ek./(xTL,/-s,^1) | det(x) |

In order to avoid confusion between f(x)c = f (x) (complex conjugation applied tothe value f(x)) and f°(x) (complex conjugation applied to the Fourier coefficient,i.e., a(n,f°) = a(n,f)c), we write the latter action as fc. Then we claim that

f I V ( x ) = %(det(x))-1f(xxN-1) = N-k/2W(?t)fc(x) and g I xf1 = r//2W((p)gc

for the constants W(A,) and W((p) with absolute value 1, where

xN = G GL2(Af). In fact, by the same computation as in (1.1a), we

see that (f I XN"1)! = f(XN(z))j(XN,z)'k regarding XN on the right-hand side as anelement in GL2(Q). Then Proposition 5.5.1 shows the fact when N is ap-power. A key point is that T*(p) = XNT(P)XN"X for p outside N andXNS(N)XN-1 = S(N). The general case follows from the semi-simplicity of theHecke algebra hk(Fo(N),%;Q(%)) by the same argument which proves


(M(X) 0s]Proposition 5.5.1. Note that TL = TNP with (3= for M(A,)N = L.

Then f | xLA(x) = x tde tWy^xf rV 1 ) ) = f I TN^xp"1) = N"k/2fc(xp-1)

using the fact that 1 = %(N) = %(Nf)x<~(N) = %(Nf). This shows that

a(n;f | TL"1) = N-k/2W(X)a(n/M(X),f)c = N-k/2W(X)X(T(n/M(X)))c and

a(n;g I TL 1 ) = J-//2W((p)a(n/M((p),g)c = J'//2W((p)9(T(n/M((p)))c

fM(cp) 0\ (y x\for M((p) = L/J. Thus (2a) is equal to, for p1 = and w = I I,

(2b) W(^)cW(9)2-2s+k+/-1L-3s+2k-2+2/N^2r/ /2^(-l)G(^1)rc(k^

xJYfc(wp-1)cgc(wp-1)E'k.a(z,/-s,^1)yk-2dxdy (z = x+iy)

= W(?i)cW(9)2"sN-k/2J-//2L-3s+2k-2+2ôo(-l)G(^1)rc(k-s

xLL(k+/-2s,%V"1) XI=i MT(n/M(X)))9c(T(n/M((p)))ns+1-k"/,

where we agree that X(T(n)) = cpc(T(n)) = 0 if n is not an integer. Since L isthe least common multiple of N and J and since L = M(?i)N = M(cp)J, M(k)and M((p) are mutually prime. Then n/M(k) and n/M((p) are both integers if andonly if n is divisible by M(X)M((p). Moreover M(k) \ J and M((p) | N, andhence we have T(pe) = T(p)e for p dividing the level (see (5.3.4a) and [M,Lemma 4.5.7]). Therefore, we have

X(T(nM(q>))) = ^(T(n))X(T(M(q>))) and cp(T(nM(X))) = \(T(n))X(T(M(X))).

Thus we see that

(2c) X ^ i MT(n/M(?i)))(pc(T(n/M((p)))n-s

n=1 ?i(T(nM((p)))(pc(T(nM(?i)))n-s

X ^ 1 MT(n))(pc(T(n))ns.

Combining (2a,b,c), we get

2-Tc(s-/+l)rc(s)L(s,?tc®cp)

(^

x(M(X)M(9))s+1-k-/rc(k-s)rc(k+M-s)L(k+M-s,>,<8)9c).

Using the fact that £(-1) = (-l)k"7, W(XC)W(X) = (-l)k (Proposition 5.5.1),M(X) = L/N and M(9) = L/J and(Exercise 2.3.5), we get

9.5. Functional equation of Rankin products 309

Theorem 1 (Functional equation). Suppose (la,b,c). Then for the least commonmultiple L of N and J, we have

Goo(s)L(s,A,c®q>) = W O ^ X L N J ) - 8 * ^ ^

where Goo(s) = r c (s- /+l)r c (s) and

By our assumptions (lb,c), we know that M(^) | J and M((p) | N. Under thiscircumstance, it is known that | W(A,c®q>) | = 1 (see [M, Th.4.6.17]). Since weknow that Goo(s)L(s,^c®(p) is holomorphic if X * q> and

Re(s-k+l) > -f <=> R e( s ) > ^T

by Theorem 2.1. Since k > / > 0, this is enough to show that Goo(s)L(s,A,c®(p)

is entire on the whole s-plane if X*<p. When X - (p, by the same reasoning,

the only singularity of rc(s-k+l)rc(s)L(s,^c®A,) is at s = k and k-1 . Thus

we have

Corollary 1. Suppose (la,b,c). Then Goo(s)L(s,^c®(p) is an entire function ofs unless X = (p. If X = (p, Goo(s)L(s,^c®X) has two simple poles at s = kand k-1.

Chapter 10. Three variable p-adic Rankin products

In this chapter, we first prove Shimura's algebraicity theorem for Rankinproduct L-functions L(s,A,<8>(p). Then we construct three variable p-adic Rankinproducts extending the result obtained in §7.4. As for the algebraicity theorem, wefollow the treatment in [Sh3], [Sh4] and also [H5, §6], [H7]. We only treat thecase of GL2(Q). For further study of this type of algebraicity questions for thealgebraic group GL(2) over general fields, we refer to Shimura's papers [Sh5,Sh8, Shll , Shl2] and [H8] for totally real fields and [H9] for fields containingCM fields. As for the p-adic L-functions, we generalize the method developed in[H7]. Another method of dealing with this problem can be found in [H5]. Thegeneral case of GL(2) over totally real fields is treated in [H8]. There is one moremethod of getting p-adic continuation of L(s,X,®(p) along the cyclotomic line (i.e.varying s) found by Panchishkin [Pa].

§10.1. Differential operators of Maass and ShimuraWe study here the differential operators 8k introduced by MaaB and later studiedby Shimura acting on C°°-class modular forms f on 0<\ for a complex number kand for y = Im(z)

§ k = 2 T C / ^ +2iy') a n d 5 k = 8k+2r"2 §k+25k, 8£f = f.We also define

(lb) e = -T—y2— and d = - — —.2KI dz 2KI dz

It is easy to check that(lc) 8kf = y"kd(ykf) and 8T

k = y-k-2r(y2dy-2)ryk+2.

Now define a new "weight k" action (denoted by f i-> f||kOO of a e Goo+ asfollows:

f ||koc(z) = det(a)k/2f(a(z))j(a,z)'k = det(a)1"(k/2)f | ka(z).Then the operators 8k and e have the following automorphic property: if f is ofCr-class,

(2) 8£(f | k a) = (8rkf)||k+2ra and er(f||ka) = (erf)||k-2ra for all a e G^+.

The proof is a simple computation. We only give a detail account for 8. We may

assume that det(oc) = 1 and need to prove 8k(f||kOO = (8kf)|k+2^« Note that

~a(z )= j (a , z ) " z , and

) ' k A and y(a(z)) = y(z)|j(a,z)|"2.

10.1. Differential operators of MaaB and Shimura 311

We see, writing j = j(a,z) and noting y(a(z)) =-^r thatjj

2irf{8k(£|kaH8rkf)|k+2a} = -

C(z))..kr, (2iyc , cz+d\l _ 0

iy J I \ c z + d + cz+d Jj - U«2iy

Exercise 1. Prove the second formula of (2).

We now claim that the following two formulas holds:

« •

Since the proof for these two formulas is basically the same, we only give anargument for the second formula. We proceed by induction on r. When r = l ,the right-hand side equals

which shows the formula when r = 1. Now assume that r > 0 and that the

formula is true for dr. Then

Note that d = 8 k + 2 j + ^ 1 and d(4jcy)j-r = (r-j)(4jcy)j-r-1. Thus4jty

= t (n^j^{(r-j)(4Jtyy-r-18kf+(47tyy-r5k+1f+(k+2j)(4Jcyy-r-14f}.

j=0 A 0 + k )

The coefficient of (47cy)j"(r+1)4f i s S i v ^ n by, when 1 < j < r,

j+k)-1)} _ /r+l\

= V j )When j = 0, it is given by

/r\F(k+r) /r\r(k+r) /r+l\r(k+r+l)r W r(k) + r ^ r(k) " ^ °

Similarly, when j = r+1, it is given by

Or(k+r) _ ^r+l^r(k+r+l)

T(k+r) ~

This shows the validity of (3).

312 10: Three variable p-adic Rankin products

Exercise 2. Give a detailed proof of the first formula of (3).

If f is a holomorphic function on 9{ having q-expansion of the form

f (z) = ^T-\ a(n/N;f)qn (q = e(z/N)) for a positive integer N,then as a power series of q, f(q) converges absolutely on a small diskD r = { | q | <r} . Taking a smaller disk De, f(q)/q gives a continuous functionof q on the closure of De which is compact. Hence | f(q)/q I is bounded, i.e.I f(z) I < Cexp(-27ty/N) as y —» «> for a constant C. In other words, f de-

creases exponentially at /°o. If f is a holomorphic cusp form for a congruencesubgroup F of SL2(Z), the function f is therefore rapidly decreasing. Moreover

3rf— is also exponentially decreasing as y —> °°. Since (8£.f) is a polynomial inoz(4jcy)"1 with holomorphic function coefficients as above, (8£.f) still decreases

exponentially as y —> «\ Since 5^(f ||kCx) = (8Tkf)||k+2rOc, applying the above

argument to f||k(X for a e SL2(Z) in place of f, we know that f||k(x decreases

exponentially as y —» °°. Thus we know that(4a) (S^f) is rapidly decreasing as a modular form for Y of weight k+2r if f

is a holomorphic cusp form of weight k for Y.Similarly we can prove that(4b) (5£f) is slowly increasing as a modular form for Y of weight k+2r if f

is a holomorphic modular form of weight k for Y.

Let f = Zj=0(47cy)"Jfj be a C°°-function on 9l Suppose that the functions fj are

all holomorphic. We claim that(5a) fj = O for all j if f = 0.

We prove this by induction on r (see [ShlO, §2]). When r = 0, there is nothing

to prove. Applying —, we get 0 = Z^=1-(27CjV-T)(47iy)"^1fj. Then dividingdz J

out by (47iy)"2 and applying the induction assumption, we get fj = 0 for j > 0.This shows, at the same time, 0 = f = fo. In particular,

Iterating this operation r times, we arrive, if f is of degree r in (47cy)4, at

(5b) 8rf = crfr with cr = (167i2)-rr!.

In particular, if f is an arbitrary C°°-function on 0{ with er+1f = 0, thenfr = Cr'Vf is a holomorphic function, and er(f-(47cy)"rfr) = 0 by (5b). Thus byinduction on r, we see the following fact for a C°°-function f onProp.2.4]):


(5c) er+1f = 0 <=> there exist holomorphic functions fj fj = 1, 2, ..., r)

such that ^ J

where fj is uniquely determined by f. Now we suppose that f is a modular formof weight k+2r for a congruence subgroup T satisfying the equivalent conditions

ii fa b^(5c). Then f||k+2rY = f for Y = J G r implies

r r r

x I " ' * y / i "~ • \ \ **yJ i / | | k + 2 r i — x \ **j) iViv /J\ *->*y*) \\*£JT\X\

j=0 j=0 j=0

j=0

= X(47iy)-Jfj(Y(z))(cz+d)-k-2r+J{c(z-2/y)+d}J

j=0 m=0

= (47ty)-rfr||kY+ S ( 4 i c y ) Xm=0 j=m

Noting that the second sum in the last formula is a polynomial in (47cy)"1 of degree

less than r, we conclude from (5a) that fr||kY=rr for all ye T comparing ther(k)

coefficient in (4jiy)"r. Now by (3), —-—77(8^) is a C°°-class modular form

of weight k+2r which is a polynomial in (47iy)"1 of degree r whose coefficientr(k)in (47iy)"r is fr. Thus f-CS^hj.) for hr = ——— fr is of degree r-1. Repeating

1 (r + k)

this process, we can write, if k > 1 and r > 0,

(6) f = Xr=o8^2r-2jhJ i f f satisfies (5c) ([Sh3], [ShlO, Prop.3.4]),

where hj is a holomorphic modular form of weight k+2r-2j. The modular formshj are uniquely determined by f. In this argument proving (6), we implicitly

F(k)assumed that k > 1; otherwise, hr = —-—— fr may not be well defined (i.e.

1 (r + k)F(k)

——— may have a pole at non-positive integers k). Now we have by (2)I (r + k)er+1(f||k+2ra) = (er + 1f) |k .2a = 0 for as GL2(Q)nGoo+ Then by (5c), we

can write f ||k+2rOc = Zj=08k+2r-2jnj for holomorphic modular forms hj. On the

other hand, by (2)

j h J l | k + 2 r a =J=0 U=0 J j=0


By the uniqueness of h'j, we see that hj = hj||k+2r-2j0c. This shows that

(7) f is slowly increasing (resp. rapidly decreasing) if and only if hj isholomorphic at every cusp (resp. a cusp form).

Let fAk (To(N),%) be the space of C°° modular forms f satisfying:

(Nl) f is slowly increasing;(N2) er+1f = 0;

(N3) flkY = Z(d)f for y = (* J) e r o (N) ,

where % is a Dirichlet character modulo N. We also consider the subspace!A££(ro(N),%) of 5\£k(ro(N),x) consisting of rapidly decreasing forms.Modular forms in these spaces are called nearly holomorphic modular forms. Wehave proven

Theorem 1. Suppose that r > 0 and k > 1. Then we have

The isomorphisms are given by fh-» (hj) as in (6). Moreover theseisomorphisms are equivariant under the " ||" -action of Goo+.

We write the projection map f H» ho as

(8a) H : ^ +

which is called the holomorphic projection. Since we have a Galois action

f H> f° ( a e Aut(C)) taking fWk(r0(N),x) to a4(ro(N),x°) (see §§5.3 and

5.4), we can define a Galois action on fAtk+2r(ro(N),x) so that f° corresponds

(hja) under the isomorphism of Theorem 1. Then by definition, we in particular

have, if k > 1 and r > 0 ,

(8b) H(f°) = (H(f))a for all f e ^ + 2 r ( r 0 ( N ) , x ) .

Note here that d(E°° anqn) = S°° na n q n , i.e. d = q—. This impliesn=0 n=0 Q(J

d(f°) = (df)a for the naive Galois action on q-expansion coefficients. By ourdefinition, the monomial "(4rcy)~m" is invariant under the Galois action. Since 5kis the sum of the multiplication by -k(47iy)"1 and d, we see that

(8c) 8k(f°) = (8kf)° for f e ^ + 2 r ( r 0 ( N ) , x ) .


We want to compute the adjoint of 8k and 8 under the Peters son inner product.

We start with a general argument. Consider a compactly supported C°°-function <|>

and aC°°-function \\f. Let

T7 o 2 3 A ^ f k , 3 . _ 2 3 T- • 2 Ê = 2y x —, A = 2{^?+-T-}, Ex = yz—, Ey = ly 37,

dz ^y 3z 3x ^ oy

and Ax = —, Ay = - / { - + — } .dx J y ay

Then E = Ex+Ey and A = Ax+Ay. Writing \|/c(z) = (\|/(z))c for complexconjugation c, we have, for y(z) = Im(z),

^ ^ V y + k < l ) y y + ( l ) ây dy dy

This shows

0

0 = -/

Thus we have the following adjoint formula:

(9)

Let F be a congruence subgroup of SL2(Z). Let <|> be a C°°-class modular form

of weight k. We take a sufficiently fine (locally finite) open covering FVtf=

UjeiUj on 0< so that Uj is simply connected. Let n : ^ - > I\H be the pro-

jection. We choose a connected component Uj* in TE'ÛJ). Thus n induces an

isomorphism: Uj* = Uj. Then we take a partition of unity of class C°°:

1 = EJGIXJ; here, %j is a C°°-function with Supp(%j) c Uj. We regard %j as

a function on U*j under the identification induced by TC. Then %j(|): Uj* -> C

is a C°°-function whose support is contained in Uj*. Thus we can extend Xj4> t 0

all iH by 0 outside Uj*. Then for any C°°-class modular form \\f of weight

k+2 for F, we see from (9) that

(10) J r w J G l j

X / k 2 = (4>,e\|0


as long as £jeiJ#Sk(%j<|>)Vcykdxdy is absolutely convergent. When either <|> or\j/ is rapidly decreasing and the other is slowly increasing, we can always choosethe covering {Uj} so that the above sum is absolutely convergent. Let us see thatthis is so. We take out a small neighborhood of each cusp in T\H and write therest as Yo. Then Yo is relatively compact. We choose a closed neighborhoodD s of each cusp s so that Y o r i D s * 0 , r w = LJDSUYO and i : DO =

SÔôo) (for sufficiently large M>0) inducing i*(y"2dxdy) = t~2dtd0 for the

variable 8 on S1 and t on [M,oo). Here i is given by an element a of

SL2(Z) such that cc(s) = /«> (then t = Im(a(z)) and G = Re(cc(z))). Now

take a covering S1=ViUV2 so that Vi is isomorphic to an open interval. Then

Uy = USfiij = ViX(M+j-8,M+j+8) for (1/2) < 8 < 1 covers Ds. If F(t,0)

decreases exponentially as t —> <*>, we see easily that Zijluj jFt"2dtd0 converges

absolutely. We then choose a finite open covering Yo = UaUa- Then the

covering {Us>ij, Ua} does the job. Thus we have

Theorem 2. Suppose either f e ^ ( r o ( N ) , z ) or g e ^k+2(ro(N),%) is

rapidly decreasing. Then we have (8kf,g) = (f,£g)-

Corollary 1. Suppose that f e 5k(r0(N),%) and g e fl£k(ro(N),x). If

r < k/2, then (f,g) = (f,H(g)).

Proof. Writing g = H(g)+^=18^_2jhj, we see that ( f ^ . ^ j ) = (ef, S^jhj) = 0

since ef = 0. This implies (f,g) = (f,H(g)).

Now we see that)"27i/8k+s( ,

(CZ+d), ) = ^ - ( c z + d ) - k I cz+d | -2s-(k+s)c(cz+d)"k-11 cz+d | "2s

I cz+d 12s (z-zc)

= •^±|<cz+d)-k-1(cic+d) I cz+d I "2s = ±|-(cz+d)-k-21 cz+d I -2^\(z-z ) (z-zc)

By (lc), we have Sfcf = y"kd(ykf) and hence

(11) Bk+

Thus iterating the formula

(12a) 8k+s((cz+d)-k | cz+d I "2s) = (-47cy)-1(k+s)(cz+d)-k-21 cz+d I "2(s-1),

we have

10.2. The algebraicity theorem for Rankin products 317

(12b) 8^+s((cz+d)"k | cz+d | "2s) = (-47ty)-r r ( S + k + r ) (cz+d)-k-2 r I cz+d | "2(s"r).r(s+k)

Recalling the definition in §9.3,

E'k,N(z,S,X) = y S S ( m 1 k ' ' ^

we know that

(13) (-47c)r-^ ^5Ê ' k , L ( z , s ,%) ) = E'k+2r,L(z,s-r,%).T(s+k+r)

§10.2. The algebraicity theorem for Rankin productsWe now recall (9.4.7) and (9.4.10):

(1) (4rc)-T(s)L(s,r®q>) = 2-1(gE1k.a(s-k+l,xV1).f)r0(L),

where we have used the notation in §9.4, which we recall briefly:

X : hk(r0(N),%;Z[%]) -> C (resp. q> : h/(ro(J),v;Z[v]) -> C) i s a Z f t ] -

algebra (resp. Z[\\f]-algebra) homomorphism; f and g are associated normalized

eigenforms, f = E~ A,(T(n))qn and g = I^_1(p(T(n))q11; L is the least common

multiple of N and J. We consider the set (P consisting of all Z[%']-algebrahomomorphisms X* : hk(ro(N'),%';Z[%']) -> C varying N1 and %' butfixing k. For X* e (P, the integer N1 is called the level of X\ An elementX : hk(Fo(N),%;Z[x]) -> C e (P is called primitive if N is minimal among thelevels of the homomorphisms in the following set:

{^'e 2>U'(T(p)) = X(T(p)) for all but finitely many primes p}.The primitive element X as above is uniquely determined by X1 and is called aprimitive homomorphism associated to X\ The level of the primitive element A,is called the conductor of X\ We note that

5t'(T(n)) = A,(T(n)) if n is prime to the level of X\

The modular form f associated to a primitive X is called a primitive form. Thisnotion of primitive forms coincides with the one given in [M,p.l64], and we referto [M, 4.6.12-14] for the proof of the above facts. We now assume that X isprimitive. If the reader is not familiar with the notion of primitive forms, he maymake the stronger assumption that % is primitive of conductor N; in this case, fis automatically primitive in the above sense [M, §4.6]. Under the assumption ofprimitivity, we have

(2) fc||kTN = W(?ic)f for W(kc)e Cx with W(XC)W(X) = %(-l),


(o -nwhere TN = N 0 • This assertion is proven in Proposition 5.5.1 when % is

primitive modulo pr and N = pr for a prime p. A proof of the general case canbe found in [M, Th.4.6.15]. Hereafter we simply write (f,h)L for (f,h)ro(L)-

Then we have, for any smooth modular form h on Fo(L) with character %*

which is slowly increasing at every cusp of Fo(L),

(3a)

This follows from the fact shown below (5.3.9): (h||kTL,f) = (h,fIlk^L1) and

XL1 = -tL- In particular,

W(X)W(kf(f,f)N = (f lk*N,f |k*N)N = ( f^Nand thus(3b) W(X)W(?i)c= 1.

We write T L / N : ^ k ( r 0 (L ) ,%*) -» ^k(r o (N) ,%*) for the adjoint offL/N 0^

[Fo(N)pro(L)] with p = I I. We use the same symbol TL/N to denote

the corresponding operator Mk(L,%) -> Mk(N,%). Thus

(3c) (h | TL/N,f)N = (h,f I [ro(N)pro(L)])L.We see that

a b"| . fa Lb/Nîj p = [ J andhence ro(N)d JThis implies that

ro(N)pro(L) = ro(N)p and h | [ro(N)pro(L)](z) = (L/N)k"1h(Lz/N).Then using these formulas, we have2(4JC)-T(S)L(S,?LC®(P) = (gE ' t aCs -k+LzV 1 ) .^

xi/-1) ||k-/xL),f ||kxL)L

= (L/N)1-(k/2\(g||/xL)(E'k./,L(s-k+l,xx|/-1)lk-ftL),(f lktN) I tro(N)pro(L)])L

|TL/N,f||kXN)N

x({(gc(Lz/J)(E'k./,L(s-k+l,xr1)||k-rXL)}lTL/N,fc)N,where we have used the following formula:

I, I, ,. fL / J CA „,g||/XL = g||/Xj||^ Q J = (L/J)//2W(9)gc(Lz/J).

As seen in §§9.1 and 9.2 ((9.2.4a,b)), we can compute the Fourier expansion of) and Gk-/,L(0>£) explicitly. To recall this, we put


where the norm character N only has symbolic meaning but later we consider N

to be the cyclotomic character of Gal(Q/Q). Then we have, for C = zk

T(k)and a Dirichlet character ^ modulo L

(4a) Gk,L(0£) = C{8k,12-1LL(0^)-8k,28îd(8TC I y I A)-1cp(L)L-1+GL(^k)},and for C = ik2kLkA%

(4b) Gk,L(l-k£) = C{8k,28Ltl(87C | y | A

where 8kj is the Kronecker symbol and S id = 1 or 0 according as t, = idor not. Now we write h* = h : GL2(A) —> C for the adelic modular formcorresponding to h (see (Ml) in §9.1), i.e.,

h(au) = %L(u)h(Uoo(0)Jk(Uc«,0"1 for u e S(L)Goo+ and a e GL2(Q).As seen in §9.3 (noting h||ktL = L1-(k/2)h | kXL), we have

(5a) ( E ' u ( zand by (1.13),

(5b) E!k.a(z,s-r,%¥-1) = (-

Solving the equations s-r = m-k+1 and s = l-k+/+2r, we conclude from

= x(-l)W(kc) that for an integer m with / < m = / + r < ^

(6a) p1)||k./TL)} |TL/N,f||kxN)N

1 ) | k + / . 2 m t0

L)

Now we note that

k+/We see from (4b) for an integer / < m = /+r < -y- that

(6b) r(m+1 -l)T(m)L(mX® (?)= (L/N)1"(k/2)2"1(47c)mr(m+l-/)

x({(g||/XL)(E!k-/,L(m-k+l,x\|/-1)|k_/TL)} |TL/N,f ||kxN)N

= \|/(- i)Lm+1-kN(k/2H J-//2(-47c)m-/(47c)mW(9)W(X,c)

x({g(Lz/J)(8^//_2mGk+/-2m,L(z,l-k-/+2m,%"V))}

= t({g(Lz/J)(8^//_2 mE'L(rV^k + /"2 m))}


where t = /k+/2k-/+2m7C2m-/+1L/-mN(k/2)-1J"//2W((p)W(?ic)

and E ' L ( x V ^ k + / " 2 m ) = 8k+/.2m,25L,i(87i I y I A)-1+EL(x'ViVk+/"2m).

Now we treat the other half of the critical values L(m,A,c<E>(p) for m with

k+/l+-"2" < m < k. We solve the equations s-r = m-k+1 and s = 0 and get

m = k-l-r with r > 0. We have

(7a) 2(47t)-mr(m)L(m,?tc<g>(p)

= (L/N)1-(k/2)({(g||/TL)(E'k.a(m-k+l,xV'1)||k-nL)}

2.k4X^^Xr*) II2m+2-k-/tL) )

where we have used the following formula:

Then we see from W(X)C = %(-l)W(Xc) that

(7b) r(m+1 -l

= 2k-/+2m/k+/7C2m+l-/L/-mN(k/2)-lJ-//2w((p)w()lc)

-1))} |TL/N,fc)N,

where G ' L ( ^ k ) = 6k,i2-1LL(0^)-5k,25îd(87i I y I ^ "

Lemma 1. Let % be a Dirichlet character modulo N. Suppose that all the prime

factors of L divide N. Then for all h e 44(ro(L),%) and a e Aut(C), we

have TL/N(h)a = TL/N(ha).

Proof. Since T = TL/N is the adjoint of [ro(N)|3ro(L)] for (3 = 1 Q J ,

we see for two modular forms (|) of weight k for T = Fo(L) and \\f of weightk for T = T0(N) that

Jrv^V I kPyk"2dxdy = J ^ | T¥yk'2dxdy.

Using this formula, we compute T directly:

I kPyk"2dxdy = (L/N)k-1Jrx^)Xi/(p(z))yk-2dxdy

= (L/N)-1Jprp.1N^)(pl(z))\|/(z)yk-2dxdy


I kPlY(z)V(z)yk~2dxdy.

Thus we see that 0 IT = I y e prp.lxr,(t) | k(3ly. Note that p r

prp^nr = {(* *] e ro(N) I b e (L/NJZJ =Thus choosing a common representative set R for

(pl)-1rplnr\rt = prp-1nriNTi

we havero(L)pTo(N) = lJYERro(L)ply and ro(N)pTo(N) =

This shows that the operator T = TL/N coincides with T(L/N) onand, if all the prime divisors of L divide N, then T is equal to T(L/N) on thelarger space f^4(ro(L),%). In fact, if all prime divisors of L divide N, for any

fa b^Y = , in F , taking 0 < u < L/N such that au = -b mod (L/N), we see

asthat Y(0 x I e p r p ^ n r . Thus we can take R =\ I Q x 11 0 < u < L / m

the common representative set, and we see TL/N = T(L/N): a(n,f | T) = a(nL/N,f).This shows the lemma if all the prime divisors of L divide N.

The assertion of the above lemma is in fact true without assuming any condition.However, to prove it, we need a fairly long argument either from the theory ofprimitive forms [M, §4.6] (see [Sh3] for the actual argument) or from thecohomology theory studied in §7.2. We shall prove it in the general case by acohomological argument. First we introduce the space of N-old forms. Since

p t ^ r o W p t ^ r o C N t ) fo r p t = ( j J j w i t h 0 < t e Z ,

for any modular form f e ^k(To(N),%), we see that ft(z) = f(tz) = t1-kf | pt is

an element of ^k(ro(tN),%). Then we define the sub space S^(T o(L),%) of

N-old forms in 5k(ro(L)>%) to be a subspace spanned by modular forms in the

set: {ft(z) = f ( t z ) | 0 < t | L/N and f e 5k(ro(N),%)}.

Lemma 2. Let % be a Dirichlet character modulo N. Then for all

h E 5kT(r0(L),%) and a e Aut(C), we have TL/N(h)° = TL/N(ha).

Proof. What we need to prove is TL/N(ft)a = TL/N(fta) for any divisor t of L/Nand all f e 5k(Fo(N),%). Writing the operator fh->f|kpt as [t], we see that[L/N] = IIp[pe] according to the prime decomposition L/N = EL | L/NP6* ^ n e n bv

definition of TL/N» we have TL/N = np|L / NTpe, where Tpe is the adjoint of[pe] : 5k(Fo(L/pe),%) —> 5k(ro(L),%) which has the same effect as, for example,the adjoint of


[pe] : A(ro(N),x) -> 5k(r0(Npe),x).Thus we may assume that L/N is a prime power without losing generality. ThenTpe = Tp,e0Tp,e-i0-*-°Tp,i, where Tpo- is the adjoint of

[p] : ^ O W p J - 1 ) , * ) -> 5k(r0(NpJ),z).

Thus we may assume that e = 1 because fpk | Tpo- = fpk-i if j > 0 and k > 0.

If p IN, the assertion is already proven by Lemma 1. Thus we may assume that

p^N. Then choosing x e SL2(Z) by the strong approximation theorem (Lemma

(o -n6.1.1) so that x = 11 0 I modp and x e Fo(N), we can take the following

set as the representative set R as in the proof of Lemma 1:i ] I o < u <

Then I Q Jx = I I = x(5p modp shows that f I Tp = f I TN(p) if

f G 5k(ro(N),%). On the other hand, we see that

OVl n

This shows that fp | Tp = (l+pk"1)f. This finishes the proof.

The following fact is a key to our argument:

Proposition 1. Suppose that k > 2. Then the orthogonal projection

K% : 5k(r0(L),%) -> 5^(ro(L),x)

w rational; that is, we have 7Txa(f°) = {7Cx(f)}° for all a e Aut(C).

Proof. Note that (3tXL = tXL/t for any divisor t of L. This shows that^(%) = ^(TQ(L),X) is stable under XL. Note that the Hecke operators T(n) oflevel N and L are different on 5k(Fo(N),x) if n has a prime factor q such thatq divides L but q is prime to N. To indicate this difference, we write TiXn)for Hecke operators of level L if necessary. We write L = NoLo such that Lois prime to N and all the prime factors of No divides N. If n and tLo arerelatively prime, we see

a(m,ft | TL(n)) = £ % L ( b ^ a ^ f ) = a(m,(f | TN(n))t)0<b|(m,n) b l

« f t | T L ( n ) = (f |TN(n)) t

where the subscript "L" to % is given to indicate that x is a Dirichlet charactermodulo L (i.e. XL(^0 = 0 if b and L are not relatively prime). This followsfrom the two facts

11 mn/b2 <=> 11 m and x L ( b ) = ° <=*


under the assumption that n and N are relatively prime. When t is a primepower qr |Lo with r> 1, then

a(m,fqr I TL(q)) = a(mq,fqr) = aOm/q^f) = a(m,fqr-i).If r = 0, a(m,f | TL(q)) = a(mq,f) = a(m,f | TN(q))-%N(q)qk-1a(rn,fq). Thesefacts show that S(%) is stable under TL(n) for all n > 0. Since 5(%) is stableunder XL and TL is an automorphism of S(%), 5(%) is stable underTLTLWTL"1 = Ti/n)* (= the adjoint of Ti/n) under the Petersson inner product).Let SH%) be. the orthogonal complement of 5(%) in 5k(Fo(L),%) under thePetersson inner product. The stability of S(%) under TL(n)* and TL(n) for alln shows the stability of 5X(%) under these operators. Then, for anyQ(%)-subalgebra A of C, we let /^(A) (resp. /t(A)) denote the A-algebra gen-erated over A in EndcCS-Kx)) (resp. Endc(5(%))) by TL(n) for all n. By theduality between 5k(F0(L),x;A) and h(A) = hk(r0(L),%;A) (Theorem 6.3.2),we know that h(C) = /t-L(C)®/t(C) as an algebra direct sum. Now we canthink of corresponding subspaces H5X(A) and H5(A) in Hp(Fo(L),L(n,%;A)).Thus |3t induces a morphism

[pd = [ro(N)ptro(L)] : H^roOTJ^njfcA)) -* Hi>(ro(L),L(njc;A)),and H5(A) is defined to be the sum of Im([pt]) over all positive divisors ofL/N. Then from the fact that

H1P(r0(N),L(n,x;Q(%)))(8)Q(x)C = H1

P(r0(N),L(n>x;C)),

we know that H5(Q(%))®Q(X)C = H5(C). We can define H5X(A) to be the or-thogonal complement of H5(A) under the pairing (6.2.3a) in the full cohomologygroup Hp(ro(L),L(n,x;A)). The formula (6.2.3b) then tells us that the Eichler-Shimura isomorphism induces isomorphisms of Hecke modules,

UsHC) = SHx)®SHjf)c and H5(C) = 5(X)05(XC)C,where, for example, SH%°)° = { f (z) | f € 51(%c)} for complex conjugation c.This implies (see the proof of Theorem 6.3.2)

6HC) = AX(Q(X))®Q(%)C and /t(C) = ACQ(X))<8>Q(Z)C.

Thus again by the duality theorem, we know that

SHr,Q(X))®Q(x)C = SHX), 5(X;Q(X))®Q(Z)C = S(%) and

5k(ro(L),x;Q(x)) = 5x(x;Q(X))e5(%;Q(x)),

where SH%;A) = 5k(r0(L),x;A)ri51(x) and 5(x;A) = 5k(r0(L),x;A)n5(x).This shows the rationality of nx over Q(x)- Now we want to show that fora e Gal(Q(x)/Q), the following diagram is commutative:


KX : 5k(r0(L),x;Q(%)) -» 5k(ro(L),x;Q(x))(*) l a 4 a

KXO : 5k(r0(L),xa;Q(X)) -

To see this, we consider the Galois action of a E Gal(Q(%)/Q) on the

ro(N)-module L(n,%;Q(%)), which induces an isomorphism

a : H1P(ro(N),L(n,x;Q(%)))=H1p(ro(N),L(n,xa;Q(%))),

which in turn induces a Q-algebra isomorphism

a* : hk(ro(N)oca;Q(x)) = hk(r0(N) JC;

Here a* takes T(n) to T(n) = a"1T(n)a and coincides with a"1 on Q(%). Itis easy to check that (ha*,f)a = (h,fa) for h e hk (r 0(N),x°;Q(z)) andf e 5k(r0(N),%;Q(x)), because a* takes T(n) to T(n). Then again by theduality, we know the commutativity of (*), which shows the proposition.

Corollary 1. For f e ^ k+2r( ro(L) ,%) (k > 1 and r > 0) and

a G Aut(C), we have (f | TL/N)a = f° ITL/N.

Proof. Using Theorem 1.1, we can writef = E r ^

Then, with the notation of the proof of Lemma 1, we see that

= i ; = o I Y e R (L/N)2J(8i+2r_2j(hj|TL/N)).

Note that TL/N is the adjoint of [L/N]: 5k(ro(N),%) -> 5k(ro(L),%) and thusKer(TL/N) contains Ker(jcz) in Proposition 1. Thus TL/N = T L / N 0 ^ - Then byLemma 2 and Proposition 1, we see that

(hj I TL/N)a = (%(hj) I TL/N)° = (7tx(hja) | TUN) = hja ITL/N.Then by (1.8c), we know that

(f|TL/N)°= {^ = 0 I Y 6

Now we shall prove the following algebraicity theorem for Rankin products:


Theorem 1 (Shimura [Sh3,4]). Let X : hk(r0(N),%;Z[%]) -> C (resp.cp : h/(ro(J),\|/;Z[\|/]) -» C) be a Z[%]-algebra (resp. Z[y]-algebra) homo-

morphism and let f = E°°_ ^(T(n))qn e 5k(ro(N),%). Suppose that X and (p

are both primitive. Then for all integers m with I <m <k, we have

and for all ae Gal(Q/Q), S(m,?ic<g>(p)a = S(m,A,ac<gXpa).

Proof. We write ( , ) for the Petersson inner product on 5k(Fo(N),%c) andwrite ( , ) : hk(r0(N),xc;Q)x5k(r0(N),%c;Q) -^ Q for the pairing given by(h,f) = a(l,f I h). Under the primitivity assumption on X, it is known thatXc : hk(r0(N),xc;Q(A,c)) -> Q(XC) has a section in the category of Q-algebras;i.e., there exists an algebra homomorphism s : Q(^c) -» hk(ro(N),%c;Q(A,c))such that X°s = id and B = Ker(X) is a Q-subalgebra of hk(r0(N),%c;Q(X,c)).Thus hk(ro(N),xc;Q(^c)) = Q(A,)Uc0B with an idempotent l^c When % isprimitive (modulo N), hk(ro(N),%c;Q(^c)) is semi-simple and thus this is ob-vious. The general case is shown in [M, Th.4.6.12]. We consider the linear forml\ : <|>i-* (Uc,(|)> on 5k(r0(N),%c;Q). The action of a on L(n,%c;Q) inducesan isomorphism a : HJ>(ro(N),L(n,%c;Q)) -> H^roCNJ.LCn^Q)) . We seethat aT(n)a-1 = T(n) and thus induces an algebra isomorphism

a :

Then we see by definition that (ha,<|)a) = (h,<|))0. In particular, we have

a ) , i .e. , hc(<\>)° = lXco(<ba).

As a linear form, l%c is the unique one satisfying / c(<)) | T(n)) = X(T(n))c/xc((t))and /xp(fc) = 1 because of the duality between the Hecke algebra and the space ofmodular forms (Theorem 6.3.2). We consider another linear form <(> h-» (<|>,fc).Since

(<|> I T(n),fc) = (<|),fc | T(n)*) = (

l\c is a constant multiple of <|> h-> (<|),fc), i.e. l\c(§) = C((|),fc). We compute C.We see that 1 = /xc(fc) = c(fc,fc) = C(f,f) and thus

Here the equality (fc,fc) = (f>f) can be shown via the change of variablez h-> -z . Writing


C i lr /

• = 1 8 rr2"""' L, .,; 2m UN l 2 k + r

m)) I TL/N) if / S m S -«-,

we know from the above formula that $ has algebraic Fourier coefficients and

Here we have used the fact fca = (f°)c shown after the proof of Corollary 5.4.3

when N is a prime power. The assertion in the general case also follows from thesame argument by [M, Th.4.6.12 and (4.6.17)] under the primitivity assumptionon X. Then by Lemma 1 and (1.8b,c), we have, for £ = %"V>

tfWm^^L^))^) if l+ < m < k,

" [H(a(L/J)(5^|E'(Â^k+/2m)) | TL/N) if / < m

This shows Ll-mWlS(m,Xc®<?f = L/"mN-1S(m,A,ac(8)(pa), which proves thelast assertion. The first assertion follows from the last assertion sinceQ(XA) <= Q(^c) and Q(\)/) <= Q(q>) because for example

pkA = X(T(p)2)-?t(T(p2)) (see (5.3.4a) and §6.3).

Applying the above theorem to the primitive algebra homomorphismq>®co : h/(ro(Jl)>\|/c°2'Z[xl/co2]) —» C for a Dirichlet character co modulo Csuch that (pco(T(n)) = co(n)(p(T(n)) for all n prime to CJ, we have

Corollary 2. Let the notation be as in the theorem. Then for each Dirichlet char-acter co modulo C, write T for the level of the primitive algebra homomorphismq><8>CG associated to g | co. (T is a divisor of the least common multiple of J andC2.j Then we have, for all integers m with I < m < k,

( l ) m + /

_and for all GGGal(Q/Q), S(m,A,c<g>(p<g>co)°: = S(m,X,oc®q>a®a)a).

§10.3. Two variable A-adic Eisenstein seriesWe fix a prime p and put p = 4 if p = 2 and p = p otherwise. Let O bethe integer ring of a finite extension K of Qp. Let t, and % be two Dirichletcharacters modulo p. We assume that %(-l) = 1. We agree to put in force thefollowing convention:

10.3. Two variable A-adic Eisenstein series 327

^(n) = %(n) = 0 when p | n,

even if % or £ is trivial. We also consider the continuous character

K = KX : W = 1+pZp -> Ax = O[[X]]X given by K(Z) = (1+X)s(z) for s(z)

given by u s ^ = z (u = 1+p). We have added the suffix "X" to K because we

want to write Kz for the "same" character having values in Az = O[[Z]] obtained

from K by replacing X by Z. We consider the following formal q-expansions:

E(%iVk)(q) =

where i is the trivial character modulo p, Ep(%Nk) is the q-expansion intro-duced in §10.2 and (n) = nco(n)"1 e W. Thus the operation f 11 is just takingout terms involving q11 for n divisible by p, i.e. f | i = f-(f | T(p))(pz). Thisseries has the following property:

(El) E(^%)(uk-l,e(u)ur-l) = dr(E(%co"kA^k)|e^co-r) for k > l and r > 0 ,

where d = qrz = — — and e is a finite order character of W and the above

identity is the identity of power series in O[e][[q]]. Here we denote by O[e] the

subring of Q p generated over O by the values of e. In fact, for each character

e : W —> Q px of finite order, regarding e as a character of Z p

x by

e(n) = e«n», we have K«n))(e(u)uk-1) =nkeco"k(n) and thus

= d r{ IZ=i ^co-r(n)Xo<b|nbk-1%co"k(b)qn} = d'

Here abusing the symbols a little, we have written KQ for co iV. For eachcharacter e : W —> Q p

x of finite order, we consider the form E(^,x)(X,e(u)-l).Then by (El), we have

E(^,x)(uk-l,e(u)-l) = E(%Kok) I ê € f*4(ro(pap),e^2%)for sufficiently large a independent of k.

Now we take a finite extension M of the quotient field L of A = O[[Y]] andwrite J for the integral closure of A = O[[Y]] in M. Take a J-adic formG e S(\|/,J). We define a convolution product


Write %P = e'%co"k and \|/Q = e'^co'7 for Q E J4.(J) and P e ,#(A) whenk /

vel~2T) | ^co"r),

k /PflA = Pk)£' and QflA = P/>e». Then, if 0 < r < -y-, we have

and if ^ - < r < k-Z

G*E($A|T1x)(P,Q,e(u)ur-l) = G(Q)dr(E(VQ-1%Pe-2co2riVk-/-2r) | êo)-r)

= G(Q)d-r-1+k-/d2r+/-k+1(E(\|/Q-1xPe-2co2riVk-/-2r)

= G(Q)dk-/-r-1(G(\|/Q-1xPe-2co2rN2r"k+/+

Then viewing G*E(£,\|/'1%)(e(u)-l) as a formal q-expansion with coefficients in

A<§>oiJ for a fixed finite order character e : W -» Q px , we know that

G*E(^,\(/"1x)(e(u)-l) satisfies, if k>Z,

G*E(§,\|T1x)(P>Q,e(u)-l) = G(Q)(E(TI) | §e) € ^k(r 0 (pa)^2xP) ,

where T] = VQ^Zpe ' 2 © 2 1 ^" 7 and a = max(f(P),f(Q))+2f(e) for theconductors pf^P), p f ^ and pfê^ of %P, \|/Q and e, respectively. Thus we knowfrom (7.4.5) that

Here the suffix "a" indicates that we regard the character (£2%)a = £2% as de-

fined modulo p a ; thus, the character (^2%)a may not be primitive. By the

definition of completed tensor products, S(%eo;A)®oJ is the completion of

U § under the m-adic topology for the maximal ideal m of A(§> ad.

Lemma 1. Let S be the space of H = S°° a(n;H)(Z)qn withn=l

a(n;H)(Z)e (A®^)[[Z]]satisfying H(e(u)-1) e S(%oo;A)®oJ[£(u)] for all finite order characters e 6>/

W. Then, for H e S, we /zave H e S(%oo;A)®oJ[[Zl] and hence H(e(u)ur-1)

G SCXoojA)®^- Moreover, we have

H | e(e(u)ur-l) = H(e(u)ur-1) | efor all finite order characters e and r > 0.

Proof. By the same argument as the one given below (7.4.5), we may assume that

J = Zp[[Y]] and A = Zp[[X]]. Then A®oJ = Zp[[X,Y]]. Let £n be a primitive

pn-th root of unity. We put <t>n = ria(Z+l-£nO) e Z[Z], where ^n° runs over all

conjugates of £n over Q. Then we know that con = (Z+l)pn-l = IIo<m<n$m'

10.3. Two variable A-adic Eisenstein series 329

Thus H H» ^n=0H(Cn-l) defines a morphism cpn : S -> ^n=0S(Xoo;A)® J[Cn]- If

(pn(H) = O, a(n;H) is divisible by (Z+l-£m) in Zp[Cm][[X,Y,Z]] for all m < n .

Thus H e conS. This shows that Ker(cpn) = conS. Since HnCOnS = {0}, we

know that S injects intoKm Im((pn) = lim (S/conS).n

The space S(Xoo;A)<§>oJ[[Z]]/(cGn) consists of Hn = £~=1 a(n;Hn)qn with

a(n;Hn)e Zp[[X,Y]][Z]/(con) such that Hn(Cm-l) e S(%~,A)®0j&m] for all

0 < m < n. Thus S/conS is a subspace of S(%00;A)®0J[[Z]]/(con). Thus we

have a natural map of S into lim S(Xoo;A)(g)oJ[[Z]]/(cGn), which is equal ton

S(Xoo;A)%>aJ[[Z\l Since S(x~;A)<§)oJ[[Z]] is a subspace of Zp[[X,Y,Z]][[q]],the map is injective, and hence we can regard S as a subspace ofS(Xoo;A)<§)oJ[[Z]]. Then the projector

e : SOcÂ) -> S ^ x ^ A ) (Proposition 7.3.1)extends linearly to

e : S(Xoo;A)<§>oJ[[Z]] -* S ^ x ^

By definition, e commutes with the specialization map: H h-> H | z=z for anyz e Qp. In particular, H(e(u)ur-1) |e = H|e(e(u)ur-l) in Zp[e][[X,Y]][[q]].

Corollary 1. Suppose that G(Q) has q-expansion coefficients in Q and

G(Q)e 5k(r0(pp),\l/Q;Q). Then the two limits

lim>G(Q)dr(E(\|/Q-1Xp8-2co2riVk"/-2r) I £eco"r) I T(p)n! for 0 < r < ^ ,

Mm G(Q)dk-/-r-1(G(\i/Q-1xPe"2co2r^2r-k+/+2) I êco"r) I T(p)n! for ^ < r < k-/,

exist in O[e][[q]] under the p-adic topology, where

Moreover, writing the above limit as h, we have, in O[e][[q]],

h = (G*E(^,xi/-1X) I e)(P,Q,e(u)ur-l),

which is an element in 5k(ro(pap),Xp^2; Q) for a such that

Proof. We start with a general argument. Let A be a Q-subalgebra of C and

g e ^ ( r o C p ^ x y V ^ A ) . Write g = 2™0{ATzyY% with g j e A[[q]].

Take f e jy£?' (ro(pp),\|/;A) and write f = I™0(4jcy)-jfj. We consider

g( 5rk. f | e^) e *$?££ (ro(p

p),x;A).If k'+/ > 2m+2m', we can write uniquely


f I e£) = H(g(8£ I <

for hj G 5k-+/-2j(ro(pP),X;A) and H(g(8£f I e$)) e 5k-+/(ro(pP),z;A). Then

equating the constant terms of (*) as a polynomial in (4?cy)'1, we see from (1.3)

that

(1) godr'(fo I e§) = H(g(5r

k.f I

This shows that H(g(8£ f I e£)) = godr'(fo I e ^ - Z ^ ' ^ h j . Now assume that

A is a subalgebra of QflK. Then we can consider the identity (1) as an identity

in QP[[q]]. Since H(g(8k'f I e§)) e 5k'+/(ro(pP),%^2;Qp), we see, under the

p-adic topology, that

H(g(5rk. f | e$)) I e = Km {g(8£ (f | e£)) I T(p)n!},

where T(p) acts on q-expansions as in the corollary. Note that, for

^ e P"YO[[q]] ( 0 < y e Z),

d<t> I T(p)n ! = £ ~ = 1 a(mpn!;d^))qn = £ ~ = 1 mpn!a(mpn!;(t))qn -> 0

as n —>• eo. Since hj e p"YO[[q]] for sufficiently large y by Theorem 6.3.2,we conclude that

(2) H(g(8£f I e^)) I e = {godr'(fo I e?)} I e.

By (2.4a,b), we find that E^N*) and G(^N^) are modular forms except when

j = 2 and ^ = i d . The forms E(N2) and G(N2) show up in the formula

when either k = /+2r+2 or k = 2r+/. In this case, m = 0 and m1 = 1 by

(2.4a,b); thus, we can find f e ^2( r o(p) ,Q) such that f0 = E(A^2) or

G(N2). Hence the existence of the limit follows from the above argument. When

k'+/ > 2m, H(f(8J g I e^)) | e is always a classical modular form. Thus to finish

the proof, we only need to check the identity h = (G*E(£,r|) | e)(P,Q,e(u)ur-l) for

T| = y\fAX- Taking f as above and writing g = G(Q), we recall (1):

G*E(^\|r1x)(P,Q,e(u)ur-l) = gdr'(f0 I e§) = H(g(5rk,f \e$))+^=T+T'd"hy

By Lemma 1, (G*E(^,\|/-1%)(Z) | z=e(u)ur-i) I e = (G*E(^,\j/-1%) | e)(Z) | z=e(u)ur-i.

Thus (G*E(§f>|r1x) I e)(Z) I z=e(u)ur-i I e = ( G ^ E ^ , ^ ^ ) I e)(Z) I z=e(u)ur-i. Note

that G*E£,\fh) I e € S ^ f o ^ A ) ® ^ ^ ] ] . For F G S((X^2)C O;A), we have

F I e = lim F | T(p)n! under the w-adic topology for the maximal ideal of A. In

fact, assuming F E S((%^2)a;A) for some a > 0, we have by definition,

10.4. Three variable p-adic Rankin products 331

F | e(uk-l) = f(uk-l) | e = l imF(uM) I T(p)n! under the p-adic topology on

5k(ro(pa),%^2co"k) for k > a with a sufficiently large a. This implies, writing

p a = I I " Pb, F | e = lim lim (F mod Pa>a) | T(p)n!, which is equal to the

limit under the m-adic topology. Thus we see that

lX) I e = Km (G*E(§,y I T(p)n!.(

This implies(G*E(§,\|r1x) I e)(P,Q,e(u)ur-l) = ,1m(G*Eft,>|r1z)(P,Q,e(u)ur-l)) I T(p)n!,

which is equal to h. Here the last limit is taken in O[e][[q]] under the p-adictopology.

§10.4. Three variable p-adic Rankin productsWe take two normalized eigenforms G e S(\|/;J) and F G Sord(%;l). Note thatSord(X;l) = 0 for p less than 11 (see §7.6). Thus we may assume that p > 5.We define the A-algebra homomorphisms X: hord(x;A) —> I and (p: h(\|/;A) —» Jby

F | T(n) = A,(T(n))F and GI T(n) = <p(T(n))G.

By extending scalars if necessary, we may assume that

(Ir) the ring O is integrally closed in I and J.

Then I <§> c>J[[Z]] is an integral domain finite and flat over O[[X,Y,Z]]. We write

A = I <§> oJ[[Z]] and its field of fractions as M = Frac(A). Then we consider

Sord(%;l)<§>oJ[[Z]] = Sord(%;l ®oJ[[Z]]). As constructed in §7.4, we have an

inner product ( , ) | : Sord(%;l)xSord(%;l)-> K for the quotient field K of I.

We extend this product linearly to an inner product

( , )A : Sord(%;A)xSord0c;A) -> M.

Now we define Lp(?t<g>(pc) = (F,e(G*E(id,\i/"1x)))A e M. We study the values

Lp(X<8Xpc)(P,Q,R) for (P,Q,R) e *(l)x*(J)x*(A). We write PflA = Pk,£.,

QflA = P/)£», R = Pr>E and Xp = eXC0 k» ¥p = e'V®"'* ^ e further write

^ P : hk(r0(N),xP;Z[xP]) -^ Q and 9 Q : hk(r0(pp),\)/Q;Z[\|/Q]) -> Q

for the algebra homomorphism given by

F(P) | T(n) = XP(T(n))F(P) (resp. G(Q) I T(n) = <pQ(T(n))G(Q))


as long as Q is admissible relative to G. By the ordinarily of F, we have

Xp is either trivial with N = p or primitive modulo N.

To compute the value Lp(A,®(pc)(P,Q,R), we assume the following condition one:(P) G(Q) I eco"r is a primitive form of exact level J for a p-power J.

Although the condition (P) is hard to verify without using representation theory([Ge] and [C]), let us explain a little bit about this assumption, referring to [H5, II,Lemma 5.2] for details. By the definition of primitive forms given at the begin-ning of §10.2, we can always find a primitive form g of level J° associated withG(Q) | eco"r. We know that g | T(n) = eco"r(n)?ip(T(n))g for all n prime to p. Ifp I J° and g | T(p) = 0, we see that g = G(Q) I eco'r because eco"r(p)^P(T(p)) = 0.For almost all non-trivial e, J° is divisible by p and g I T(p) = 0. Thus theassumption (P) is known to be true for almost all e. When G(Q) is ordinary oflevel J (and thus G is automatically ordinary), the condition P is equivalent to

(P1) eco"r is neither XJ/Q"1 nor ip if \|/Q is non-trivial, and eco"r * ip if \\fp is

trivial, where ip is the trivial character modulo p.

We state our main result in this section:

Theorem 1 (Three variable interpolation). Let X : hord(%;l) -> I be an

I -algebra homomorphism associated with a normalized eigenform F e Sord(%,l)and G be a normalized eigenform in S(\j/;J). For each admissible point Q of

#(J) , we write (pQ : hk(Q)(r0(pP),eQ\|/0)"k(Q);O[e]) -> Q p for the O[e]-algebra

homomorphism associated with G(Q). Then we have a unique p-adic L-function

Lp(X®(pc) in the quotient field of l<§>oJ[[Z]] with the following evaluation

property: for P e A(l) with k = k(P), admissible Q e #(J) with I = k(Q)and R = Pr>e e A(A) satisfying the condition (P) and 0 < r < k-l, we have

S(P)Lp(A,<g>(pc)(P,Q,R)

(27cOk+/+2r^1'k(F(P)°,F(P)°) '

where F(P)° is the primitive form associated with F(P), No is the level ofJ is the level of the primitive form G(Q)|eco~r and

1 if either %p is non-trivial or k = 2,

S(P) = i J ^ _ } - i ( 1 . _E -i otherwtse.MT(p))2 XP(T(p))2


Here we understand W(k?) = 1 if %p is trivial and k > 2. The relation be-tween this three variable p-adic L-function and those obtained already (Theorems7.4.1 and 7.4.2) will be clarified in the following section. We can even computethe value Lp(^(8>(pc)(P,Q,R) without assuming (P). However the computationadds additional technical difficulty. Thus here we only present the result under(P). We refer to [H5, II, Th.5.1d] for details in the general case.

We start proving the above theorem. We want to relate the valueLp(A,®(pc)(P,Q,R) with the complex L-value L(SOAP®<PQC®£) for an integer sodetermined by R. We divide our argument into the following two cases:

Case I: 0 < r < ^ and Case II: ^ y ^ < r < k-/.

By definition, writing A for Qp(^Pc®(pQ®e), we have by Corollary 3.1

(1) Lp(?t<g>(pc)(P,Q,R)

= f (F(P),e(G(Q)dr(E(\|/Q-1%Pe-2co2rNk-/-2r) | eo)-r)))A in Case I,1 (F(P),e(G(Q)dk-/-r-1(G(\|/Q"1xPe"2co2riV2r+/-k+2) | eo)-r)))A in Case II.

Now we prove a lemma:

Lemma 1. Let e : (Z/p^Z)* —> Ox be a character. Assume that

lim (f(g I e)) I T(p)n! converges to a power series e(f(g | e)) Radically in 0[[q]]

for f and ge O[ [q ] ] , where I ~ = 1 a n q n I T(p) = I ~ = 1 a p n q n and

STânq111 e = IT=ie(n)anqn. Then lirn^f I e)g) I T(p)n! converges to a power

series e((f I e)g) and satisfies e(f(g | e)) = e(-l)e((f | e)g).

Proof. We define I JT_ anqn | p = Supn | an | p. Then the p-adic convergence is

just the convergence under the norm I | p in 0[[q]]. Writing a(n,f) for the co-efficient in q11 of f, we first prove that

a(npr,f(g|e)) = e(-l)a(npr,(f|e)g) if r>y .This is just a computation,a(npr,f(g|e))-e(-l)a(npr,(f|e)g)

= X Je(npr-j)a(j,f)a(npr-j,g) - e ( - l )£ Jr£(j)aa,f)a(npr-j,g) = 0,

because e(npr-j) = e(-j) = e(-l)e(j) if r > y. This shows

(f(g | e)) | T(p)r = e(-l)((f | e)g) | T(p)r.

This implies that ]JmJ(f \ e)g) I T(p)n! and Krnjfig I e)) I T(p)n! converge at the

same time and e(f(g 18)) = e(-l)e((f I e)g).


Note the fact that e(G(Q)drg) = Km (G(Q)drg) | T(p)n! in O[[ql] under the p-

adic topology. Then, writing simply E = E(£ A^"'"2*) and

F = 8j,12-1LL(0^)-5j,25îd(87C I y I A)-1(P(L)L-1+G(^j)

for £ = \|/Q"^pe"2co2r and j = 2r+/-k+2, we know, from (1) and the fact that

eco-r(-l) = (-l)r,

Lpa<g>cpc)(P,Q,R)

= (-l)rxj (p(p)'e(G(Q) I eco-rdrE(^k- /-2 r)))A in Case I,

1 (F(P),e(G(Q) | eco-rdk-/-r-1G(^2r+/"k+2)))A in Case II,

J (F(P),e(H(G(Q) | eco-r5rk_/_2rE)))A in Case I,

X [ (F(P),e(H(G(Q) | eco-r8|r-i7_rkl2E!)))A in Case II,

where we have used (3.2) and the fact (2.4a,b) that E and E1 are classicalmodular forms to obtain the last equality. Since e is self-adjoint under the pairing( , )A and F(P) | e = F(P), we have

T n r, cwD n m , 1 M 1 (F(P),H(G(Q) I eco"r5rk_/_2rE))A in Case I,

Lp(A,<8Xpc)(P,Q,R) = (-1) xs , f . \P ^ [(F(P),H(G(Q)|eo)-r5k

r-i7_rki2E'))A in Case II.

Write L for the least common multiple of N = p a (the level of F(P)) and

j = pP+2Y (the level of G(Q) I eco"r), and put p6 = L/N. Then the self-adjoint-

ness of T(p6) shows that

(2) Lp(?t<S>(pc)(P,Q,R)

8 [(F(P),H(G(Q)|eco-r8l_/-2rE)|T(p5))A in Case I,= (-l)\(T(p))"8x v i I *P (F(P),H(G(Q) eco-r5^7_rki2E') T(p8))A in Case II,

1 A (T( ))-5x{ ( F ^ ' H ( G ( Q > I «»"r8k-/-2rB) I T(p8))c in Case I,P P [(F(P))H(G(Q)|eo)-r5|r-i7_rki2E')|T(p8))c in Case II.

When either %p is primitive modulo N = p a or k = 2, as we have already seenin (7.4.1a) (and in a remark after Corollary 7.2.1 when k = 2) that

(F(P),g)oo (g,F(P)ch) (g,F(P))(F(P)jF(p))oo - ( F ( p ) j F ( p ) c | T ) - (F(P),F(P))'

because F(P)C | x = W(A,C)F(P). We now compute (F(P),g)c in terms of the

complex Peters son product ( , ) when %p is trivial. The above formula is true

except for the last term. In fact, the linear form(F(P) )


satisfies L<>T(n) = XP(T(n))L and L(F(P)) = 1. Thus by the duality theorem

(Theorem 6.3.2), we find that L(g) = (F(P),g)c. Therefore we need to compute

i f° -^F(P)C I x for x = Q . As shown in Corollary 7.2.1 (or its proof), if k > 2,there exists a unique normalized eigenform F(P)° e 5k(SL2(Z)) such thatF(P)° | T(n) = XP(T(n))F(P)° for n prime to p and F(P)° | T(p) = (cc+p)F(P)°for a = A.p(T(p)) and P = j>k'lX?(J(p))'1. We note here that pc = a becausea+P is a real number (because it is an eigenvalue of the self-adjoint operatorT(p)). Moreover

F(P) = F(P)°-a-1F(P)°lk[o i}We call this form F(P)° the primitive form associated with F(P). Either when %p

is primitive modulo p a or when k = 2, we simply put F(P)° = F(P), i.e., F(P)is already primitive. We put W(k) = 1 in this case, W(k) e C because

F(P)° 11 1 Q I = F(P)°. Then we see, noting that (F(P)°)C = F(P)°, that

F(P)c |x = F(P) c | ( ° ^

= F(P)°lk[0 1J-ap-1F(P)o = -ap-1(F(P)o-a-1pF(P)°lk[0 J ) .Writing ( , )p for the Petersson inner product of level p and ( , ) for that of

fP Q\level 1, we now compute, for 8 = I Q A and f = F(P)°,

(F(P),F(P)C | x)p = -pp'^f-a^f I S.f-a^pf I k8)p.Now we use the fact that

(f,f)p = (SL2(Z):r0(p))(f,f) = (l+p)(f,f), (f,f| k8)p = (f | T(p),f) = (a+p)(f,f),(f I k8,f)p = (a+p)(f,f) (see the proof of Lemma 2.1),

(f Ik8,f Ik8)p = (f I kx,f |kx)p = Pk2(fl|kx,f||kx)p = pk"2(l+p)(f,f),

where Tp(p) in the last formula indicates the Hecke operator of level p. Thisshows, if xp is trivial and k > 2, that

(3) (F(P),F(P)C | x)p =

We now compute (F(P),H(G(Q) | eco^S^^Ê) | T(p 6 ) ) c in Case I. We suppose

either k = 2 or %p is non-trivial. We recall (2.6a) replacing X and cp there by

Xpc and (pQ

c:

(4) 2(47c)-/-T(/+r)L(/+r,Xp®(pQc(8)8-1cor) = (L/N)1-(k/2)(-4Tc)T(r+l)-1%p(-l)W(?ip)

X({(G(Q)C I e"1^| | /TL)(^. / .2 r(E'k. / . 2 r ,L(z,j ,^1) ||k./-2rXL))} I T(P5),F(P))N


where j = l-k+/+2r, \ = Xp^Qê^co21 and L is the least common multiple of

N = p a and J = p^+2y which is the level of G(Q) I eco"r. As already seen in the

proof of Lemma 2.1, the operator TL/N is nothing but T(p ). We want to relate

the value (F(P),H(G(Q) I eco-r6^.2rE) I T(p5))c to that of (4). We have

(5) (F(P)°,F(P)°)(F(P),H(G(Q) I e c o " ^ . ^ ) I T(p5))c

= (H(G(Q) | eco-^ .Ê) | T(p6),F(P))

| T(ps),F(P))

l/XL^.Ê} | T(p5),F(P)).

We need to compute G(Q) I eco"r || /XL. Under (P), we have (by (2.2))

(6)

= (L/J)//2W(cpQ(8)eco-r)(G(Q)c I e'

for a constant W((pQ®eco~r) with absolute value 1. Then (5) is equal to

(-l)/(L/J)//2W(9Q®eco-r)({G(Q)c I e^co^z/J)} ||flL5}E(§ tfj) I T(p5),F(P)),

where j = k-/-2r. Then recalling the formula (2.4b),

(7) (E'k./.2r,L(z,l-k+/+2r,^1) ||kXL)*

and the fact that

D(s,F(P),gc(pvz)) = ]T~=1 e W

= Xp(T(p))vp-vsD(s,F(P),gc),

we see from (4) that (5) is equal to

xr(r+l)r(/+r)L(/+r,?ip(S)(pQc<8)e"1cor).

Thus finally we get, if either %p is non-trivial or k = 2 and if (P) holds,

(8) Lp(?i<g>(pc)(P,Q,R)

= (-l)/Xp(T(N))^P(T(J))"1J(//2)+rN1"(k/2)W(Xp)"1W((pQ®eco-r)

r(r+l)r(/+r)L(/+rAp(g)(poc(g)£-1CQr)

(27c/)k+/+2r7C1-k(F(P)°,F(P)°)


We will show that the above formula holds also in Case II if either %p is

non-trivial or k = 2 and if (P) holds.

Now we consider the case where %p is trivial and k > 2, but we still remain in

Case I. We then need to compute

(H(G(Q) I eco-r8rk./-2rE) I T(p5),F(P)c I x)P

= p(k/2)-i(G(Q) | e c Q - ' I ^ L S ^ E ) | T(p5),F(P)c||kT)p

Q

x({(G(Q)c |e^coOCLz/Dl/TL^.^} I T(p5),F(P)c||kx)p

1-k+/+2rL-(k-2r)/2+/+17C"V(^2)'1r//2W((pQ®e^

x({(G(Q)c | B-lGf)£f)HxL^.Ê'k./.^LCza-k+Mr,^1)||xL} | T(p5),F(P)c||x)p.

By (2.6a), we have, for g = (G(Q)C | e"V)(L2yj)

W - 2 r j ^ ^

We then know, noting that N = p and k is even, that(H(G(Q) | eco-^ .Ê) | T(p5),F(P)c I <c)p = (-Dîlni)

xW((pQ(8)eco"r)r(/+r)r(r+l)X(T(L/J))L(/+r,^P(B)(pQc(8)e-1cor).

By (2) and (3), this shows, if k > 2 and %p is trivial, under (P), that

(9) Lpa<8Xpc)(P,Q,R) = (Ay

x ( 1 ykA y i ( 1 pk-2

\ 2x ( 1 y ( 1

\P(T(p))2 \P(T(p))2 (27r/)k+/+2r7C1-k(F(P)°,F(P)°)

In this case, for XP° : hk(SL2(Z);Z) -> Q given byF(P)°|T(n) = V(T(n))F(P)°,

we note that W(?ip°) = 1. We will see later that (9) also holds without any

change in Case II if %p is trivial and k > 2.

We now deal with Case II. As before, we first assume that either %p is trivial or

k = 2. We compute

(F(P)°,F(P)°)(F(P),H(G(Q) | eco-^^.'kÊ1) I T(p5))c

= (H(G(Q) | e c o - ^ . ^ E ' ) | T(p5),F(P))N

= (-l)/(L/J)//2W((pQ®eco-r)

x({(G(Q)c I e-VXLz/J)|/rLG(Q) I e Q - ^ ^ E 1 } I T(p6),F(P))N.


Now by (2.7b), solving k-l-m = k-/-r-l and 2m+2-k-/ = 2r+/-k+2, we havem = /+r and

^(T(L/J))(L/J)-/-T(r+l)r(/+r)L(/+r,^p®(pQc<8>e-1cor)

= 2k+/+2r/k+/JC/+2I+lL-r-(//2)N(k^)-lwap)

x({((G(Q)c Ie-lG?)0.z/T)\\tih)(^l;£+2E')} | TL/N,F(P)°)N.

Thus we have

(F(P)°,F(P)°)(F(P),H(G(Q) | eco-r

= (-l)'+r(27cO"k'/"2rJck"1N-(k/2)+1X.p(T(L/J))Jz/2+rW((pQ®eco-r)W(X,p)-1

1

which shows that the formula (8) again holds in Case II if %p is non-trivial or k =

2. We now assume that k > 2 and Xp is trivial. Thus N = p. We compute

(H(G(Q) | eco-r6t!tk+2E') I T(P8),F(P)C I T)N

= (H(G(Q) | eco-r8k;|tk+2E') I T(P8),F(P)C

= C({(G(Q)C|e-ÔCLz/Dl/XLS^-Ê'} |T(p5),F(P)c||kT)N,

for C = ( - ^ ' ( L / ^ ' V ^ ^ W C ^ e c o " 1 ) . By (2.7a), for g = (G(Q)C I e V

2(47c)-/-r(L/J)-/-rXp(T(L/J))r(r+l)r(/+r)L(/+rAp€>q)Qc<8)e-1O)r)

= (L/N)1-(k/2)(-4jt)k-1-/"T(/+2r+2-k)

where we have used (2.4a) and (2.5a) combined with

(E'k>L(z,s,^1) | |T L )* = (.l)k+/r(/-k+2r+2)-1(27i0/"k+2r+22L-(/"k+2r+2)/2El.

This shows that

(H(G(Q) | e c o ^ ^ E 1 ) I T(p5),F(P)c | x)N

= (-l)/+r(27c/)'k"/'2r7ck-1J(//2)+rW(9Q(8)eco-r)Xp(T(L/J))

xr(r+l)r(/+r)L(/+r,A,p(8)(pQc®e"1cor).

Thus again, in Case II, under (P), we know that (9) remains true if %p is trivial

and k > 2. This finishes the proof.

10.5. Relation to two variable p-adic Rankin products 339

§10.5. Relation to two variable p-adic Rankin productsIn this section, we clarify the relation between three variable p-adic Rankin productand the two variable one constructed in §7.4. To relate two p-adic L-functions, weneed to perform a supplementary computation to ease a little bit the condition (P) in§4. For that, we use the notation introduced in §4. We assume that G is ordinaryand \|/Q is non-trivial. Let ip be the trivial character modulo p. Then

G(Q) I tP = G(Q)-G(Q) I T(p)(pz) = G(Q)-(pQ(T(p))p1-/G(Q) 1,8

for 8 = I 0 j I. Then writing the exact level of G(Q) as Jo and assuming that

0 -1G(Q) is primitive and Jo is divisible by p, we see for J = Jop and x =

that

G(Q) Upjl/x = J1-(//2){G(Q)-(pQ(T(p))p1-/G(Q)

= p(//2)W((pQ)G(Q)c(pz)-(pQ(T(p))p-//2W((pQ)G(Q)c.

Performing the same computation as in §4 replacing the formula (4.6) by

G(Q) I tpl/TL = (L/J)//2{p1-(^W«pQ)G(Q)c I /8-pk-2J0(k/2)-1W((pQ)G(Q)c}(^),

we have

(1) Lp(X<g>(pc)(P,Q,R)

= (-l)/Xp(T(N))?ip(T(Jo))-1Jo(//2)+rN1-(k/2)W(V"1W(cpQ)

r(r+l)r(/+r)E(/+r)L(/+r,X,p(g)(poc)

(27cOk+/+2r7t1"k(F(P)°,F(P)0) '

where E(s) = 1--S-l \

p 1

>Q(T(p)Ap(T(p))j

Let us prove (1). We only deal with Case I because Case II can be done similarly.We have from (4.5) that

(F(P)°,F(P)0)(F(P),H(G(Q) I ipSkj.&E) I T(p8))c

. ^ } ITL/N,F(P))

Q Q

x([{G(Q)c(Lz/J)-(pQ(T(p))-1p'G(Q)c(Lz/Jo)} 1 ^ . , . ^ ] I TL/N,F(P)) ,

which is equal to (by (4.4) and (4.7))


xr(/+r)r(r+l)L(/+r,A,p®(pQc)

xL/+rp"/(Xp(T(L/J))(L/J)-/-r-(pQ(T(p))cp?lp(T(L/Jo))(L/Jo)-/-r)

==(-l)/(-l)r7Ck"1(27iO"k"/"2rW(9Q)W(^p)"1Jo//2+rN1"(k/2)>.p(^

x(l-(pQ(T(p))-c^p(T(p))-1p/+r-1)r(/+r)r(r+l)L(/+r,)ip®9Qc).

Then by (4.2), we have the formula (1).

We now assume

(Ql) ^ V Q ^ X P is primitive modulo L,

(Q2) %p is non-trivial and eco"r is trivial,

(Q3) \|/Q is non-trivial.

We write the level of the primitive form G(Q) as Jo, which is divisible by p by(Q3). We have defined in §7.4 the two variable Rankin product Lp(A.c®(p). Wehad the following evaluation formula (Theorem 7.4.2):

Lp(Xc®(p)(P,Q)

= L<k-%(T(N/L))(pQ(T(L/N)>

We recall the functional equation (Theorem 9.5.1):

(27c)"2k+1+/r(k-/)r(k-l)L(k-l,>,pC(8)(pQ) = Q

where

w = (-l)/W(V)W(9Q)G(%p-VQ)^p(T(L/Jo))L1-V2(pQ(T(L/N))cN/-(k/2).

This shows that

Lp(A.c<g>9)(P,Q) = (-l)k+/W(V)W((pQ)^p(T(L/Jo))L/-1Jo//2(pQ(T(L/N))cN/-(k/2)

(27tOK+'jc1"K(F(P)°)F(P)°)By (1), we know that

Lp(^®(pc)(P,Q,P0)

= E(/)Lp(Xc®cp)(P,Q),

10.5. Relation to two variable p-adic Rankin products 341

where E(Z) = (1 - 2 ) = (1- ° F ). Here we note that9Q(T(p)Ap(T(p)) V T ( ) )

j ^ j a n d E ( p Q ) = (1_M3M)^ T h u s writing^>(T(p))

Lp(?i<8>(pc)p0 for the element of the quotient field of I <§> QJ given by evaluating the

variable R of Lp( <8>(pc) at Po, we expect to have

Lp(k®(pc)p0 = E.Lp(A,c<g>(p) in F r a c ( l ® o J ) .

To see this, we note the following fact valid in a more general setting. Let A bean integral domain and !P b e a set of prime ideals with the property that

ripG2>P= {0}. Then the congruence a = (J m o d P for all P e fP ( a ,

p G A) implies the identity a = (3. If T is a set of infinitely many primes of

height 1 in A, this % satisfies the above property flpepP = {0} because for

Pi , ..., Pr G (P, P i O • -flPr is contained in the r-th power of the maximal ideal of

A. Thus we need to prove

L e m m a 1. Let

<P= {(P,Q) G * ( l ) x * ( J ) | p and Q satisfy (Ql-3) and k > / }.

Regard <2 as a set of prime ideals of I ® aJ. Then the intersection of all ideals in

2 is null.

Proof. Since I ®oJ is a finite extension of 0[[X,Y]] = A<§)0A, we mayassume that I = J = A. Then A<8)0A can be identified as a measure space ofWxW. Then, as seen in §3.6, the evaluation of power series O at (Pk,e,P/,ef)corresponds to the integration O h-> Jxke(x)y/e'(y)dji^(x,y). For fixed finiteorder characters e and e', the functions {xke(x)y/e'(y) I k > /} spans a densesubspace inside the space of continuous functions on WxW. In fact, by thevariable change (x,y) h-> (xy'^y) = (s,t), this set equals the set of primes withrespect to the variables (s,t) given by

{(Pjiee'-l,Put)lj>0 and / > 0 } ,

which corresponds to the space spanned by {s:'ee'"1(s)y/e'(y) I j > 0, / > 0},which is obviously dense. Since the measure is determined by the dense subspace,we get the lemma, because (P contains a set of the above type for a suitable choice

of e and e'.

The above lemma shows the desired identity Lp(^®(pc)p0 = E»Lp(Xc<8>(p),

because Lp(X®(pc)p0(P,Q) = (E»Lp(?ic®(p))(P,Q) implies


Lp(?i<g>(pc)p0 = E-Lp(Xc®(p) mod (P,Q)

for all (P,Q) e (P in the lemma inside the subring of Frac(l <8>oJ) *n which thedenominators of Lp(A,®(pc)p0 and Lp(Xc®<p) are inverted. Thus we have

Theorem 1. We have Lp(A,®(pc)Po = E*Lp(Xc<8>(p) in the field of fractions of

I <§>oJ. That is, we have as a function of X(\)xX(J)

where Lp(X®q>c) is the function in Theorem 4.1 and Lp(kc®(p) is the functiongiven in Theorem 7.4.2.

Note that E = (1-7* —) is contained in I <§>oJ and hence holomorphicMT(p))

everywhere. When (p = X, it has a zero along the diagonal line (i.e. E isdivisible by X-Y). On the other hand, the two variable L-function Lp( c®A,) hasa simple pole at the diagonal. Thus we know that Lp(A,®(pc)(P,Q,Po) does nothave any singularity at the diagonal line. Since we can compute the valueLp(A,®(pc)(P,Q,Po) even for non-primitive X?, we can remove the condition (P3)in the evaluation formula in Theorem 7.4.2:

Corollary 1. Let the notation be as in Theorem 10.4.1. Assume that G(Q) is

primitive and %P" V Q ^ primitive. Then for the two variable L-f unction

in Theorem 7.4.2, we have

S(P)Lp(?tc<x)(p)(P,Q)

{1 if Xp is primitive,

f, MT(p))c

otherwise.

Proof. We know the following formula by (1) when X? is primitive and byTheorem 4.1 (combined with the same computation which yields (1)) when X? isnot primitive:

S(P)Lpa®(pc)(P,Q,R) = (-l)/

r(r+l)r(/+r)E(/+r)L(/+r,?ip®(poc)

10.6. Concluding remarks 343

Here note that L(s,Xp®(pQC) = Ei(s)L(s,?ipo®(pQC) for the primitive algebrahomomorphism Xp° associated with X?, where

J 1 if X is p r i m i t i v e ,l ( S ) = |(l-?ip(T(p))-1(pQC(T(p))pk-1-s) otherwise.

We can also write, if X is primitive,

Then by the functional equation, we see that

(27t)-2k+1+/r(k-/)r(k-1 )L(k-1 ,(V)c®(pQ)

This shows that

S(P)Lp(?t<g>(pc)(P,Q,Po)

Then the result follows from Theorem 1.

§10.6. Concluding remarksIn this final section, we try to give some indications for further reading. It iscertainly affected by prejudice on the author's part and not at all exhaustive. In thisbook, we have discussed the theory of the critical values for L-functions of twoalgebraic groups GL(1) and GL(2) defined over the base field Q. AboutL-functions for more general algebraic groups and related subjects (especially "thetheory of automorphic representations"), the reader may take a look at some articlesin [CM] and [CT] and other articles and books quoted there. We have provedmany algebraicity results for the critical values of L-functions in the sense ofDeligne and Shimura [D]. The reader who is interested in such algebraicity resultsfor L-functions of GL(2) and more general algebraic groups should consultShimura's papers quoted in the references, especially [Shll, Shl2]. We shouldalso note a paper of Blasius [B] on this subject, which gives a proof of a conjectureof Deligne about CM periods and the critical values of Hecke L-functions of CMfields. For the philosophical and geometric background of such critical values andtheir transcendental factors, we refer to [D] and some expository articles in [RSS]and [CT]. There exists a largely conjectural theory (due to Beilinson and others)


for L-values at integers outside the critical range, although we have not touchedanything about that in the text. The reader may consult [RSS] and some expositorypapers in [CT] for these topics. We have also proved the existence of many p-adicL-functions out of the algebraicity results of these L-values. Such theory forGL(1) and GL(2) is now available for a large number of base fields, inparticular, totally real fields and fields containing CM fields. As for abelian p-adicL-functions for CM fields, the reader will find every detail in an excellent article ofKatz [K5], and for such p-adic L-functions with auxiliary conductor outside p, anexposition is given in [HT2]. As for p-adic L-functions for GL(2), somegeneralization of the results in Chapters 7 and 10 can be found in my paper [H8]and [Pa]. We have only discussed the automorphic side of the theory of p-adicL-functions in the text. As for the geometric side (i.e., the theory of motivicp-adic L-functions), see the articles of Coates and Greenberg in [CT] and articlesquoted there. After having constructed p-adic L-functions, it is natural to ask theirmeaning. A partial answer to this question is supplied by the so-called "mainconjectures" of an appropriate Iwasawa theory. As for the original Iwasawaconjecture proved by Mazur and Wiles ([MW] and [Wi2]), see the books ofWashington [Wa] and Lang [L]. For the motivic (or geometric) side of suchtheory, see [Mz2] and [CT], [RSS]. For the automorphic side, see [MzS], [MTT],[Wi2], [MT] and [HT1-3]. Finally, as for the A-adic Galois representationdescribed in §7.5, generalizations of the result in §7.5 to totally real base fields canbe found in [Wil] and [H10]. The existence of such large Galois representationscan be understood well from the perspective of deformation theory of Galoisrepresentations developed by Mazur [Mz3] (see also [HT3]).

Appendix: Summary of homology and cohomology theory

In this appendix, we give a summary of the theory of cohomology andhomology on complex and real manifolds, in particular, Riemann surfaces inorder to make the text self contained. However we will not give detailed proofsfor all of the material presented here. For sheaf cohomology, we refer to [Bd]and for group cohomology to [Bw] and for singular cohomology to [HiW] for de-tails.

Let X be a compact Riemann surface. We remove from X a set S of finitelymany points and write Y for the resulting open Riemann surface. We fix a basepoint y e Y and let F be the fundamental group 7Cy(Y). Let R be a commu-tative ring and for any left R[F]-module M, we define the cohomology groupHi(F,M)(=Ext?m(R,M)) as follows. Let

be an exact sequence of R[F]-modules, where on R, F acts trivially. When the

Fi are all R[F]-free modules, we call F an R[F]-free (acyclic) resolution of R.Then by applying the contravariant functor HomR[p](*,M), we have a complex

d\* 82*

0 -> HomR[pj(Fo, M) •-» HomR[rj(Fi, M) -» HomR[rj(F2, M) -»••• .

Then we define H°(r, M) = Ker(3i*) = HomR[r](R, M) = M r andHq(F,M)=Ker0q+i*)/Im0q*) for q > 0 .

The independence of the cohomology group of the choice of the free resolutionfollows from

Lemma A.I. If we have another resolution (which may not be R[F]-free),

then there exists a morphism (p : F -» F' of complexes extending the identity onR and (p is unique up to homotopy equivalence.

Proof. We can define inductively an R[F]-homomorphism <pj : Fj —»F j as

follows. For j = 0, taking a basis fa of Fo over R[F] and picking f a e F'o so

that e'(fa) = e(fa)» we define (po(fa) = fa and extend this map to Fo by

R[F]-linearity. Then by definition, (po satisfies £'°(po = £• Suppose we have

constructed (po» •••> <pj satisfying cpi_io3i = 3'i°cpi for all 0 < i < j . Then we see

from (Pj-i°3j = 3j°cpj that 0 = (pj_i°3j°3j+i = djO(pjo9j+1) and hence Im((pj°3j+i)

d Ker(3 j) = Im(3 j+i). Choosing a basis xa of Fj+i and picking ya e F j+i so

that 3'j+i(ya) = <Pj°9j+i(xa), we define (pj+i(xa) = ya and extend it to FJ+1 by

R[F]-linearity. Then we see that (pj°3j+i = d j+i°(Pj+i. Now we have a morphism

of the complex (p = ((pj). If there are two morphisms \|/ and \ j / ' : F —>F!

extending the identity map on R, we now show the existence of an

346 Appendix

RpTI-morphism 5j : Fj_i —> F'j such that \|/ 'j-xj/j = 8JO3J+3J+I°8J+I . Since

e'oOj/'o-xj/o) = id-id = 0,

Im(3'i) = Ker(e') z> Imty'o-Yo).Then picking any ya such that Vo(fa)"Vo(fa) = 3'i(ya), we define 5i(fa) = ya

for a basis fa of FQ and extend it by R[r]-linearity to FQ. Thus\|/'o-\(/o = 3'i°5i. Suppose that we have 8i for i = l , . . . , j . Then fromVj-i-Yj-i = 8j-i°3j-i+3 j°5j, we see that

^ j°(vrVj-5j°aj) = d j°v'j-3 jo\|/j-a jogjoaj= 31jo\|/lj-3

1jo\|fj-(\|f

tj.1-\|rj.1-8j.io3j.1)o3j = 0.

This shows Im(3j+i) = Ker(3j) => Im(\|/j-\|/j-8jo3j). Taking a basis xa of Fj, we

can find y a e F'j+i such that 3 j+i(ya) = (Yj-Vj-8j°3j)(xa), and we define

8j+i(xa) = ya and extend it to Fj by R[F]-linearity. Then we have

V'j-Vj = 8jo3j+3fj+io8j+i.

Thus \|/' is homotopy equivalent to \|/.

Supposing F1 is also R[F]-free and interchanging the role of F and F , we finda morphism 9 ' : F —> F extending the identity on R. Making F = F' andreplacing \\f by the identity map and \|/' by (p°(p\ we know that cp induces anisomorphism between the cohomology groups of F and F . Thus the coho-mology group is independent of the choice of the resolution.

We define a standard R[F]-free resolution of R as follows. Let Fq be the tensorproduct of q+1 copies of R[F] over R and consider it as an R[F]-niodule viamultiplication by R[F] of the first factor. Then Fq is a free R[F]-module witha basis {[yi,...,yq] = l®Yi®---®yq| (yi,..., yq) e H } . Then we define3q : Fq -» Fq_i by 3i[y] = y-1 and for q > 1

3[Yi,...,Yq] =Yi[Y2, —.Yq]+XjLi (-^[Yi'—»YjYj+i.—>Yq]+(-l)q[Yi,....Yq-i]

and extend it R[r]-linearly on Fq. One can compute that 3q_i°3q = 0. Defining

e : Fo = R[F] —> R by e(LySLyy) = Zyay, we see also that eod\ = 0. Noting that

YtYl* "-> Yq] f° r (Y»Yi»--»Yq) G Tq+1 gives an R-basis of Fq, we define an

R-linear map Dq : Fq -> Fq+i by Dq(Y[Y2, ..., Yq]) = [%Yi. .... Yq]- T h e n w e s e e

easily that

3qDq-i+Dq3q_i = id (q > 1) and 3iDo+e = id.This shows that F is an R|T]-free acyclic resolution of R. Let C = C(TM) bethe space of functions on F1 into M and put C°(r,M) = M. Note that anyR[F]-linear map from Fq to M is determined by its values on the standard basis[Yi» •••> Yq] a n d hence HomR[rj(Fq, M) = Cq(r,M). Then the differential map

Appendix 347

3:C 1 ->C 1 + 1 induced by d on F is given by 3u(y) = (y-l)u for U G M if

i = 0, and if i > 0,

Then H ^ M ) = Z^FJv iyB^M) where Z^FJVl) = Ker(3 : C1 -> Ci+1) andB ^ M ) = Im(3 : C1"1 -> C1). Thus we again get

H°(r,M) = M r = {x G M | yx = x for all y e F}.Any element in ZÎ^M) (resp. B^FjM)) is called an i-cocycle (resp. ani-coboundary). In particular a 1-cocycle u : F - » M is a map satisfyingu(y8) = u(y) + yu(5) for all y,8 G F and a 1-coboundary u is a map of the formu(y) = (y-l)x for any X G M independent of y. This shows that u ( l )=0 ,u(y1) = -yûCy), uCySy"1) = u(y) + 711(8) - ySyûfy) for cocycle u, and

H ^ M ) = Hom(F,M) = Hom(Fab,M) if F acts trivially on M.

Let H be the universal covering space of Y and n : H -> Y be the projection.For each SG S, we consider ns e F which corresponds to the path starting fromy turning around the point s in the counterclockwise direction, and returning toy. Let r s = {7ts

me r | m e Z}. We consider the set of all conjugates of 7CS forall s G S in F, which will be denoted by P. We define the paraboliccohomology group by

H|>(r,M) = ZJ^(T,M)/B1(r,M), H|(F,M) = Z2(F,M)/B^(F,M),

where zJ>(T,M) = {u e Zl(TM) I U(TC) G (TC-1)M for all KG P},

B|(F,M) = {3u I UG Cl(JTM) with U(TC) G (TC-1)M for all n G P}.Now the restriction of each 1-cocycle u from F to F^ = {7Cm|7CG Z} (n G P)yields a morphism res^: H^FjM) —> H^r^,]^). Therefore if u is a 1-cocycle ofTn, then

u(7Cr) = (l+7C+7i;2+ ••• +7Cr"1)u(7c) andu(7i"r) = -nMn1) = -7C"r(l+7C+7C2+ ••• +7ir4)u(7c) for r > 0 .

Thus the cocycle u is determined by the value U(TC) at the generator 7C. IfU(K) = (7i-l)x for some x G M, then

u(7Cr) = (1+7C+TC2+ ••• +7ir"1)(7C-l)x = (TIM)X and

U(7fr) = -7fr(l+7C+7t2+ — +7Cr-1)(7l-l)x = (7C-r-l)x.Thus H ^ F S J M ) = M/(TC-1)M and we now know the exact sequence

0 -> HJ>(F,M) -> H 1 ^ ) -> line? H^F^M),where the last map is given by nTires^. Actually, if U(TI)G (TC-1)M, then for any

348 Appendix

!u(Y) = u(y)

V(Y)Thus, in fact, the conditions defining the parabolic cohomology group are finitelymany, and we have

0 -> Hj,(r,M) -> H ^ M ) -> eserNpH1^,]^).

We now consider another description of Hp(r,M) by using a simplicial com-plex. Let Yo be an open Riemann surface obtained from X by excluding asmall disk around each point s e S without overlaps. Let us take the pull backHo of Yo to H, which is a simply connected open subset of H. We make asimplicial complex K with the underlying space Ho such that

(Tl) Every element of T induces a simplicial map of K onto itself,

(T2) for each cusp s e S, the boundary of the excluded disk is the image of a1-chain ts of K,

(T3) there exists a fundamental domain of Oo in 9J§ whose closure consistsof finitely many simplices in K.

We can construct such a complex by first taking a fundamental domain of <X>oand then making a finite simplicial complex £ with the property (T2) of the clo-sure of the fundamental domain and finally shifting this simplex to cover all Hoby elements of P. We consider the chain complex (Ai,9,a) over R constructedfrom K; thus, Ai is a free R-module generated by i-chains of K. We have anexact sequence (by the simply-connectedness of Ho)

0 -> A2 —^-» Ai —^-> Ao —2-» R -> 0,

where 3 is the usual boundary map and a(Xz c(z)z) = Szc(z) e R. Wesometimes identify S with the set of generators {TIS} of F s for s e S. Wedefine Ai(M) = HomR[rj(Ai,M). Thus we have another complex

0 -> HomR[r](R,M) —-2-* A0(M) —^-> Ai(M) — ^ A2(M) -» 0,

which may not be exact. DefineZ^K.M) = Ker(3 : Ai(M) -> Ai+i(M))

andBi(K,M) = im(a:Ai.i(M)-Âi(M)) and ffCK.M) = Zi(K,M)/Bi(K,M).

We further define

Z|>(K,M) = {ue ZX(K,M) I u(ts) e (TTS-1)M for all S G S } ,

B^(K,M) = {3u | u(ts)e (7Cs-l)M for all S G S ) ,

HJp(K,M) = Zj>(K,M)/B1(K,M), Hp(K,M) =

Proposition 1 (Shimura [Sh, Prop.8.1]). There is a canonical isomorphism

Appendix 349

Hi(K,M)=Hi(T,M),where "*" indicates P or f/ze wswa/ cohomology.

Proof. We shall give two proofs of this fact. By construction,

A : 0 -> A2 — —> Ai —^-> Ao ——> R -^ 0 gives an R[F]-free resolution of

R and thus we can compute H^FJM) by using this resolution, which yields the

above isomorphism for H1. For each n e P with n = y sY"1, y(ts) is a simplex

of K. Identify R with the universal covering space of the image of y(ts) in Oo

and let i^ : R - » H be the induced map. Then the closure y(ts) gives a

triangulation of a fundamental domain of F XR in i^R). Thus we can define

Ai(rc) to be a free RIT^-module generated by i-simplices in y(ts) and we have

an RfFJ-free resolution:

A(7i): 0 -> AI(TC) —^-> A0(7i) —-2-* R -> 0.We have a natural inclusion i^: A(%) -» A induced by i^. Thus we havei : ©TCepAfa) -> A. We have a natural action of F on ©^pAiCTi) which justpermutes its simplices. Writing F={F q } (resp. F(TC) = {Fq(7c)}) for thestandard R[F]-free (resp. RfFJ-free) resolution of R, we have a commutativediagram,

i iA >F,

where the vertical maps are the natural inclusions and the horizontal maps are themaps extending the identity on R as constructed in the beginning of this section.Then applying the functor HomR[rj(*, M) to this diagram and computing the

cohomology, we have another commutative diagram,H1(F,M)ês esH1(F7 r s ,M)

i iH 1 ( K , M ) - ^ © S G S H 1 ( K ( T C S ) , M ) ,

where Hq(K(7i),M) is the cohomology group of HomR[r7I](A(7i), M). Since the

vertical arrows are isomorphisms and the parabolic cohomology group is defined

by the kernel of the horizontal maps, we see that Hp(K,M) = Hp(F,M).

We now give another (more explicit) proof given in [Sh, §8.1]. We compute thecohomology group H^FjM) by the homogeneous chain complex; that is, wedefine (Q,3,a) as follows. For i > 0, Q is the R-free module generated by allthe ordered sets (yo,yi, ..., Ji) of i+1 elements of F, the differential3 : Q -» Q_i is defined by

3(Yo,yi,..., Yi)= X j = 0

350 Appendix

the augmentation a is given by a(Eiri(Yi)) = Siri and F acts on Q by Y(Yo>Yi>

Yi). T h e n

gives an R[F]-free resolution of R. In fact, we can easily check that this com-plex is isomorphic to the standard one F by the R[F] -linear isomorphism i given

by i((Yo,Yi, •••> Yq)) = Yo[Yo"1YiJYi"1Y2, • ••, Yq-fVqL Thus by puttingCi(M) = HomR[T](Ci,M), the cohomology group of the complex

0 —2-* Co(M) —^-> Ci(M) —d—> ...gives the cohomology group HÎ^M). As already remarked, we can define anisomorphism of complex (C(M),3) to (Q(M),3) by defining u' e Ci(M) out ofu e C(M) by

u'((Yo, Yi> ••., Yi)) = YouCYo'Vi, Y f S , . . . , (Yi-i)"SO.The isomorphism of H^FjM) onto H^KjM) can be constructed as follows. Wetake a finite set of i-simplices Si in K so that any simplex in K can be writtenuniquely as Y(S) f°r YG T. In particular, we can include ts in S\ and qs inSo if 3ts = qs-7Cs(qs). Then we define a map fo : Ao -» Co by fo(Y(s)) = (Y) foreach 0-simplex s in So, and then a°fo = a on Ao. Thus a(fo(3s)) = 0 for alls G Si. Because of the exactness of Ci —> Co -> R -> 0, we can find fi(s) inAi so that 3fi(s) = fo(3s). We may assume fi(ts) = (l,7ts). Then we define forany ye F, fi(Ys) = 7fi(s) and extend this map R-linearly to the whole of Ai.Since we have defined fi so that if 3s = 0 for S E K, then 3fi(s) = 0 and wehave 3fi(3s) = 0 for se S2. Now we can find f2(s) e C2 so that 3f2(s) = fi(3s)by the exactness of C2 -> Ci -> Co. Then by the R[F]-linearity, we extend f2to an R[F]-linear map of A2 into C2. The morphism f=(fo,fi,f2) from(Ai,3,a) to (Ci,3,a) we have constructed satisfies

(i) a°f0 = a, f°3 = 3°f and f°Y = Y°f f(>r Y G r»(ii) fi(ts) = (l,7Cs) for se S.By interchanging the role of (Aud) and (Q,3), we can similarly constructg : (Q,3,a) —> (Ai,3,a) satisfying(i') a°go = a, g°3 = 3°g and g°Y = Y°g f°r Y G r ,

(ii') gi((l>7is)) = ts + (7Cs-l)bs with 1-chain bs such that 3bs = qo-qs forafixed 0-simplex qo in So independent of s e S.

In fact, first fixing a 0-simplex qo, we define go((Y)) = Y(qo)- Then, by defini-tion, a°go = a. Thus we can find gi((Y>8)) such that

3gi((Y.8» = goO(Y>8)) = go((5)-(Y)) = 5(qo)-Y(qo).Then we can extend gi to the whole space Ci by R[F]-linearity. In particular

Note that 3ts = qs-7Cs(qs)- Thus by taking bs e A\ so

Appendix 351

that 3bs = qs-qo (this is possible since a(qs-qo) = 0). Then we have3(ts+0rs-l)bs) = qo-TCs(qo), and we may define gi((l,7Cs)) = ts+(7Cs-l)bs, becausegi((l>TCs)) can be any element x in Ai such that 3x = go(3(l,7Cs)). The maps fand g induce morphisms

f* : HJ>(T,M) -> HJ>(K,M) and g* : H1>(K,M) -> H ^ M ) .In fact, if u : F -> M is a 1-cocycle, the corresponding homogeneous chain u' isgiven by u'((l,y)) = u(y). In particular if U(TCS) = (ns-l)x, then f*u'(ts) = u'°f(ts) =uf((l,fts)) = (7is-l)x, and thus f* takes parabolic cohomology classes to paraboliccohomology classes. Similarly, if u e HomRÂ^M) with u(ts) e (TCS-1)M,

then

g*u((l,7C,)) = u(ts + (7Cs-Dbs) e (7CS-1)M.Since (Q,3) and (Aj,3) are R[F]-free resolutions, and f and g induce theidentity on R, f°g and g°f are homotopy equivalent to the identity. That is,there are R[F]-linear maps U : Q —» Ci+i and V : Ai —> Ai+i such thatfog. id = 3U+U3 and g°f - id = 3V+V9. The map U can be defined as follows.Since f(g((y))) = (y), we have fo°go = id, and we simply put Uo = UI Co = 0- Wehave

3(f(g(d,y)))-(i,y)) = f(gOd,y)))-3(i,y) = obecause fo°go = id. Thus we can find U((l,y)) so that

Then we extend U by R|T]-linearity to Ci. By definition, we havefi°gi - id = 3U+U3. Similarly, we see that

a(f2(g2((l,y,5)))-(l,y,5)-UO(l,y,5)))= fi(giO(l,y,8)))-3(l,y,8)-3UO(l,y,8)) = 0

because 3U((5,y)) = f(g((5,y)))-(8,y) and 3(l,y,8) = (y,8)-(l,8)+(l,y). Thus wecan find U(l,y,8) so that

3U(l,y,8) = f2(g2((l,y,8)))-(l,y,8)-UO(l,y,8)).

Then we have U satisfying f°g - id = 3U+U3. In this way, we continue to

define U inductively (actually, it is sufficient to have U defined as above be-

cause Hp(F,M) = 0 if i > 2). As for V, we proceed as follows. Since

a(g(f(s))-s) = 0 by definition, we can find V : Ao -> Ai so that3V(s) = g(f(s))-s for all s e Ao.

Then we consider 3(gi(fi(s))-s-V(3s)) = go(fo(3s))-3s-3V(3s) = 0 by the de-finition of V. Now we can define V(s) for s e Ai so that 3V(s) =gi(fi(s))-s-V(3s) and continue to define V inductively. Then for each 1-cocycleu e Zi(M), we see that u°f°g-u = u3U+uU3 = 9uU e Bi(M). Thus g*f* = idand similarly f*g* = id on the cohomology groups. As already seen, theypreserve parabolic classes and hence induce isomorphisms on paraboliccohomology groups.

352 Appendix

Let So be a subset of S and T be the disjoint union of the image of ts in Y

for s G So. Let Kj be the subcomplex of K generated by all translations of tsby T. We put KT = K/KT. Consider the free R-module A?1 (resp. AT,i)

generated by the i-simplex of KT (resp. KT). Then we write H§ (F, M) for

the cohomology group of HomR[r](A^,M). When S = So, we write Hq(F,M)

for

Proposition 2 (Boundary exact sequence). We have a long exact sequence

0 -> H°SQ (T, M) -> H°(T, M) -> 0SGsoHO(r7v M) -> H ^ (F, M)

-> H^r, M) -»esesoH 1 ^ , M) -> njo (r, M) -» H2(r, M) -> o.

/ * particular, B°SQ (F, M) = 0 z/ So * 0 , and H* (r, M) = H | ( F , M).

Proof. We need the following well known fact:(1) If 0 ^ A - » B — > C — » 0 is an exact sequence of complexes, then we

have a long exact sequence••—> Hq(A) -> KP(B) -> Hq(C) -> Hq+1(A) -> HP+1(B) ->•••.

This is checked by applying the snake lemma to the commutative diagram:Ap/9p-i(Ap.i) -> Bp/ap.i(Bp_i) -> Cp/ap.i(Cp.i) -> 0 (exact)

id id id0 -> Ker(ap+i | A p + 1) -> Ker(3p+i | Bp+1) -> Ker(3p+11 Cp+1) (exact).

We have an exact sequence 0 —» Ay —> A —> A —> 0. Since A T is also

R[F]-free, this sequence splits as R[F]-module, and we have another exact se-

quence

0 —> HomR[r](AT, M) -> HoniR[r](A, M) -^ HomR[r](AT, M) -^ 0.

Then applying (1), we get the long exact sequence. The vanishing of H s (F, M)

(when So ?* 0 ) follows from the injectivity of

M r = H°(F, M) -+ S s e S o H 0 ^ , M) = 0sesoMr7ts.

The last assertion follows directly from the definition .of Hp(F, M).

We now define the notion of sheaves on a smooth manifold Z of real dimensionn. Here the word "smooth" means that Z is of C~-class. Let O(Z) be the cat-egory of all open subsets of Z; that is, objects of O(Z) are open subsets of Zand Homo(Z)(U,V) is either the inclusion map U —> V or empty according asU is contained in V or not. A presheaf F on Z is a contravariant functor onO(Z) having values in the category of abelian groups. Thus for each open set Uin Z, F(U) is an abelian group and if U 3 V, we have a natural restriction mapresu / v : F(U) -> F(V) satisfying resu/u = id and resv/wcresu/v = resu / w if U z>V D W . A presheaf is called a sheaf if the following axiom is satisfied:

Appendix 353

(S) If for a given open covering U = UiVi, s i e F(Vi) satisfies

resVi/Vinv.(si) = resVj/VinVj(Sj) for all i and j with ViflVj * 0 , we have a

unique element s e F(U) such that resu/Vi(s) = S{.

If U = U[=1 Vi is an open covering, adding Vo = U to this covering, the con-

dition (S) implies that s e F(U) is uniquely determined by resu/Vi(s) for all

i = 1, ..., r. A morphism (]) of a sheaf F into another G is defined to be a

morphism of contravariant functors. That is, for each open set U, we have a

morphism (|)(U) : F(U) -^ G(U), and for every open subset V of U, the

following diagram is commutative:4>(U)

F(U) -> G(U)

4/ resy/y i> reSy^y

F(V) -> G(V).4>(V)

If U is an open subset of Z, then O(U) is a subcategory of O(Z). Thus we can

restrict the sheaf to O(U). The restriction of F to U will be written as F lu -

If 71 : T —> Z is a surjective morphism of smooth manifolds, we can define a

sheaf associated with T by

T(U) = {s : U —> T I s is continuous and izos = id on U}.

It is easy to verify the condition (S) for the usual restriction map

res u / v ( s ) = s | v if U D V. Let A be any abelian group with the discrete

topology. We consider T = ZxA with the product topology. Then if an open set

U in Z is connected, any continuous section on U of K : T —> Z is a constant

function, and hence T(U) = A. This sheaf T is called the constant sheaf A. A

sheaf on Z is called locally constant if for each point z e Z, we can find an

open neighborhood U of z such that F | u is a constant sheaf. Returning to the

situation in Lemma 1, we can give plenty of examples of such locally constant

sheaves. For any F-module M, we can define T = F \HxM letting F act on

H x M by y(z,m) = (7(z),ym). We put the discrete topology on M and put the

quotient topology on T. Then the projection n : T —> Y = F \H gives a sheaf T,

which we write ML For a small open set Uo of H such that

yUoflUo * 0 <=> y= 1, writing U for the image of Uo in Y, we see easily that

7C"1(U) = UxM and hence M l u is a constant sheaf M. For every point Z E Y ,

we can always find such a neighborhood U, and hence M is a locally constant

sheaf.

Now we introduce the Cech cohomology group of a presheaf F. For an open

covering Zl = {Ua}aGi* Z = UaeiUa and a = (ao,. . . ,ar) e Ir+1, we write

354 Appendix

Ua = UaorV*riU(Xr. Then we define C ^ F ) to be a module of functions

s : Ir+1 -> IJaGiF(Ua) satisfying s(a) e F(Ua). Then we define

by, for a = (ao,...,ar+i)e Ir+2, (3s)(a) = Xj=o a(j)

where a® = {oq e a | i * j} and we understand F(0) = 0. For example,

d(s)aPy = sPy I Uap rSay I UapY+Sap I Uapy w n e n r = hand when 3(s)ap = sp I Uap-sa I Uap,

3(3s)apy = dspy I Uapr3say I Uapyf 3sa(51 Uapy

= Sy I UaprSp I Uapy- { Sy I U a p r Sa I Ua(3y} +Sp I UapySa | Uapy = 0.

This shows d2°d\ = 0. We can similarly check 3r+io3r = 0 for general r. We

write the cohomology group Hq(C(^;F)) as Hq(Z;1i;F):

Hq(Z;^;F) =Ker(3r)/Im(ar4).

Another open covering 1 = {Vp}p€ j is called a refinement of U if there exists amap p : J —> I such that UpQ 3 Vj for all j . In this case, we write U > V, Wedefine U and V to be equivalent if U > V and V> U Let C be the set ofequivalence classes of all coverings of Z (C is a set because the set of allcoverings is a subset of the power set of the power set of Z). The partial order-ing ">" induces a filtered ordering on C In fact, if Zl and <V are two cover-ings, then (uxrU= {UiHVjJêixj is a common refinement. Suppose^- {Vp}pej is a refinement of Zl with the map p : J —» I. Then p induces amap pr+1 : Jr+1 -* Ir+1. For each s e C ^ F ) , we can define p*s e Cr(^;F) asfollows:

p*s(p) =resupr+i(P)/Vps(pr+1(P)).

It is obvious that p* commutes with the differentials and therefore induces amorphism of cohomology groups:

Lemma 2. The morphism p* does not depend on the choice of the map

p: J -> I.

Sketch of a proof. Let I be another map from J into I having the same prop-erty as p. We show that p* and I* are homotopy equivalent. Let5 : CqCU;F) -» Cq+1(^,F) be a map defined by

(8S)(P) =

Then by a direct computation, we get i-p = 63+38.

By Lemma 2, if U> V, we have the canonical homomorphism

Appendix 355

) : Hq(Z;?i;F)

If 1/ and V are equivalent, by the uniqueness i(Zl,ty°i('V,ZL) and

i(y,<li)oi(?i,'^ have to be the identity, because i(11,tL) is the identity. Thus

Hq(Z;Ti;F) is determined by the class of U in C. Similarly, if U > V> <W, then

1(^,^)01(11,^ = 1(11,14/). Since C is filtered, we have an injective system

{ Hq(Z; ?i;F),iCU,<K)} ^ c. We then define

lim11

Giving a 0-cocycle s in C°(t/;F) is tantamount to giving Si G F(Ui) for each

i G I such that resui/uij(si) = resuj/uij(Sj). Thus if F is a sheaf, we have a unique

s G F(Z) such that resz/ui(s) = Si. This shows

(2) H°(Z; <U;F) = F(Z) if F is a sheaf.

Proposition 3. Suppose that Z is simply connected and î={Ui}iei is a

covering of Z with a countable I such that (*) Ua for each a G Ir+1 w ezY/zer

empty or simply connected for all r G N. //* resu/v is surjective for all V c U

with V connected (this assumption holds if F is constant), then Hq(Z;£/;F) = 0

Proof. We fix (5e I and write U = Up. We consider F | U andUI u = {UnUa I ae 1}. Then we can define, for the fixed p G I

8 r : CT+l(<U I F;F I u) -» Cezi I u;F I u) by 5s(a) = s(pUa) G F(UnUa) = F(Ua).

Then 38s(a) = I[=0(-l)iresuaG)/^^ and

59s(a) = 3s(pUa) = sCaVl^-lVresu^yuasCpLJa^^ = s(a)-35s(a ). Thusid = 83+38. So if s is an r-cocycle (r > 0) in Cr+1(1/;F), we can find t suchthat s I u = 3t. The cochain t is uniquely determined by s moduloSCC'ZilujFlu)- By the same argument applied to U' = Up with UflU1 * 0 , wefind t' so that 3t' = s | u1 on U'. Modifying t1 by an element in 3Cr(*Zi | u;F I \j),we may assume that resu/unu't = resuyunu't', because the restriction map3Cr(^lu;Flu)->9Cr(^/lunui;Flunu1) is surjective. Thus t extends to UJU1.By continuing this process, by the simple connectedness of Z, we can extend tto UiUi = Z. This shows the result.

For a presheaf F, we study the presheaf F # : U H H°(U,F | U). The restrictionmap resu/v of F induces that of F#. By (2), if F is a sheaf, then F* = F. Let^ = (Uihei be an open covering of U. To each s e F(U), assigning the0-cocycle s(i) = resu/u/s) e C°(T/;F), we have acohomology class [s] in F#(U).Thus we have a natural morphism of presheaves i : F —> F#. If s G F CU) and if

356 Appendix

resu/Ui(s) = 0 for all i, s itself vanishes because the image of s in

H0(U;^;F|u) is 0. Thus se F*(U) is determined by its local data. Let si for

all i G I be sections in F*(Ui) satisfying resu/Ui (Si) = resu/Ui-Csj) or a ^ * anc^

j . Then we take a sufficiently fine open covering U\ = {l^lke J of Ui and

choose a 0-cocycle s1 e C°(Zli;F\u) representing sj. Then writing p : IxJ —> I

for the projection, we regard <]/= {l4)(i,k)eixJ as a refinement of 11. Then the

cochain: (i,k) H> s\k) is a 0-cocycle in C°(V ;F I u)- Writing s for the

cohomology class of this cocycle, we have plainly S[ = resu/Ui(s) for all i e I.

Thus F* is a sheaf. If $ : F -» G is a morphism of presheaves and if G is a

sheaf, <|> induces 0* : F* —> G# = G. It is obvious that $ = ( ^ i and (|>* is

characterized by this property because any s e F#(U) can be described by localsections. Thus F# satisfies the following universal property:

(3) for each morphism 0 of presheaves from F into a sheaf G, there exists aunique <|>* : F* —» G satisfying 0 = (Jî.

By this universality, F* is uniquely determined by F (up to isomorphisms). Wecall F# the sheaf generated by F.

We say a sheaf (resp. a presheaf) is a sheaf (resp. a presheaf) of R-modules for acommutative ring R if F(U) is an R-module for every open set U andresu/v • F(U) —» F(V) is an R-linear map if U z> V. For any presheaf F ofR-modules, the sheaf F# generated by F is naturally a sheaf of R-modules. IfF and G are two sheaves of R-modules, then we write F ® R G for the sheafgenerated by the presheaf: U H F(U)<E>RG(U). The map (x,y) i-» x®ycomposed with the canonical map i is again denoted by the same symbol:(x,y) h-> x®y for (x,y) e F(U)xG(U). It is easy to verify the following universalproperty (see the description after Corollary 1.1.1):

(4) If there is a morphism of sheaves of R-modules (|) : FxG —» H such that<KU) is R-bilinear and <|>(U)(x,A,y) = (|>(U)(Ax,y) for all XeR, then thereexists a unique morphism of sheaves of R-modules <(>* : F®RG —> H such

that <|>*(U)(x<g>y) = <|)(x,y).

If F is a subsheaf of a sheaf G, we can define a presheaf U \-> F(U)\G(U). Wewrite FXG for the sheaf generated by the above presheaf. We have a naturalmorphism G -> G/H. On the other hand, it is plain from the sheaf axiom (S)that, if (p : F —> G is a morphism of sheaves,

Ker(cp): U -> Ker(cp(U)) is again a sheaf.

Appendix 357

A sequence of morphisms of sheaves F — — > G — ^ H is called exact ifIm(a)# = Ker(P), where Im(a)# is the sheaf generated by the presheafU H> Im(a(U)). For any presheaf F and x e Z, we define the stalk Fx at x byFx = limF(U), where the transition map is given by resu / v , and the order on the

set of open sets containing x is given by the inclusion relation. Then it is easy tocheck from the definition that

(5a) F#x = Fx for all X G Z , and F —> G -» H is exact as sheaves if and only if

IlxezFx -> IIxezGx -» IlxezHx is exact as abelian groups.

In other words,

(5b) if a : F —»G is a surjective morphism of sheaves, then for each g e G(Z),

there exists an open covering {Ui)iei with fi e F(Ui) such that

oc(fi) = resz/Ui(g) for all i.

For each section f e F(U), we define Supp(f) to be the closed set in U definedby ( x e U | fx * 0}, where fx is the natural image of f in Fx.

Now we return to the original situation: X is a compact Riemann surface,Y = X-S and F is the fundamental group of Y. Let M be a F-module and Mdenote the locally constant sheaf on Y associated to M.

Theorem 1. Let U = {Ui}iGi be a covering of Y with a countable index set Isuch that (i) for each Ui, there exists a simply connected open subset Ui* in Hsuch that the projection n : H —» F\H induces an isomorphism Ui* = Ui, and(ii)for all r, Ua* = Uao*lT • -flU^* is either simply connected or empty for alla G F+1. Then there is a canonical isomorphism Hq(F,M) = Hq(Y;^;M).

Proof. We consider the open covering 11* = {y(Ui*)}yeT,iei °f H. By (i) and

(ii), if U a * 0 for a = (ao,.-.,ar+i) e Ir+1, then there exists a unique

(Yo,...,yr)e Tr+1 so that Ua* = yoUao*n-"nyrUar* * 0 . We put Ua* = 0 if

U a = 0 . Then yUa* for y e F is either empty or simply connected. Thus if

yUa* * 0 , then F(yUa*) = M. We consider the subset Ir in Ir+1 consisting of

a with non-empty Ua. Let R[F][Ir] be the formal free module generated over

R[F] by elements of Ir, and write [y,a] for the element (y,a) (ye F and a e

Ir)in R[F][U-Then we see that {[y.alJ^r.ae^ is a basis of R[F][IJ over R,

and we haveHomR(R[F][Ir],M) = Cr(^*;M) by c> i—> s

where s(y,cc) = (J)([y,a]) e F(yUa*) = M. By this isomorphism, C(ZI*;M) has anatural structure as R[F]-module. We define an R-linear map

358 Appendix

by 3[y,a] = ^=0

where a® = (ao,...,ocj_i,aj+i,...,ar). Obviously 3r is R[r]-linear. Thus wehave a complex

R[where

Then C(^*;M) = HomR(R|T][r|, M) as complexes. Since H is simply con-nected, by Proposition 3, C(£/*;M) is exact and thus R[r][I] is an R[rj-freeresolution of R. It is obvious that

HomR[r](R[r][U], M) = C(*Z;M) as complexes.Thus by Lemma 1, we get the desired isomorphism.

Since we can take a simply connected polygon as the fundamental domain of Y,for any covering <V of Y, we can find a refinement U satisfying the assumptionof Theorem 1. Thus we see the following

Corollary 1. Under the same notation as in Theorem 1, we have a canonical iso-

morphism Hq(r,M) = Hq(Y,M).

Exercise 1. (i) Let U be an open subset in Z and suppose U = U ^ U i is a

disjoint union of connected open sets Ui. Show that H°(U,C) = Ch.

(ii) Show Hr(R,C) =0 if r > 0./•••\ ot. TTr/r»/r* #-<\ f C if r = 0 and 1,Cm) show Hr(R/z,o={ 0 otherwise;

Let F be a locally constant sheaf on Z having values in the category of finitedimensional vector spaces over C. We consider the sheaf $tpT (on Z) ofsmooth differential forms of degree r with values in F. Thus .%r(U) is thespace of C°°-r-forms defined on the open set U with values in F(U); so,-#Fr = -#cr®cF in the sense of the tensor product of sheaves (4). Since the exte-rior derivation d is defined locally on J%c and locally .%r(U) = J3cr®F(U), dis well defined on %$ and we have a resolution of the sheaf F:

The above sequence is exact in the sense of sheaves (5a,b) by Poincare's lemma,which tells us the validity of (5b) for dq : ^tpq ~> ^q+i = Ker(dq+i) on everysimply connected open set. Let U - {Ui}i€i be an open covering of Z. Let{(j)j}je j be a partition of unity subordinate to the covering U Thus there is amap a : J —> I such that ty is a smooth function on Z with Ua(j) => Supp((|)j),all but a finite number of (|)j vanish at each point x € Z and Xj<t>j(x) = 1 fc>r all

Appendix 359

x e Z. Such a partition of unity is known to exist. Then we put for all cocycles

s e C(U;J^P), s'(P) = £j(|>js(o(j)UP) for p e Ir. Then

k=0 j j k=0

Since 3s(aG)LJa) = s(a)-Ifc=0(-l)Va(j)Ua(k)) = 0, we see that

This shows that

(6) Hq(Z; <U\fts?) = 0 for every ZL

Let H^R(Z, F) be the cohomology group of the complex

MZ): 0 -* ^°(Z)Û V(Z)-^ F2 (Z) -Û- .

The group H^R(Z, F) is called the de Rham cohomology group. Then we have

Theorem 2. If the covering U of Z satisfies the condition of Proposition 3,

then there is a natural isomorphism H£R(Z, F) = Hq(Z; <U;%EP).

Proof. For any sheaf, it is easy to see that the presheaf:

U H> C(1l | u;F I u) = 0 q C q ( ft I u;F I u)

is actually a sheaf of complexes, where U \ u = {UiflU l i e I} . We still write this

sheaf of complexes as C(ft;F). From the exact sequence:

we get another exact sequence of sheaves of complexes:

(*) 0 -> C(ft;^) -> C(ft;^pq) ~> C(ft ;Vi) ^ 0.

In fact, for each simply connected open set U, Poincare's lemma tells us thesurjectivity of d : .%q(U) —» 2^+i(U). Since every non-empty Ua (a e f+1) issimply connected, (*) is not only exact as sheaves but also exact as complex ofC-vector-spaces. Then we have the long exact sequence (1) attached to (*):

0 H ^

Since the restriction map of ^Fq(U) to ^pq(V) for V c U is surjective, we

can apply Proposition 3 and know that Hp(Z;ft;.%q) = 0 for all p > 0. ThisshowsHq(Z;1/;F) = H\ ^

360 Appendix

A sheaf F is called flabby if for every inclusion of open sets V c U,resu / v : F(U) -> F(V) is surjective. We define abelian groups F(F) and FC(F)by F(F) = F(Z) and Fc(F)={fe F(F)|Supp(f) is compact}.

Lemma 3. If 0 —> F -> G -> H -» 0 is an exact sequence of sheaves on Z.Suppose that F is flabby. Then 0 -> F(F) -> F(G) ->F(H) ->0 and 0-> FC(F)—» FC(G) —» FC(H) —> 0 are exacf sequences of abelian groups. In particular, ifF attd G are flabby, then H is flabby.

Proof. We only need to check surjectivity of a : FC(G) —> FC(H) and

F(G) -> F(H). Let s e F(H). Let C be the collection of all pairs (t,U)

(t€ G(U)) such that cc(U)(t) = resz/u(s). We give an order on C so that

(t,U) > (t',Uf) if U D U 1 and res^Ct) = t'. Evidently C is inductively ordered.

Thus there is a maximal element (U,t) in C by Zorn's lemma. Suppose U ^ z .

Let x G Z-U and take a small neighborhood V in Z of x. If V is sufficiently

small, we can find v e G(V) SO that (V,v) e C by the surjectivity (see (5b)).

Then resu/upV(t)-resv/VpU(v) e Ker(a)(UflV). Thus by flabbiness of F, we can

find f e F(ULJV) such thatresuuv/unvff) = resu/unvW"resv^/nu(y)-

This implies that t and f+v coincide on UflV. Then by the sheaf axiom, wecan find t' e G(ULJV) such that

resuuv/v(tf) = f+v and resuuv/u(tf) = t

Then (ULJV, t') is larger than (U,t) contradicting the maximality of (U,t). ThusU = Z and oc(t) = s. This shows the assertion for F. When s e FC(H), thenV = Z-Supp(s) is an open set and resz/v(s) = resz/v(oc(t)) = 0. Thus we can findt' e F(F) such that Supp(t-t') c Supp(s). Then t-t1 e FC(G) and cc(t-t') = s.This shows the assertion for Fc.

Let F be an arbitrary sheaf. LetFl: 0 ->F -> Fl0 -> Hi -> Fl2 -^ •••

be an exact sequence of sheaves where the Fli are all flabby. Such a complex iscalled a flabby (acyclic) resolution. There is a standard flabby resolution F1(F)of F: We define F1°(F)(U) = nxeuFx. Then F1°(F) is plainly a flabby sheaf.Then the diagonal map: fh-> Ilxeufx takes injectively F into F1°(F) by (S).After defining Flk(F) and 3k_i: F\kA(F) -^Flk(F), we just define

Flk+1(F) = Fl°(Coker(3k_i)) and ^ : Rk(F) -» Coker(3k_i) -> Flk+1(F),

where the last arrow is the natural diagonal map. We define the sheaf coho-mology group Hq(Z,F) (resp. the compactly supported sheaf cohomology group

Appendix 361

Hq(Z,F)) to be the cohomology group of the complex F(F1(F)) (resp.

rc(Fl(F))).

Proposition 4. There exists a canonical isomorphism Hq(Z,F) = Hq(Z;F).

Proof. By Lemma 3, Hq(Z,F) = Hq (Z,F) = 0 if F is flabby. First suppose thatF is flabby. Then the sheaf Cq(î;F) is flabby. Consider the exact sequence ofsheaves 0 -» Z^ -» Cq(<£i;F) -» Z^+i -» 0 for Zq = Ker(3q). This induces anexact sequences of complexes

0 -> H(j^) -> F1(C(^;F)) -> Fl(2^+i) -» 0.Note that F^C^^F) ) is flabby. Applying (1), we have a long exact sequence:

Hp(Z,Cq(^;F)) -> Hp(Z,z;q+1) -> H ^ Z , ^ ) -> H ^ ^ ^ C ^ ^ F ) ) .Since both ends of the above sequence vanish because of flabbiness ofwe have

This shows0 = Hq(Z,Zo) = H^CZ,^) = • • • s H^Z^q.i) = Hq(Z; ^;F).

Thus we see that(7) If F is flabby, then Hq(Z;ft;F) = 0 for all covering U

Now we treat the general case. Consider the exact sequence of sheaves:0 -> Z'q -> Flq(F) -> Z'q+i -> 0 for Z'q = Ker(3q).

Note that Flq(F) is flabby. From the long exact sequence of Cech cohomology,we have an exact sequence

0 = Hq(Z,Flq(F)) -> Hq(Z,Zq+i) -> Hq+1(Z,^q) -> Hq+1(Z,Flq(F)) = 0.Since both ends of the above sequence vanish because of flabbiness of C(1i;F),we have Hq(Z,£q+i) = Hq+1(Z,Zq). This shows

Hq(Z,F) = Hq(Z,Z'o) £ Uq-\Z,Z'i) = — = ttl(Z,Zq_i) = H^ZJF).

In fact, we can compute the sheaf cohomology group Hq(Z,F) of F using anyflabby resolution 0 —> F —> Fl of F. Out of the sheaf exact sequence

0 -> Z^ -> Flq -> Zî -> 0for Zq = Ker(3q), we have another exact sequence of complexes

0 -> Fl(Z^) -> Fl(Rq) -> H(2^+i) -> 0.

Applying (1) to this, we have, for H* denoting any one of Hq and Hq,

Hq(Z,F) = Hq(Z,2o) = H r ^ Z , ^ ) £ - £ H2(Zf^) = Hq(Fl).Thus we have

(8) For any flabby resolution 0 -> F -> Fl,

Hq(Fl) £ Hq(Z,F) £ Hq(Z,F) and Hq(R) = Hq(Z,F).

362 Appendix

Suppose F is a presheaf with surjective resu / v for every inclusion V c U and

F satisfies a part of (S):

(S!) If for each covering {Ui}, a given set of elements Si e F(Ui) satisfiesresz/uiJUjSi = reszyujJUjSj* there exists s e F(U) such that resz/u^s) = Sifor all i;

then we see easily that F# is flabby.

Now assume that Z has a triangulation K. We now introduce the subdivisionprocess of complexes. Let A2 = (a,b,c) be an r-simplex. We define a subdi-vision Sd(A2) of A2 by all the simplices in Figure (ii):

c1

b1

c c

Figure (i) Figure (ii)

Here h is the barycenter of A. Thus writing h*(xi,...,xr) = (h,xi,...,xr), we seethat Sd(a) = a, Sd(b,c) = (b,a')+(a',c) for the barycenter a' of the line segment(b,c) and Sd(A) = h*Sd(3A). By the last formula, we can inductively define thesubdivision operator Sd for all r-simplices. We apply this process ofsubdivision to each simplex of the complex K = Ko. We write the resultingsimplex as Ki = Sd(Ko). We continue this process n times and write the n-thsubdivision obtained by Kn = Sdn(K) = Sd(Kn_i). Finally we define Koo

= limKa. Let R be a commutative algebra with identity. For any open set U

on Z, we can consider a module Aq(U) of formal linear combinations£aA€RaAA f°r aA e R for i-simplices A in KeoflU. Then we have a covariantfunctor Aq : U h-> Aq(U) with natural inclusion map Incu/v * Aq(V) —» Aq(U) ifU 3 V. Thus we have a presheaf HomR(Aq,F): U H> HomR(Aq(U), F(U)). If Fis locally constant, HorriR(Aq,F) satisfies (S) and hence it is a sheaf andHoiriR(Aq,F) gives a flabby resolution of F. We can thus compute Hq(Z,F) andHq(Z,F) using HomR(Aq,F).

We now return to the situation in Proposition 1: Z = Ho and F = M. Let Aq>a

be the R-free module generated by q-simplices in Ka and putSq(Ka;M)=HomR(Ai,a;M) with T-action (ft | y)(A) =

Then we define Hq(Ka,M) by the cohomology group of the complex

Appendix 363

0 -> S°(Ka;M)r -» S^K^M)1" -* S2(Ka;M)r -» 0.

We want to show that Sd induces an isomorphism Hq(Ka,M) = HFQKa+i.M).Let â be the triangulation of Yo induced by Ka. Each vertex x e £a+i is abarycenter of a unique simplex A of £ a by the definition of the subdivision.Choose one vertex y of A and define cp(x) = y. We know that if (x,y) is a1-simplexof fat+i, then either x or y, say x, is the barycenter of itself. Hencey is the barycenter of (x,z). Then cp(x) = x and either (p(y) = x or (p(y) = z. Ineither case (cp(x),(p(y)) is a simplex of £a+i . Here we allow repetition ofvertices: (x,x) implies the 0-simplex x. Similarly, if (x,y,z) is a 2-simplex of£a+i, then looking at figure (ii), we may assume that x is the barycenter of x

itself, y is the barycenter of (x,a) e £ a and z is the barycenter of(x,a,b) e kjx. Thus we define cp(x) = x, cp(y) = x or z and (p(z) = x, a or b. Inany choice, (cp(x),(p(y),(p(z)) is a simplex of £a- Thus 9 induces a simplicialmap and hence a continuous map on Yo into itself, which we denote by the sameletter (p. We want to show that (p is homotopically equivalent to the identity.For each x e Yo, by definition, both (p(x) and x belong to the closure of a2-simplex t(x). Moving Yo into R2 by the homeomorphism given by the tri-angulation £, we may assume that Yo is embedded in R . Then we define F :[0,l]xY0 -» Y0 by F(u,x) = (l-u)(x-a)+u((p(x)-a)+a if t(x) = (a,b,c) for threevectors a, b and c in R2. Obviously F is continuous and hence gives thehomotopy equivalence between (p and the identity. Extending the simplicialmap cp to (p : Aqj(X+i —» Aq>a by R[F]-linearity, we have an R[F]-morphism (pwhich induces an isomorphism

(p* : Hq(Ka+bM) = Hq(Ka,M) satisfying (p*oSd = Sdocp* = id.

We now identify Hq(Ka, M) with Hq(K, M) by Sda. Since Yo is isomorphicto Y as a smooth manifold, we can compactify Y identifying Y with theclosure of Yo in X. We write Ys for this compactification. Note that Ys andX are different; that is, X has one point at the cusp but the boundary of Ys atthe cusp is a circle S1 isomorphic to a small circle around the cusp in X. Wecan think of something in between Y and Ys depending on So for a given sub-set So of S. We remove the boundary at s e So from Ys and write it asYs-S°. Note here that Y 0 = Y. Since Sq(Koo;M)r are the global sections of thesheaf HomR(Aq,M) on Yo, we have

Proposition 5. Let the notation be as in Propositions 1 and 2 and Theorem LThen we have canonical isomorphisms

Hq(Ys-s°,M) = Hô(F,M) for all subsets So in S and Hq(Y,M) = Hq(K,M).

Then by the boundary exact sequence in Proposition 2, we have

364 Appendix

Corollary 2. Let the notation be as in Proposition 2. We have a long exact se-quence

0 -> H°(YS"S°,M) -» H°(Y,M) -> 0sesoHoOsY*,M) -> ^(YS-S°,M)

-> H^YJMD -> e sôHâsY*^) -> if (Ys"So,M) -> H2(Y,M) -» 0,

3SY* is the boundary around s.

We now briefly recall the Poincare duality for Y. Our proof of the duality ismerely a sketch, and the reader should consult standard texts (for example [HiW,4.4.13]) for details. Suppose that R is a field and M is a finite dimensionalvector space over R. Let <|> e HomR(Ao)Ct>M*) be a cocycle for M* =HoiriR(M,R), where a is a fixed integer (which we make large if necessary).We suppose that <|) is compactly supported on Y. Then we can find a smallopen neighborhood of 3YS which does not meet the support of $. We take acocycle COG HorriR(A2,oJvI). Then we define for each i-simplex A of fox,

<(J),co)(A) = <<|>(A),©(A)> for the dual pairing ( , ) : M*xM -> R.

Since A is simply connected, <|> is a constant function on A. Here we write

<|>(A) for this value of ty on A. It is obvious that ((|),co) e HomR(A2,a,R) and

Supp(((),co) is compact in Y. We see easily that ((|),3co) = 3(<|),co) and hence

(<]),co) is a 2-cocycle. It is well known (see Proposition 6.1.1 for a proof) that

H2 (Y,R) = R. That is, we have a pairing

(,):H°(Y,M*)xH^(Y,M)->R.

Suppose (0,0)) = dr[§ for all $ with compactly supported r^ e HorriR(Ai,R)depending on 0. We fix a basis {ei, ..., er} of M and take its dual basis{ei*,..., er*} of M*. We write <|> = Z i t e * and co = ZiO)iei. Then we candefine a 1-chain ^ e HomR(A1,M) by £i(Ai) = (n0ie.*(Ai)ei* on each1-simplex Ai in the A. Put S; = £ & . Then we see from construction that3^ = co on A. We extend this process of defining £ to 2-simplices adjacent toA. Then inductively, we get £ such that 3 ^ = co. Thus the pairing isnon-degenerate on HC(Y,MJ. Similarly, we can show that the pairing isnon-degenerate on H (Y,M*) much more easily because there is no-restrictionfor co to be a 2-cocycle (i.e. Y is two dimensional). Thus we have

Proposition 6. Suppose that R is afield and M is finite dimensional. Then the

pairing ( , ) : H°(Y,M*)xH^(Y,M) -> R is a perfect duality.

References

Books

[BGR] S. Bosch, U. Giintzer and R. Remmert, Non-Archimedean analysis,Springer, 1984

[Bourl] N. Bourbaki, Alg&bre commutative, Hermann, Paris, 1961, 1964, 1965and 1983

[Bour2] N. Bourbaki, Topologie generate, Hermann, Paris, 1960,1965[Bour3] N. Bourbaki, Alg&bre, Hermann, Paris, 1958-[Bd] Bredon, Sheaf theory, McGraw-Hill, 1967[Bw] K. S. Brown, Cohomology of groups, Graduate Texts in Math. 87,

Springer, 1982[CM] L. Clozel and J. S. Milne, Automorphic forms, Shimura varieties and

L-functions, Perspectives in Math. 10,11 (1990), Academic Press[CT] J. Coates and M. J. Taylor, L-functions and arithmetic, LMS Lecture notes

series 153 (1991), Cambridge Univ. Press[dS] E. de Shalit, Iwasawa theory of elliptic curves with complex multiplication,

Perspectives in Math. 3 (1987), Academic Press[FT] A. Frohlich and M. J. Taylor, Algebraic number theory, Cambridge studies

in advanced mathematics 27, Cambridge University Press, 1991[G] R. Godement, Notes on Jacquet-Langlands theory,[Gv] F. Q. Gouvea, Arithmetic ofp-adic modular forms, Lecture Notes in Math.

1304 (1988), Springer[Ge] S. S. Gelbart, Automorphic forms on adele groups, Ann. of Math. Studies

83, Princeton University Press, 1975[Ha] R. Hartshorne, Algebraic geometry, Graduate Texts in Math. 52, Springer,

1877[HiW] P. J. Hilton and S. Wylie, Homology Theory, Cambridge University

Press, 1960[HiSt] P. J. Hilton and U. Stammbach, A course in homological algebra, Graduate

Texts in Math. 4, Springer, 1971[Hor] L. Hormander, An introduction to complex analysis in several variables,

North-Holland, 1973[Iwl] K. Iwasawa, Lectures on p-adic L-functions, Ann. of Math. Studies 74,

Princeton Univ. Press, 1972[J] H. Jacquet, Automorphic forms on GL(2), II, Lecture notes in Math. 278,

1972, Springer[JL] H. Jacquet and R. P. Langlands, Automorphic forms on GL(2), Lecture

notes in Math. 114, Springer[Kb] N. Koblitz, p-adic numbers, p-adic analysis, and zeta functions, Graduate

Texts in Math. 58, Springer, 1977

366 References

[Ko] K. Kodaira, Complex manifolds and deformation of complex structure,Springer, 1986

[L] S. Lang, Cyclotomic fields, Graduate Texts in Math. 59, Springer, 1978[M] T. Miyake, Modular forms, Springer, 1989[Mml] D. Mumford, Abelian varieties, Oxford Univ. Press, 1974[N] J. Neukirch, Class field theory, Springer, 1986[NZ] I. Niven and H.S. Zuckerman, An introduction to the theory of numbers,

John Wiley & Sons, 1980 (4th Edition)[P] L. S. Pontryagin, Theory of continuous groups (Russian), Moscow, 1954[Pa] A.A. Panchishkin, Non-Archimedean L-functions of Siegel and Hilbert

modular forms, Lecture notes in Math. 1471, 1991, Springer[RSS] M. Rapoport, N. Schappacher and P. Schneider (ed.), Beilinson's

conjectures on special values of L-functions, Perspectives in Math. 4,Academic Press, 1988

[Sh] G. Shimura, Introduction to the arithmetic theory of automorphic functions,Princeton Univ. Press, 1971

[Wa] L. C. Washington, Introduction to cyclotomic fields, Graduate Texts inMath. 83, Springer, 1980

[Wl] A. Weil, Basic number theory, Springer, 1974[W2] A. Weil, Elliptic functions according to Eisenstein andKronecker, Springer,

1976[W3] A. Weil, Dirichlet series and automorphic forms, Lecture notes in Math. 189,

Springer, 1971

Articles

[Ba] D. Barsky, Fonctions zeta p-adiques d'une classe de rayon des corpstotalement reels, Groupe d'etude d'analyse ultrametrique 1977-78; errata,1978-79

[B] D. Blasius, On the critical values of Hecke L-series, Ann. of Math. 124(1986), 23-63

[Cl] H. Carayol, Sur les representations /-adiques associees aux formesmodulaires, Ann. Scient. Ec. Norm. Sup. 4e serie, 19 (1986), 409-468

[C] W. Casselman, On some results of Atkin and Lehner, Math. Ann. 201(1973), 301-314

[CN] P. Cassou-Nogues, Valeurs aux entiers negatifs des fonctions zetap-adiques, Inventiones Math. 51 (1979), 29-59

[Co] P. Colmez, Residu en s = 1 des fonctions zeta p-adiques, InventionesMath. 91 (1988), 371-389

[D] P. Deligne, Formes modulaires et representations /-adiques, Sem. Bourbaki,exp.355, 1969

References 367

[Dl] P. Deligne, Valeurs de fonctions L et periodes d'integrales, Proc. Symp.Pure Math. 33 (1979), Part 2, 313-346

[DR] P. Deligne and K. A. Ribet, Values of abelian L-functions at negativeintegers over totally real fields, Inventiones Math. 59 (1980), 227-286

[DS] P. Deligne and J.-P. Serre, Formes modulaires de poids 1, Ann. Sclent. Ec.Norm. Sup. 7 (1974), 507-530.

[DuS] G. F. D. Duff and D. C. Spencer, Harmonic tensors on Riemannianmanifolds with boundary, Ann. of Math. 56 (1952), 128-156

[GJ] S. Gelbart and H. Jacquet, A relation between automorphic representations ofGL(2) and GL(3), Ann. Sclent. Ec. Norm. Sup. 4e serie 11 (1978),471-542

[GS] R. Greenberg and G. Stevens, p-adic L-functions and p-adic periods ofmodular forms, preprint

[Ha] G. Harder, Eisenstein cohomology of arithmetic groups. The case GL2,Inventiones Math. 89 (1987), 37-118

[HI] H. Hida, On congruence divisors of cusp forms as factors of the specialvalues of their zeta functions, Inventiones Math. 64 (1981), 221-262

[H2] H. Hida, Congruences of cusp forms and Hecke algebras, Sem. de Theoriedes Nombres, Paris 1983-84, Progress in Math. 59 (1985), 133-146

[H3] H. Hida, Iwasawa modules attached to congruences of cusp forms, Ann.Scient. Ec. Norm. Sup. 4e serie 19 (1986), 213-273

[H4] H. Hida, Galois representations into GL2(Zp[[X]]) attached to ordinarycusp forms, Inv. Math. 85 (1986), 546-613

[H5] H. Hida, A p-adic measure attached to the zeta functions associated with twoelliptic modular forms, I, II. I: Inventiones Math. 79 (1985), 159-195; II:Ann. Institut Fourier 38 No.3 (1988), 1-83

[H6] H. Hida, p-adic L-functions for base change lifts of GL2 to GL3, In:Automorphic forms, Shlmura varieties, and L functions, II, Perspectives inMathematics 11 (1990), Academic Press, pp.93-142

[H7] H. Hida, Le produit de Petersson et de Rankin p-adique, Sem. Theorie desNombres, Paris 1988-89, Progress in Math. 91 (1990), 87-102

[H8] H. Hida, On p-adic L-functions of GL(2)xGL(2) over totally real fields,Ann. Institut Fourier 41, 2 (1991), 311-391

[H9] H. Hida, On the critical values of L-functions of GL(2) and GL(2)xGL(2),in preparation

[H10] H. Hida, Nearly ordinary Hecke algebras and Galois representations ofseveral variables, in supplement to the Amer. J. Math, with the title:Algebraic analysis, geometry and number theory, Johns Hopkins Univ.Press, 1989, pp.115-134

[HT1] H. Hida and J. Tilouine, Katz p-adic L-functions, congruence modules anddeformation of Galois representations, Proc. LMS Symposium on"L-functions and arithmetic", Durham, England, July 1989, LMS Lecturenotes series 153 (1991), Cambridge Univ. Press, pp.271-293

368 References

[HT2] H. Hida and J. Tilouine, Anti-cyclotomic Katz p-adic L-functions andcongruence modules, Ann. Ec. Norm. Sup., 4e serie, 25 (1992)

[HT3] H. Hida and J. Tilouine, On the anticyclotomic main conjecture for CMfileds, preprint

[Kl] N. M. Katz, p-adic properties of modular schemes and modular forms, inModular functions of one variable, Lecture Notes in Math. 350 (1973),Springer, pp.69-190

[K2] N. M. Katz, p-adic L-functions via moduli of elliptic curves, Proc. Symp.Pure. Math. 29 (1975), 479-506

[K3] N. M. Katz, p-adic interpolation of real analytic Eisenstein series, Ann. ofMath. 104(1976), 459-571

[K4] N. M. Katz, Formal groups and p-adic interpolation, Asterisque 41-42(1977), 55-65

[K5] N. M. Katz, p-adic L-functions of CM fields, Inventiones Math. 49 (1978),199-297

[K6] N. M. Katz, Another look at p-adic L-functions for totally real fields, Math.Ann. 255 (1981), 33-43

[Ki] K. Kitagawa, On standard p-adic L-functions of families of elliptic cuspforms, preprint

[KL] T. Kubota and H. W. Leopoldt, Eine p-adische Theorie der Zetawerte. I.Einfiihrung der p-adischen Dirichletschen L-funktionen, J. reine angew.Math. 214/215 (1964), 328-339

[La] R. P. Langlands, Modular forms and /-adic representations, In Modularfunctions of one variable II, Lecture Notes in Math. 349, 1973, Springer,pp.361-500

[Md] Y. Maeda, Generalized Bernoulli numbers and congruence of modularforms, Duke Math J. 57 (1988), 673-696

[Ma] K. Mahler, An interpolation series for continuous functions of a p-adicvariable, J. reine angew. Math. 199 (1958), 23-34

[Mnl] Y. I. Manin, Periods of cusp forms, and p-adic Hecke series, Math.USSR-Sb.21 (1973), 371-393

[Mn2] Y. I. Manin, The values of p-adic Hecke series at integer points of thecritical strips, Math. USSR-Sb. 22 (1974), 631-637

[MM] Y. Matsushima and S. Murakami, On vector bundle valued harmonic formsand automorphic forms on symmetric Riemannian manifolds, Ann. ofMath. 78 (1963), 365-416

[MSh] Y. Matsushima and G. Shimura, On the cohomology groups attached tocertain vector valued differential forms on the product of upper half planes,Ann. of Math. 78 (1963), 417-449

[Mzl] B. Mazur, Courbes elliptiques et symboles modulaires, Sem. Bourbaki,expose 414 (1972, juin)

[Mz2] B. Mazur, Rational points of abelian varieties with values in towers ofnumber fields, Inventiones Math. 18 (1972), 183-266

References 369

[Mz3] B. Mazur, Deforming Galois representations, Proc. of workshop on Galoisgroups over Q, Springer, 1989, pp.385-437

[MzS] B. Mazur and H.P.F. Swinnerton-Dyer, Arithmetic of Weil curves,Inventiones Math. 25 (1974), 1-61

[MT] B. Mazur and J. Tilouine, Representations galoisiennes, differentielles deKahler et "conjectures principales", Publ. I.H.E.S. 71 (1990), 65-103

[MTT] B. Mazur, J. Tate and J. Teitelbaum, On p-adic analogues of theconjectures of Birch and Swinnerton-Dyer, Inventiones Math. 84 (1986),1-48

[MW] B. Mazur and A. Wiles, Class fields of abelian extensions of Q,Inventiones Math. 76 (1984), 179-330

[Ri] K. A. Ribet, A modular construction of unramified p-extensions of Q(JLLP),

Inventiones Math. 34 (1976), 151-162[R] K. Rubin, The "main conjectures" of Iwasawa theory for imaginary quadratic

fields, Inventiones Math. 103 (1991), 25-68[Sch] C.-G. Schmidt, p-adic measures attached to automorphic representations of

GL(3), Invent. Math. 92 (1988), 597-631[Se] J.-P. Serre, Formes modulaires et fonctions zeta p-adiques, In Modular

functions of one variable HI, Lecture Notes in Math. 350 (1973),Springer, pp.191-268.

[Shi] G. Shimura, Sur les integrates attaches aux formes automorphes, J. Math.Soc. Japan 11 (1959),291-311

[Sh2] G. Shimura, On the holomorphy of certain Dirichlet series, Proc. LondonMath. Soc. 31 (1975), 79-98

[Sh3] G. Shimura, The special values of the zeta functions associated with cuspforms, Comm. Pure Appl. Math. 29 (1976), 783-804

[Sh4] G. Shimura, On the periods of modular forms, Math. Ann. 229 (1977),211-221

[Sh5] G. Shimura, The special values of the zeta functions associated with Hilbertmodular forms, Duke Math. J. 45 (1978), 637-679

[Sh6] G. Shimura, On certain zeta functions attached to two Hilbert modularforms: I. The case of Hecke characters, Ann. of Math.114 (1981),127-164; II. The case of automorphic forms on a quaternion algebra, ibid.569-607

[Sh7] G. Shimura, Confluent hypergeometric functions on tube domains, Math.Ann. 260 (1982), 269-302

[Sh8] G. Shimura, Algebraic relations between critical values of zeta functions andinner products, Amer. J. Math. 104 (1983), 253-285

[Sh9] G. Shimura, On Eisenstein series, Duke Math. J. 50 (1983), 417-476[ShlO] G. Shimura, On a class of nearly holomorphic automorphic forms, Ann. of

Math. 123 (1986), 347-406[Shll] G. Shimura, On the critical values of certain Dirichlet series and the periods

of automorphic forms, Inventiones Math. 94 (1988), 245-305

370 References

[Shi2] G. Shimura, On the fundamental periods of automorphic forms ofarithmetic type, Inventiones Math. 102 (1990), 399-428

[Stl] T. Shintani, On evaluation of zeta functions of totally real algebraic numberfields at non-positive integers, J. Fac. Sci. Univ. Tokyo, Sec.IA 23(1976), 393-417

[St2] T. Shintani, On a Kronecker limit formula for real quadratic fields, J. Fac.Sci. Univ. Tokyo, Sec.IA 24 (1977), 167-199

[St3] T. Shintani, On values at s = 1 of certain L-functions of totally realalgebraic number fields, In Algebraic number theory, Proc. Int. Symp.Kyoto, 1976, pp. 201-212

[St4] T. Shintani, On certain ray class invariants of real quadratic fields, J. Math.Soc. Japan, 30 (1977), 139-167

[St5] T. Shintani, A remark on zeta functions of algebraic number fields, InAutomorphic forms, representation theory and arithmetic, Tata Institute ofFundamental Research, Bombay, 1979, pp.255-260

[St6] T. Shintani, A proof of the classical Kronecker limit formula, Tokyo J.Math. 3 (1980), 191-199

[Si] C. L. Siegel, Berechnung von Zeta-funktionen an ganzzahligen Stellen,Nachr. Akad. Wiss. Gottingen 1969, 87-102

[Wil] A. Wiles, On ordinary X-adic representations associated to modular forms,Inventiones Math. 94 (1988), 529-573

[Wi2] A. Wiles, The Iwasawa conjecture for totally real fields, Ann. of Math. 131(1990), 493-540

Answers to selected exercises

§1 .2 .1. We only need to show that (2nl(m):&+(m)) is finite, because I(m)lTC\I(m)

injects into I/(P. We see by definition that the natural map 0-{O} —»fP, which

takes each integer a e O to the principal ideal generated by a, induces a surjec-

tion from the finite group (O/m)x onto #\l(m)l<P(m). Hence {^\I{m):T{ni)) is

finite. The map F —> RrxCt given by F 3 a h-» (a^) induces an injection

of 2(m)/2+(m) -> (Rx)7(R>o)r, where R > 0 = {x e R | x > 0}. Thus

(2(m):(P+(m)) < 2r. This shows the desired assertion.

2. (b) What we need to show is that the integer ring O is not a unique factor-ization domain. Note that 0={x+yV-5 | x,y e Z}. We have a decomposi-tion 3-7 = 21 = (1+2V-5 ) (1 -2 -N/ -5 ) . All these factors cannot be factored into aproduct of two non-units. For example, if 3 = a(3 with non-unit a and (3,then 9 = N(a)N($) and hence Af(oc) = 3 because otherwise (3 becomes aunit. This is impossible because 5 mod 3 is not a square. Similarly, one canshow that 7, (1+2V^5) and (l'2^f^5) are all primes. Obviously,(l±2V~5)/3 and (l±2V~5)/7 are not integers, and hence the prime factorizationin O is not unique.

3 . By the Minkowski estimate, there is 0 * a e O such that

1 < | iV(a) | < | V W l ^ . Thus 1 < ^ - j - < | VE>F I. There is another way

to prove this only using an argument similar to the proof of Lemma 1.2.4: In fact,

by the lemma, if Ki Kd = I VDp I, then the set

..,Kd) = { 0 * a e o\ | a ( i ) | < K{ for all i}

is not empty. In particular, for any 8 > 1, choosing a small e > 0 so that

8K!.(K2-e) (Kd-e) = | VD^ I, then Y(8Ki,(K2-e),...,(Kd-e)) * 0 . By putting

X(Ki, . . . ,Kd) = {0 * a G O | a ( i ) | < Ki for all i > l and I oc(i) I < K i } ,

we have X(8Kx,...,Kd) z> Y(8Ki,(K2-e),...,(Kd-e)) * 0 . Then making 8 ^ 1

and keeping in mind the fact that X(Ki,.. .,1^) is a finite discrete set, we conclude

that X ( K i , . . . , K d ) * 0 . Thus for a e X ( K i , . . . , K d ) , we see that

1 < \N(a)\ <

372 Answers to selected exercises

5. For any proper subset S of {1,2,...,r+t}, interchanging the order of infinite

places, we may assume that {l,...,r+t-l} z> S. Consider the map

/ : O* -> R^1"1 given by Z(e) = (log I e(i) | 5), where 8 = 1 or 2 according as

(i) is real or complex. Then as shown by Dirichlet's theorem and its proof, l(cf)

is a discrete submodule of RT+t~l and Rr+t"V/(C^) is compact. Thus for a suffi-

ciently large M, any vector v e RT+t~l can be brought into the domain

D = {(xi)e Rr+t-l < M for all i}

by translation by an element of /(0*). We take v e R^1"1 so that vi = N for

some N > 0 and for i e S and Vi = -M for i e So-S. Then we pick

6 e Ox so that v-/(e) e D. Then -M < N-log I e(i) | 8 < M for i e S and

-M < -M-log I e( i ) I 5 < M, so log I e( i ) I 6 > N-M for all i e S and

-2M < log | e(i) | 5 < 0 for i e So-S. Thus by making N sufficiently large,

we see that log I e(i) I 5 > 0 for all i e S and log I e(i) I 5 < 0 for i e So-S

and Ilt\llog | e(i) I 5 > 0. Thus I e(i) I > 1 for all i e S and I e(i) | < 1 for

all i e S n S 0 and n ^ ' 1 1 e(i)l 6 > 1. Since iTî I e(i) I 8 = \N(e)\ = 1 , we

see that | e(r+t) | < 1. This shows the desired assertion.

§ 2 . 1 .3. (a) Note that if z stays in a compact set inside C-Z, we can find a positive

integer M such that + < Mn"2 for all n > 0, and hence|z + n z - n |

ST-i 1 ^ + i l 5 M ^ 2 ) - T h u s X L ( ^ + ^ ) converges absolutelyand locally uniformly on C-Z. Similarly, if z stays in a compact set in C-Z, we

can find e > 0 such that | ^ | > en"1. Thus J^=1 | ^ | > e ^ = 1 n"1.

Since Xr=i n"X > M + 1 ~ + T ^ X "-* +o° as m -^ +©<>, we know the divergence

of y °° because in an absolutely convergent series, any partial series is

also absolutely convergent.

(b) We shall first show the absolute convergence of Xk=iXn=i n"2kz2k-1 when

| z | <1. We see that X ^ S ^ I n" 2 ^" 1 1 < ^ = 1 k(2k)| Iz12 k l- Note

that

I £(2k) I < 1 +Jx"2kdx = 1+ = l4-(2k-l)"1 < 2.

Thus


This shows the absolute convergence of Xk=iXr=i n'^z21^"1. Since the limitvalue of the absolute convergent series does not depend on the order of summa-tion, we know that

2-rk=lAi=l n z " 2^n=l2.k=l n Z 'which was to be shown.

^4. Write F(x) for - ^ - . Then F ( - x ) = - ^ - = -^— = l-F(x), and we see1+e 1+e 1+e

) m F(x) l x = 0 for m> 0.

( d \ m I

•^j F(x) I = 0. Writingm = 2n, we know formally, by the functional equation, that

2r(2n+l)cos(n7T+(7c/2))Note that £(s) is finite at s = 2n+l but cos(nrc+(7i;/2)) = 0. Thus £(-2n) n a s

to be 0 for the validity of the above equation. Thus we know that

C(-m) = (^-)mF(x) | _0 = 0 for even positive integer m.

5. Looking at the expression

we know that (t^-) ^(t) has the expression

for a polynomial Pm(t) with integer coefficients. This can be proven byinduction. Then of course, we have

Pm(t) I -m-l z

This shows the result.

6. (b) Suppose p appears in the denominator of C(l-k). Then by Exercise 5,

for any integer a> 1, ak(ak-l)£(l-k) e Z. Thus if a is prime to p, ak-l

must be divisible by p, that is, ak = 1 mod p. We choose a so that a mod p

gives the generator of the cyclic group (Z/pZ)x. Then ak = 1 mod p if and

only if k = 0 m o d p - l .

§2 .2 .3. (b) First of all, when s = 1,

Jp(8)G(y)dy = JaD(e)G(y)dy = (2!Ci)Resyas0(G(y)) = 2TCI.On the other hand, we know that


Ress=i(e27tis-1)-1 = (2JU)"1.

Thus by (e2™-l)r(s)C(s) = JP(e)G(y)ys-1dy, we have

Ress=iC(s) = 1.

4. (a) By the functional equation, we get

Since £(2n) > 0, we know the signature of Bn.

6. We give the argument for the integral over Q/(m) since we can treat Qr(m) inthe same way. For z on Q/(m), z = -(2m+l)+yL Thus I e"z I = e2m+1 andI l-e"z | > I e"z | -1 = e2m+1-l. Therefore

i i e 2 m + 1

I I ^ ^ 2 on Q/(m) if m > 0-This gives an estimate:

|JQKm)G(z)zs-1dz| = \\(^li)n G(-(2m+l)+yO(-(2m+l)+yOs-1dy|

K2xn+i), | | o . l d <2f2m+1* ((2m+l)2

+y2)(a-1)/2dy

dyJJ-(2m+l)7t

<47t(2m+l)a - ^ 0 as m -> +«> (if a = Re(s) < 0).

§ 2 . 3 .4. By Corollary 2, LQ-n.x"1) = 0 if %(-l) = 1 and L(l-n,xA) is finite oth-erwise. Then by the functional equation

Us y) = S

G(x)(27i/N)sL(l-s,%-1) . f

2V:ir(s)sin(7is/2)

if %(-l) = - l , r(s)sin(?rs/2) is finite at s = l and hence L(s,%) is finite ats = 1 and if %(-l) = 1, the simple zero of r(s)cos(7i;s/2) is canceled out by thezero of L(l-s,%"!) at s = 1 and hence L(s,%) is finite at s = 1.

5. Since the argument is essentially the same in the two cases where %(-l) = ±1,we only treat the case of %(-l) = 1. Then by the functional equation, we see that

= 2r(s)cos(7is/2)L(s,%)

(2TC/N)SL(1-S,%-1) '

Expanding L(s,%) = Zn>mam(s-2)n with am * 0, we see that


1= L(s,X) = Z n , m a m (s-2) n and (27C/N)(-I)2m|am|r

Applying the functional equation for the Riemann zeta function (i.e. replacingG(x) by 1 and % by the trivial character in the above formula), we know that

r ( i ) = VTC because r(y) > 0 and hence G(x)G(x"1) = N = %(-l)N. When

X is real valued, X"1 = %> am e R and by the above formula, we see that

G(x) = (-l)mVN. Thus if the order of zero at y *s e v e n (in particular if

L(^,X) ^ 0), we know that G(x) = VN = ^/%(-l)N.

§ 2 . 4 .

1. (a) For a = I a I e10 e H' (I 9 I < y) and s = c+ix (a,x G R), we see

that I a"s | 2 = a"s(a)"r = I a | "2ae20T. Thus if s stays in a compact subset of C,then | a"s | < M | a | "2Re^s) for a positive constant M independent of a. When Ais not real, then

I L*(n+x)"s |

< Mllf=11 Re(aii(ni+xi)+-"+ari(nr+xr))+Im(aii(ni+xi)+'--+ari(nr+xr)) | "ai

' a i if Gi>0.

Thus we may assume that A is real by replacing A by Re(A). When A is a realmatrix, let M > 0 be the minimum of all entries of A. Then

I L*(n+xys | = n"1(aii(ni+xi)+...+ari(nr+xr))"Oi

< nîCMCm+xO+^+M^+x,))-01 < M'Tr(o>n?1(ni+...+nr))"Oi

if G{ > 0 and not all ni are zero (if <J{ is negative, then replacing M by themaximum of the entries of A, we can deduce a similar inequality). Thus we mayassume that all the entries of A are equal to 1, xi = 1 for all i and %i = 1 forall i. Then

I C(s,A,x,x) I < S n G Z + r(n 1 + -+n r ) - T r ( a ) .

We see easily that #{(ni,..., nr) e Z+r | ni+---+nr = k} < Ckr-1 for a posi-tive constant C and hence

I n e Z + r (n 1 + - + n r ) - T r ( o ) < c X ^ k ' - 1 - 7 ^ = CC(Tr(a)-r+l).

Since the Riemann zeta function ^(s) converges if Re(s) > 1, ^(s,A,x,x) con-

verges if Re(Tr(s)) > r and Re(si) > 0 for all i.

(b) It is sufficient to show that


B(r,k) = #{(m,..., nr) e Z+r | m+---+nr = k} > ck1""1

for a positive constant c, because thencC(Tr(a)-r+l) < Xn e Z + r(ni+--+nr)-T r^.

We see easily that B(2,k) = k+1 and hence we can take 1 as c in this case.Supposing that B(r-l,k) > ckr"2 for some c> 0, we prove the assertion for r.We see easily that

B(r,k) = IJLoBCr-lj) > cIJLof2 > cj*xr"2dx = C-£pwhich shows the desired assertion.

3. We have

(e47C/s-l)(e27C/s-l)r(s)2C((s,s),A,x,(l,l))

where_ exp(-xiu(a+bt)) exp(-x2u(c+dt))" l-exp(-u(a+bt)) x l-exp(-u(c+dt))_ exp(-xiu(at+b)) exp(-x2u(ct+d))- 1 . e x p( . l l ( a t + b)) x i-exp(-u(ct+d))

Since

we see that

and

Km (e4it"-l)(e2*»-l)r(s)2 = 2<2%i)\« » i n v ((n1)!)2

= the coefficient of u 2 ^ 1 ^ 1 of

= the coefficient of t11"1 of

( nk+iBk(xi)Bi(x2)jÔ^"1^ k!j!-k +j=2nt k>0,

Xiâ+P=n-1, a>0, P>ol a I (3

where we have used the binomial formula

(X+Y)s = i ;


l if n = 0,f l if n = 0, 1( J ) = js(s-l) (s-n+1) i f n > 0 (in particular, ('*) = (-l)n)

I n!for

Therefore, the final formula is

.D) = 2-VDl

§ 2 . 5 .2. Let a and w be two ideals in O, and first show that there is a e O such

that aO = aS and £ is prime to m. Write w = pi e i p/1 (ei > 0). Put

c = ap\p2 pi and c\ - c/p{. Then q Z) c, and the two ideals are distinct.Thus we can find (Xi e q-c. Then oci is in apj for j ^ i but is not contained inap[. If a is in api, then oci belongs to ap\ because all other Oj (j ^ i) areinside ap19 a contradiction. Thus a = oci+---+ar is not contained in ap\ for alli. Since a e a, we can write aO= aS and B is prime to w. Now, fixing aclass C of ideals, we show that there exists an ideal de C such that d isprime to a given ideal nu In fact, ^ can be taken inside O. First take an integralideal c in the class C and write c = CQC SO that the prime factors of CQ consistsof those of m and c is prime to m. Take arbitrary O^yeco and decom-pose yO = aco. By the above argument, we can find a e O such thataO= aB and 5 is prime to nu Then

a aBc t .J£c = — = £c

and the integral ideal <f= Be is prime to m and in the same class C as c.

3. Let C be the class of nil and take an integral ideal B in C prime to m (Wehave used Exercise 2). Then Bm = yO, because Bm is in the principal class.

4. (b) Consider the module Z2 which we consider to be made of row integerfa b\

vectors. Then, for ex = I d I e SL2(Z),

fa b\(m,n) h-> (m,n) , = (ma+nc, mb+nd)

vc aJinduces an automorphism of Z2 whose inverse is given by a'1 (a"1 again hasinteger entries). In particular, if a e F(N), a obviously induces an automor-phism of the set (a,b)+NZ2. On the other hand, we see that

(moc(z)+n) = m ^ ^ + n = (cz+d)"1((ma+nc)z+(mb+nd)).

Thus Ek,N(cc(z),s;a,b) = I ( m ,n ) e (a)b)+Z2 (ma(z)+n)"k | (ma(z)+n) I "2s


= £(m,n)e((a,b)+z2)(X (mz+n)* | (mz+n) | -2s(cz+d)k | cz+d | 2 s

= S(m,n)G((a,b)+z2) (mz+n)-k | (mz+n) | "2s(cz+d)k | cz+d | 2 s

= Ek)N(z,s;a,b)(cz+d)k I cz+d I 2s for a e T(N).

Thus Ek,N(z»s;a,b) is a modular form on F(N) of weight (k,s).

§2 .6 .1. We have a non-trivial unit e such that 0< e < 1 < e a . Then if% : Km) -> Cx is an ideal character such that x((a)) = a k a a m , then

a k a am = % ( ( a ) ) = % ( ( 8 a ) ) = (ea)k(ea)o m.

Thus e k e o m = 1. By taking logarithms of both sides, we see that

klog(e) + m/<9g(e°) = 0, which yields klog(e) = mlog(e) because eea = 1.

Since logiz) * 0, we know that k = m.

3. Writing h for the class number of Q(V~P) an (i %(a) = ( t ) ' we have1 =-hp. Since p = 3 mod 4, %(-l) = -1, %(p-a) = -%(a) and

= liP=";1)/2(Z(a)a+X(p-a)(p-a)} = 2^ 1 ) / 2 X(a) -p l l P

= - ; 1 ) / 2 X(a) = -hp.Similarly

Thus combining the above two formula, we have

where A (resp. B) is the number of quadratic residues (resp. non-residues)

modulo p between 1 and ?. By the quadratic reciprocity law, %(2) = 1 or -1

according as p = 7 or 3 mod 8. This shows (1) and (2).

§ 3 . 5 .

1. We know from (la) that J ( ^ydjiicp = (3nO(p) | T = o . We write

ft—\dn = dt . This shows

J

2. By (2b), we see that


« W ( t - D = Jtxd(q>*\|/)(x) = JJtx+ydcp(x)dV(y)

= JJtx+yd9(x)dV(y) = Jtxd(p(x)JtydV(y) =*9( t - l

3. We know that Zp[T]] is the completion of R under the /rc-adic topology for

m = (p,T) (T = t-1). Thus R = Zp[[T]]DQp(t), where the intersection is takenin the quotient field of Zp[[T]]. Since Zp[[T]] = êa*(Zp;Zp), the operator [<|>]preserves Zp[[T]] for any locally constant function $ on Zp having values inZp, because the operation [$] : d-jLL h-> tyd\i preserves the measure space. Onthe other hand, by definition [0] preserves Qp(t). Thus [<|>] preserves the inter-section R.

§ 3 . 6 .l.(ii). By definition, Rq, = R/(p(Ker(e'))R. Thus we have a sequence of groups:

0 -» HomA-aig(R(p,S)--^HomA.aig(R,S) 9* )HomA-aig(R'»S)

I! II , II0 — > Ker(<p*)(S) n* > G(S) 9* > G(S),

where K : R -» R9 is the projection map. The injectivity of TE* follows fromthe surjectivity of n. An algebra homomorphism <|> : R —> S is in Ker((p*) ifand only if the following diagram is commutative:

R1 ^ U R

ie1 1$A -> S.

Thus (|) e Ker((p*) <=> cp(Ker(e')) c Ker(<|>) <=> <|> induces (()' : Rq, -> Ssuch that 71*$' = 0. This shows the exactness at G(S).

§ 8 . 1 .1. Since the adele ring FA is the restricted direct product with respect to CV, for

v outside m, we can consider an adele a in d(m) whose v component is 1 for

almost all v and is G3V-1 for some finite number of places v outside m, where

05v is a prime element of Ov. Then 1+a in l+d(m) has C5V for some finite

number of places. On the other hand, the v-component of any idele in d(m)x is a

unit in CV for all finite places. Thus l+a£ d(m)x.

§ 8 . 2 .

1. If xn e FAX converges to x, then for sufficiently large n, (XnX' f falls in

the neighborhood U(m). Thus A,((xn)f) = X(xf). On the other hand, A,((xn)oo) =

(xjeo^ is a polynomial function on the vector space

F~ = fly realR X U, complexC (1.1.5b),


which is obviously continuous. Thus lim X,((xn)«,) = (Xoo) and hence lim X(xn)n—>©o n—>«»

= A,(x). That is, X is continuous.

3. (i) Since Cl(m) is a finite group, if one writes h for its order, for any ideal a

prime to m, a*1 = aO for a e P(w). Thus X*(a)h = X*(aO) = a^, which is

contained in the Galois closure O of F over Q. Thus X(d) is always algebraic.

If {«i}i=i h is a representative of classes modulo m, then any a can be written

as ao{ for one of the o 's. In particular,

X(a) = a^X(oi) s K = d>(X(ai) | i=l, . . . , h).

The field K is generated over O by finitely many algebraic numbers and hence is

a number field. Since X(x) = X*(xO) for x e FA(W), we know that X(x) is

contained in a number field K independent of x e FA(W).

(ii) This follows from (1.3.4b).

(iii) (A. Weil) Writing x = auax^ e FAX for a e Fx, UG U(OT), a e FA(mp)

and Xoo e Foo+, we define ^P(X) = ?L*(aO)up~ , where up is the projection of u

to n v | p F vx . If there are two expressions ocuaxoo = cc'u'a'x'oo, then apup =

oc'pu'p because (axoo)p = (a'x'oo)p = 1. On the other hand,

oc'â = u'uâ'a^x'ooXoo"1 e U(wp)FA(/rtp)Foo+nFx = P(wp).

Thus a'O = a'âaO and we see that

Thus the character Xp is well defined. The continuity can be verified as inExercise 1 replacing Foo by Fp in the argument there.

§ 8 . 3 .1. Suppose that an : K —> T is a sequence of continuous homomorphismswith an(x) = \j/(ynx) for yn e K and that an converges locally uniformly toa. We want to show that yn converges to y e K given by oc(x) = \|/(yx) forall x E K. Since ccn converges to a, for any given m > 0, ocn-a has valuesin Ii on %'mOv for sufficiently large n and hence by Lemma 1, an = a on7i~m0v. If rcr0v is the different of \|/, then a n = a on 7i"mOv means thatyn-y e 7Cm"rOv and hence making m large, we know that yn converges to y inK. Conversely, if yn converges to y in K, then reversing the above argument,we know that c^ converges to a. Thus the isomorphism of Proposition 1 is alsothe topological isomorphism.

§ 8 . 5 .1. (ii) Note that a+py1 = a(l+a'1G31Ov) and thus by definition,


) if | a | v >a

Therefore, for any locally constant function (j) factoring through (Ov/pv1)x,

Any locally constant function on CVX is of the above form for sufficiently large i

and therefore the above formula is true for any locally constant functions on CVX-

The formula holds even on Fvx because Fv

x = Ui C51OVX.

(iii). By the above formula, | x | v"1dji(x) gives a multiplicative Haar measure.

Therefore JF (|)(ax) I x | y'MjiCx) = JF (|)(x) | x | v djuiCx) for any locally constant

function (J). Any locally constant function <|) can be written (|)(x) = f(x) | x | v for

another locally constant function f, and the correspondence <|> h-» f is bijective

on the space of locally constant functions. Replacing <|) by f(x)|x|v in the above

formula, we see that

I a I vJFvf(ax)dja(x) = JFyf(ax) | a | vdn(x) = JFvf(x)dî(x),which proves the assertion.

§8 .6 .

1. Using the fact that FAX/FX = F^ /F x xR > 0 via x h-> (xf( *~.Q]), | x | A)

(where R>o = {x e R | x > 0}), we can write ^(x) = Xi(x)X2(x) for charac-

ters Xi of F(A}/FX and X2 of R>0. Since F^/F* is compact (Theorem 1.1),

?ii must have values in T, which is the unique maximal compact subgroup of Cx

(in fact Cx = TxR>o). Any continuous linear map of R into C is given by

th->cct with a e C . By identifying R with R>o via "exp", any continuous

multiplicative map of R>o into Cx is given by xf-» xs with s e C. Thus

I X(x) I = I A,2(x) I = I x I As for some s e C.

2. Let % be as in the exercise. We see in the same manner as above thatI %(x) I = I x Is for some s e C . Thus by taking %/1 % I instead of %, we

may assume that % has values in T. We already know that Homcont(R>o,T) = R

(assigning a e R to the character x h-> x10t) and Homcont(T,T) = Z (assigning

n e Z to the character x h-> xn). Since Cx = TxR>o via x h-> (x/1 x | , | x | ) ,

we thus see from this that % has the desired form.


3. For any standard function Of on FA£, we can find a rational number N ^ 0

and real number M such that | Of(x) I < M | ^Ff(Nx) | for the characteristic

function Wf of Zf. Thus we may assume that O = OfOoo and that Of is the

characteristic function of Zf. We may also assume that Ooo(x) = xkexp(-7tx2)

for 0 < k E Z. Then for £ e Q, O(£) * 0 if and only if £ e Z. This

shows that 5^GQ ^C^) = £neZ nkexp(-7cn2), whose convergence is obvious

from the convergence of geometric series Z°° exp(-7tn) and its derivatives

£°° nkexp(-7tn).

§ 9 . 1 .2. Since % : Cl(m) = AF

x/FxU(m)Foo+ -* T, %v(Owx) = 1 if v is outside m

and %(FX) = 1. If e e O*, then EG OVX for all finite v. Thus

Therefore IIv | wZv(e) = TT TT; for all e e Ox. In particular, if

0 d , then d e Ox. Thus X* 0 d

hand, j([ j ^xo . ) i )k = . Hence we have

for

• On the other

B . .

Index

Adele 239ring 239adelic Fourier expansion 276, 277adelic modular form 273

Adjoint formula (Petersson innerproduct) 315Algebraic Petersson inner product222Algebraicity

algebraicity theorem of standardL-functions of GL(2) 186algebraicity theorem for Rankinproducts 324Shimura's algebraicity theorem310

Approximation theorem 274strong approximation theorem161

Arithmetic point 220Automorphic property 126

Banach A-module77Bernoulli

number 37, 38polynomial 42generalized Bernoulli number 43

Bialgebra 90Borel-Serre compactification 187Boundary exact sequence 112, 352,363Bounded p-adic measure 78, 120

Class group 7class number 7class number formula 63

Coboundary 347Cocycle 347

Cohomology group 345compcatly supported sheafcohomology group 360Cech cohomology group 353de Rham cohomology 359group cohomology 345parabolic cohomology group 347,351singular cohomology 345

Common eigenform 146Commutative group scheme 91Congruence subgroup 125Conjugate 137Constant sheaf 353Convolution product 199, 327Coset decomposition 139Cup product 170, 183Cusp 166

equivalence classes of cusps 148cusp form 126cuspidal condition 278irregular cusp 166regular cusp 166

Cech cohomology group 353

Decomposition group 24Dedekind

domain 11zeta function 54

Derivation 92Differential operator 92, 310

invariant differential operator 93Dimension formula 160, 164, 166Dirichlet

-Hasse (a theorem of) 244character 89

384 Index

unit theorem 15class number formula 66residue formula 65

Discriminant 10, 12Discriminant function 131Distributions 119

distribution relation 119Dominated convergence theorem 36,50Dual group 250Duality (a theorem of) 142, 150, 218

Eichler-Shimuraisomorphism 160(a theorem of) 168

Eisenstein series 54, 58, 125(functional equation of) 292

Euler product 41Euler's method 25Exact 1Extension functor 3Extension module 1Flabby sheaves 360Formal completion 96Formal group 96

multiplicative group 89Fourier expansion 125, 128

adelic Fourier expansion 276, 277Fourier expansion of f |T(n) 141

Fourier transform 112, 256Fractional ideal 5Free resolution 345Frobenius

conjugacy classe 24element 23

Functional equation 26, 38, 309of Eisenstein series 292of Hecke L-function 261of Rankin products 306

Gauss sum 45Geometric interpretation 181Group

decomposition group 24functor 94

scheme 92formal group 96idele group 241inertia group 23

Haar measure 254Hecke

(a theorem of) 70, 72algebra 141, 181character 54, 63L-function 54module 177operator 110, 139ring 176universal ordinary Hecke algebra218

Hilbert modularEisenstein series 71form 72

Hodgeoperator 175theory 171

Holomorphic projection 314Homogeneous

chain complex 349polynomials 107

Homological method 107Homotopy equivalence 345Hurwitz L-function 41

Ideal group 54Idele 240, 241

group 241Imaginary quadratic field 63Inertia group 23Interpolation series 73, 75Invariant differential operator 93Involution 144Irreducibility 165Irregular cusp 166Iwasawa theory 106

Index 385

A-adiccusp form 196Eisenstein series 198form 196forms of CM type 237ordinary form 208

Leopoldt conjecture 103L-function

abelian L-function 25Dirichlet L-function 40modular L-function 125Hurwitz L-function 41Shintani L-function 47

Lipschitz-Sylvester theorem 33Locally constant function 84, 99,118Locally constant sheaf 353Long exact sequence 2

Mahler (a theorem of) 77Manifold 345Mellin transform 186Minkowski's estimate 10Mobius function 286Modular

form of weight k 125symbol 120nearly holomorphic 314

Norm 6, 7Normalized

absolute value 242eigenform 187

Number field 5Numerical 73

polynomial 75, 93One variable interpolation 225Open covering 353Ordinary 189

A-adic form 208part 202projector 202

Orthogonality relation 45, 46

p-adicanalytic function 88exponential function 21family of modular forms 195integer ring 19Dirichlet L-function 87, 124Hecke L-function 105logarithm function 21measure 78meromorphic function 88residue formula 105standard L-function of GL(2)160, 189Mellin transform 160number 17

p-fraction part 239p-ordinary 189Pairing 169Parabolic cohomology group 347,351Period 187Petersson inner product 143Place 18Poisson summation formula 256Polyhedral cone 68Presheaf 352Prime 8

factorization 11Primitive 44, 317

form 317modulo 45

Product formula 242Pseudo-representation 232

Ramanujan's A -function 131, 234Rankin product 154Rapidly decreasing 297Rational structure 131Rationality of the Dirichlet L-values138Ray class group 54, 245Real quadratic field 54Regular cusp 166

386 Index

Relatively prime 8Representation

pseudo-representation 232residual representation 229

Represented 92Residue formula 228, 271, 305Restricted direct product 240Riemann

surface 345zeta- function 25

Semi-group 175Semi-simplicity 218Shapiro's lemma 182Sheaf 352

compactly supported sheafcohomology group 360cohomology 345cohomology group 360constant sheaf 353generated by 356locally constant sheaf 353presheaf 352

Shimura's algebraicity theorem 310Shintani L-function 47Siegel-Klingen (a theorem of) 57Siegel's estimate 10Simplicial

complex 348cone 67

Singular cohomology 345Slowly increasing 297Smooth manifold 352Space of modular form 126

Standardflabby resolution 360additive character 249function 261L-function 157,186L-functions of GL(2) 186R[F]-free resolution 346

Strict ray class group 10Strong approximation theorem 161Subdivision operator 362Symmetric n-fold tensor 169

Teichmiiller character 87Tensor product 3Theta series 235Three variable interpolation 332Topology of uniform convergence250Torsion functor 3Totally real 4Trace 6

map 111operator 182

Transformation equation 137Triangulation 362Two variable interpolation 227

Universal ordinary Hecke algebra218Unramified 228Upper half complex plane 58, 125Weierstrass preparation theorem 209Weil (a theorem of) 247

Documents

LONDON MATHEMATICAL SOCIETY STUDENT TEXTSpanchish/ETE LAMA 2018-AP/Haruzo... · 14 Combinatorial group theory: a topological approach, DANIEL E. COHEN 15 Presentations of groups,