Class field theory viewed as a Langlandscorrespondence

Milo Bogaard

June 30, 2010

Bachelorscriptie wiskunde

Begeleiding: dr. Ben Moonen

KdV Instituut voor wiskunde

Faculteit der Natuurwetenschappen, Wiskunde en Informatica

Universiteit van Amsterdam

AbstractIn this thesis we formulate the main results of class field theory forlocal fields and for number fields. This theory gives a descriptionof the Galois group of the maximal abelian extension. A corollaryof these theorems is the Langlands correspondence for n = 1.This is a bijection between 1-dimensional representations of theabsolute Galois group and certain 1-dimensional representationsof a topological group which is the multiplicative group in thelocal case and the idele class group in the number field case.In the number field case we attach a complex function, calledthe Artin L-series, to a representation and using this bijectionand theorems on representations of the idele class group we showthat the Artin L-series of a 1-dimensional representation has aholomorphic extension to the complex plane.For the first chapter we follow section 23 of [1] and for the secondchapter we follow chapter VII of [5]

GegevensTitel: Class field theory viewed as a Langlands correspondenceAuteur: Milo Bogaard, Milo.Bogaard@student.uva.nl, 5743117Begeleider: dr. Ben MoonenTweede beoordelaar: dr. Jochen HeinlothEinddatum: June 30, 2010

Korteweg de Vries Instituut voor WiskundeUniversiteit van AmsterdamScience Park 904, 1098 XH Amsterdamhttp://www.science.uva.nl/math

Contents

Introduction 2

1 Local theory 31.1 Local fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.1 Absolute values . . . . . . . . . . . . . . . . . . . . . . 31.1.2 Extensions of local fields . . . . . . . . . . . . . . . . . 5

1.2 Local class field theory . . . . . . . . . . . . . . . . . . . . . . 71.2.1 Frobenius automorphisms . . . . . . . . . . . . . . . . 71.2.2 The reciprocity homomorphism . . . . . . . . . . . . . 91.2.3 The reciprocity law . . . . . . . . . . . . . . . . . . . . 101.2.4 The Weil group . . . . . . . . . . . . . . . . . . . . . . 11

1.3 L-functions and ε-factors . . . . . . . . . . . . . . . . . . . . . 131.3.1 Characters . . . . . . . . . . . . . . . . . . . . . . . . . 131.3.2 The Haar-integral . . . . . . . . . . . . . . . . . . . . . 151.3.3 L-functions and ε-factors . . . . . . . . . . . . . . . . . 161.3.4 Calculation of the ε-factor . . . . . . . . . . . . . . . . 211.3.5 The local Langlands correspondence for GL(1) . . . . . 23

2 Global theory 252.1 Number fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.1.1 Absolute values on number fields . . . . . . . . . . . . 262.1.2 Some Galois theory . . . . . . . . . . . . . . . . . . . . 27

2.2 Galois representations . . . . . . . . . . . . . . . . . . . . . . 292.2.1 Artin L-series . . . . . . . . . . . . . . . . . . . . . . . 292.2.2 The conjectures of Dedekind and Artin . . . . . . . . . 35

2.3 The idele class group . . . . . . . . . . . . . . . . . . . . . . . 362.3.1 Construction . . . . . . . . . . . . . . . . . . . . . . . 362.3.2 Galois module structure . . . . . . . . . . . . . . . . . 38

2.4 Global class field theory . . . . . . . . . . . . . . . . . . . . . 402.5 Hecke characters . . . . . . . . . . . . . . . . . . . . . . . . . 402.6 The global correspondence . . . . . . . . . . . . . . . . . . . . 42

1

Introduction

In the first chapter of this thesis we consider class field theory for a local fieldK. This theory describes the maximal abelian extension Kab of K and itsGalois group G(Kab|K). Class field theory gives an isomorphism between adense subgroup of G(Kab|K), called the Weil group, and the multiplicativegroup of K. We then consider continuous 1-dimensional representations,which are called characters. The isomorphism between the Weil group andK∗ gives a bijection between the characters of the groups. This bijection iscalled the local Langlands correspondence for GL(1).

In the case of a number field F class field theory takes the form of an iso-morphism between G(F ab|F ) and a quotient of the unit group of the adelering. The adele ring is a topological ring specifically constructed for thispurpose. As characters of the absolute Galois group of a number field fac-tor though the abelianization this gives a description of the characters ofthe absolute Galois in terms of specific characters of the idele class group.For both types of representations we can define a complex function, in theGalois case called Artin L-series and in the idele case called Hecke L-series.We will see that this bijection respects the L-series. Thus we can deriveproperties of the Artin L-series from properties of Hecke L-series. Further-more we know that Artin L-functions have nice properties with respect torepresentations induced from subgroups, so we can use Brauer’s theorem oninduced representations to answer some questions about higher dimensionalrepresentations.

As a result for a Galois extension L|F of number fields of degree n weobtain an identity

ζL(s)n

ζF (s)n= h,

where ζL and ζF are the Dedekind zeta functions of L and F are h is someholomorphic function. This shows ζL(s)

ζF (s)is holomorphic, which was already

conjectured by Richard Dedekind.

2

Chapter 1

Local theory

The goal of this chapter is to relate 1-dimensional representations of the abso-lute Galois group of a local field to those of of the multiplicative group of thisfield. The link between these types of representations is given by local classfield theory which gives an isomorphism between the multiplicative groupof the field and the Weil group which is a dense subgroup of the absoluteGalois group. We will see that the Weil group has more representations andit is easy to see which ones come from representations of the absolute Galoisgroup.

1.1 Local fields

We recall some basic properties of local fields. For the proofs the reader is re-ferred to [5]. The notation introduced in this section will be used throughoutthe first chapter.

1.1.1 Absolute values

Local fields can be defined in several ways, the most convenient way is usingproperties of an absolute value.

Definition 1.1. An absolute value on a field K is a function | · | : K → Rsuch that for all x, y ∈ K

1. |x| ≥ 0 and |x| = 0⇔ x = 0,

2. |xy| = |x||y|,

3. |x+ y| ≤ |x|+ |y|.

3

An absolute value is called nonarchimedean if also

3’ |x+ y| ≤ max|x|, |y|.

It is called discrete if the image of K∗ in R>0 is a discrete subspace.

If K is a field with nonarchimedean absolute value | · | then we denote byoK the set x ∈ K : |x| ≤ 1, which is called the valuation ring of K. Theterminology is justified by the following proposition.

Proposition 1.2. The subset oK of K is a local ring, its group of units isUK := x ∈ K : |x| = 1 and its unique maximal ideal is p := x ∈ K : |x| <1. If | · | is discrete the ideal p is principal and all the ideals of oK have theform pn.

We make some conventions regarding notation. A generator of p willusually be denoted by π and is called a prime element. The quotient oK/p iscalled the residue field. The indices denoting the field will usually be droppedwhen no confusion is possible.

Now local fields can be defined.

Definition 1.3. A local field is a field that is complete for a discrete absolutevalue with a finite residue field.

From now on in this chapter K will be used to denote a local field.For n > 0 the subgroups 1 + pn of U are called the higher units and are

denoted by U (n). The group U is sometimes denoted by U (0). The followingresults on the structure of the multiplicative group of K are used later.

Proposition 1.4. With the notation as above, for n > 0 one has isomor-phisms of groups pn/pn+1 ∼= o/p and U (n)/U (n+1) ∼= o/p. Furthermore,U/U (n) ∼= (o/pn)∗.

Proposition 1.5. For a given choice π of prime element the map Z× U →K∗ given by (n, u) 7→ πnu is a isomorphism of topological groups if Z is giventhe discrete topology and U the subspace topology inherited from K.

For a given x = πnu ∈ K the number n ∈ Z is independent of the choiceof π and is denoted by vK(x). It is called the p-adic valuation of x.

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1.1.2 Extensions of local fields

We will use the notation G(k′|k) for the Galois group of an extension offields k′|k. For a finite extension of fields L|K the norm map NL|K : L→ Kis defined by NL|K(x) = det(x) where x is considered as a linear map L→ Lof K vector spaces given by y 7→ xy.

Let K be a local field and Ω an algebraic closure. The following theoremshows that finite extensions of local fields are also local fields.

Theorem 1.6. If K is a local field for the absolute value | · |K and L a finiteextension of degree n then the map | · |L : L→ R given by

|y|L = n

√|NL|K(y)|K

is an absolute value which makes L a local field. Furthermore any extensionof | · | to L which makes L a local field is equal to | · |L.

If b is the maximal ideal of the valuation ring of L then oL/b is a finiteextension of oK/p. Now theorem 1.6 and proposition 1.2 show that there existeL|K , fL|K ∈ N such that beL|K = poL and fL|K = [oL/b : oK/p]. The numbereL|K is called the ramification index and fL|K the inertia index. Again, insituations where no confusion is possible we omit the indices indicating thefield. The ramification and inertia index satisfy the following relations.

Proposition 1.7. If M |L|K are extensions of local fields one has

1. eM |K = eM |L · eL|K,

2. fM |K = fM |L · fL|K,

3. [L : K] = eL|K · fL|K.

A finite extension such that eL|K = 1 is called unramified and an infinitealgebraic extension is called unramified if all finite subextensions are unram-ified. Unramified extensions are important in local class field theory so weprove some facts here.

Proposition 1.8. If K is a local field with residue field κ and L an algebraicextension obtained from K by adjoining a primitive n-th root of unity ζ suchthat n is not divisible by char(κ) then

1. The extension L|K is unramified and of degree f , where f is the leastnatural number such that (#κ)f ≡ 1 mod n.

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2. The Galois group G(L|K) is canonically isomorphic to G(µ|κ) where µis the residue field of L.

3. The ring of integers of L is o[ζ] and po[ζ] is its maximal ideal.

Proof. (1) Let ζ ∈ L be a primitive n-th root of unity generating L and let fbe its minimum polynomial. Because f divides Xn− 1, Gauss’s lemma givesf ∈ o[X]. Now f ∈ κ[x] is irreducible, because if gh = f then g and h arerelatively prime because gh divides xn− 1 and xn− 1 has n distinct roots inan algebraic closure of κ. Now Hensels lemma shows that if g and h are notconstant f is not irreducible. If the identification κ = Fq is made, then theresidue field of L is the smallest extension Fqf of Fq containing the nth rootsof unity. As F∗

qfis a cyclic group of order qf − 1 the field Fqf contains there

roots if and only if n|qf − 1. Part (1) now follows as n|qf − 1 if and only ifqf ≡ 1 mod n.(2) As any element of G(L|K) maps the valuation ring and its maximal idealto themselves, it defines an element of G(µ|κ). It is clear that this assignmentis a homomorphism. As the maximal ideal is mapped to itself the action ofthe Galois group commutes with quotienting out the maximal ideal, so theaction of G(L|K) on the roots of f is the same as the action on the roots off . As distinct elements of G(L|K) act differently on the roots of f they alsoact differently on the roots of f , so this mapping is injective and as the proofof part (1) shows that #G(L|K) = #G(µ|κ) it is an isomorphism.(3) We know that the residue field of L is generated by ζ as an extensionof the residue field of K. This shows oL = oK [ζ] + poL as oK modules.Nakayama’s lemma now shows that oL = oK [ζ].

The compositum of all unramified extension of K is called the maximalunramified extension of K and is denoted by Knr. Any finite subextension ofKnr is unramified and Knr is not contained in a larger unramified extension.

Proposition 1.9. Let K be a local field contained in an algebraic closure Ωthen:

1. The extension Knr|K is obtained by adjoining to K all roots of unityζn with n prime to p.

2. There are canonical isomorphisms G(Knr|K) ∼= G(κ|κ) ∼= Z.

3. We have [L ∩Knr : K] = fL|K for every finite extension L|K.

4. The extension LKnr|L is the maximal unramified extension of L.

6

Proof. Let K ′ be the extension of K obtained by adjoining to K all rootsof unity ζn with n prime to p. We first prove that for any finite extensionL|K we have [L ∩K ′ : K] = fL|K . Let µ be the residue field of L and κ theresidue field of K. Then µ is generated by a primitive n-th roof of unity ζwith gcd(n, char(κ)) = 1. By Hensels lemma the polynomial xn − 1 factorsinto linear terms over L. Let ζ be the root of xn − 1 which is mapped toζ under the quotient map oL → µ and let f be its minimum polynomialover K. By proposition 1.8 the degree of f is equal to the degree of µ|κ.As K(ζ) ⊂ K ′ this shows [L ∩ K ′ : K] ≥ fL|K . As L ∩ K ′ is unramified[L ∩ K ′ : K] = fL∩K′|K , so part 2 of proposition proposition 1.7 shows[L∩K ′ : K] = fL|K . This implies a finite extension is unramified if and onlyif it is contained in K ′, therefore K ′ = Knr, which proves part (1) and part(3)

The infinite extensionK ′ can be constructed as a tower of finite extensions.The Galois group is then the projective limit of the Galois groups of thefinite extensions. This gives an isomorphism with the Galois group of someextension of Fq. By construction this extension contains all nth roots of unity

with order prime to p and therefore is Fq. The isomorphism G(Fq|Fq) ∼= Z iswell known.

The extension LKnr is exactly the extension of L obtained by adjoining toL all n-th roots of unity such that n is relatively prime to the characteristicof the residue field of K. As this is equal to the characteristic of the residuefield of L this follows by applying part (1) to L.

Note Knr not a local field as it contains a primitive root of unity for everyprime q 6= p and thus the residue field would contain such a root and thusbe infinite. Furthermore as K ′ is the splitting field of a set of polynomialsKnr is a normal extension of K.

1.2 Local class field theory

Now we give a brief overview of local class field theory. The goal of this theoryis to describe the abelian extensions of a local field and their Galois groups.We will use this description to understanding 1-dimensional representationsof arbitrary Galois groups, as these factor through an abelian quotient group,which is the Galois group of an abelian extension.

1.2.1 Frobenius automorphisms

Throughout this section L will be a finite Galois extension of a local field Kcontained in an algebraic closure Ω. Let G = G(Ω|K) be the absolute Galois

7

group of K. We identify Knr with the maximal unramified extension of Kin Ω and define dK : G→ G(Knr|K) to be the quotient map.

If κ is the residue field of K then the Frobenius ϕK over K is the elementof G(Knr|K) which corresponds to the map x 7→ x#κ in G(κ|κ) under theisomorphism in proposition 1.9. If L is unramified it is a subextension of Knr

by proposition 1.9 so we have a quotient map G(Knr|K) → G(L|K). Wedefine the Frobenius element ϕL|K of G(L|K) to be the image of ϕK underthis map. Finally, to a finite Galois extension L|K we attach a semigroupFrob(Lnr|K) which consists of all elements of G(Lnr|K) such that the re-striction to Knr is a positive integer power of ϕK . We note that dK factorsthrough G(Lnr|K), so by abuse of notation we will also denote the inducedmap by dK .

The following propositions give some basic properties of Frob(Lnr|K).

Proposition 1.10. For σ ∈ Frob(Lnr|K) with fixed field Σ one has:

1. [Σ : K] <∞,

2. Σnr = Lnr,

3. σ = ϕΣ.

Proof. (1) By definition d(σ) = ϕjK for some j ∈ N>0, which has fixed fieldΣ ∩Knr, so [Σ ∩Knr : K] = j < ∞. This shows that the inertia index of afinite subextension is bounded by j.As any finite subextension of Lnr over L is unramified the ramification indexof a finite subextension of Σ is bounded by the ramification index of L. Byproposition 1.7 this implies that the degree of any finite subextension of Σis bounded, thus [Σ : K] <∞.(3) By the previous point G(Σnr|Σ) ∼= Z, thus the canonical surjectionG(Lnr|Σ) → G(Σnr|Σ) gives a continuous and open surjection Z → Z whichis injective as 1 generates Z. By the fundamental theorem of Galois the-ory G(Lnr|Σnr) is isomorphic to the kernel of this map and thus trivial soLnr = Σnr.(4) Let τ ⊂ κ be the residue field of Σ. As d(σ) = ϕjK the map induced onκ is the map x 7→ xj·#κ. As j = [Σ ∩Knr : K] the residue field τ of Σ hasj ·#κ elements. Thus σ is the Frobenius of Σ.

Proposition 1.11. If L|K is a finite Galois extension and σ ∈ G(L|K) thenσ can be extended to an element of Frob(Lnr|K).

8

Proof. Let σ ∈ G(L|K) and let ϕ ∈ G(Lnr|K) be such that dK(ϕ) = ϕK .The restriction of σ to L ∩Knr is a power of the Frobenius automorphism,so σ|L∩Knr = ϕnL∩Knr for some n ∈ N. By Galois theory the map

G(Lnr|Knr) ∼= G(L|L ∩Knr),

given by τ 7→ τ |L is an isomorphism. Thus there is a τ ∈ G(Lnr|Knr) suchthat τ |L = σϕ−n|L. Now define σ = τϕn. Then by construction σ|L =τϕn|L = σϕ−nϕn = σ and dK(σ) = ϕn, so σ is the required lift.

1.2.2 The reciprocity homomorphism

For a finite Galois extension L|K we will now construct a map from G(L|K)to a quotient of K∗, called the reciprocity homomorphism. Note that as K∗ isabelian this map will factor trough the Galois group of an abelian extension.This construction is done in several steps. For the proofs at each step see [5].

For the infinite extension Lnr|K we define the subgroup NLnr|K(Lnr)∗ ofK∗ as the intersection of the subgroups NM |KM

∗, where M ranges over allfinite subextensions Lnr. First we define a map from the Frob(Lnr|K) to tothe group K∗/NLnr|K(Lnr)∗

Proposition 1.12. For a finite Galois extension of local fields L|K the map

rLnr|K : Frob(Lnr|K)→ K∗/NLnr|K(Lnr)∗

given byrLnr|K(σ) = NΣ|K(πΣ) mod NLnr|K(Lnr)∗,

where Σ is the fixed field of σ and πΣ is a prime element of Σ, is a welldefined multiplicative map.

By proposition 1.11 an element σ ∈ G(L|K) can be extended to an elementof G(Lnr|K), so we can state the following theorem.

Theorem 1.13. For a finite Galois extension of local fields L|K the map

rL|K : G(L|K)→ K∗/NL|KL∗

given byrL|K(σ) = NΣ|K(πΣ) mod NL|KL

∗,

where Σ is the fixed field of an extension σ of σ to Lnr and πΣ is a primeelement of Σ, is a well defined homomorphism.

In the case of an unramified extension the reciprocity homomorphism canbe given explicitly.

Proposition 1.14. If L|K is unramified then the reciprocity map is givenby rL|K(ϕL|K) = π mod NL|KL

∗, where ϕL|K is the Frobenius of L|K and πa prime element of K∗.In this case it is an isomorphism.

9

1.2.3 The reciprocity law

We can now state the main theorems of class field theory. The reader isreferred to [5] for the proofs.

Theorem 1.15. For finite Galois extensions L|K the reciprocity homomor-phism induces an isomorphism G(L|K)ab → K∗/NL|KL

∗.

As G(L|K)ab → K∗/NL|KL∗ is an isomorphism the map K∗ → G(L|K)ab

obtained by composing the quotient map with the inverse of rL|K is a surjec-tive homomorphism with kernel NL|KL

∗. This map is called the norm residuesymbol and is denoted by ( , L|K).

The reciprocity map satisfies several functorial relations. To describe themwe need a group theoretic notion, the transfer homomorphism. For a sub-group H of G of finite index define a map Ver : Gab → Hab as follows. Firstlet R be a set of representatives of the left cosets of H. For every x ∈ G andr ∈ R there is a unique xr in H such that xr = r′xr with r′ ∈ R. We defineVer(x) =

∏r∈R xr for x ∈ Gab. It is not very difficult to verify that this does

not depend on choice of representative x or coset representatives R and thatVer is a homomorphism.

Note that the next theorem could also be given in terms of the reciprocityhomomorphism.

Theorem 1.16. Let L|K and L′|K ′ be finite Galois extensions such thatK ⊂ K ′ and L ⊂ L′. The following diagrams are commutative:

1.

(K ′)∗( ,L′|K′)//

NK′|K

G(L′|K ′)ab

K∗( ,L|K) // G(L|K)ab,

where G(L′|K ′)ab → G(L|K)ab is given by restriction.

2. Let also σ ∈ G and the map G(L|K)ab → G(σ(L)|σ(K))ab be given byconjugation then

K∗( ,L|K) //

G(L|K)ab

σ(K∗)

( ,σ(L)|σ(K))// G(σ(L)|σ(K))ab)

is commutative.

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3. If M |L|K are Galois extensions then

L∗( ,M |L) // G(M |L)

K∗( ,M |K)//

OO

G(M |K)ab

Ver

OO

is commutative.

The following theorem is called the existence theorem as the hard part ofthe proof is to show that for a given open subgroup of K∗ there is an abelianextension whose group of norms is exactly this subgroup.

Theorem 1.17. The map L 7→ NL|KL∗ gives an inclusion reversing bijection

between finite extensions of K and open subgroups of K∗ of finite index, whichsatisfies the following rules

L1 ⊆ L2 ⇔ NL1|KL∗1 ⊇ NL2|KL

∗2,

NL1L2|K(L1L2)∗ = NL1|KL∗1 ∩NL2|KL

∗2

andNL1∩L2|K(L1 ∩ L2)∗ = NL1|KL

∗1NL2|KL

∗2.

1.2.4 The Weil group

The results of the previous section can also be stated in a slightly differentway. We start by noting that for all finite algebraic extensions L and M ofK the diagram

K∗( ,L∩M |K)//

G(L ∩M |K)ab

K∗( ,M |K) // G(M |K)ab.

is commutative by theorem 1.16 so for all a ∈ K∗ the automorphisms (a, L|K)and (a,M |K) agree on L∩M . Thus the map (a,Kab|K) defined by (a,Kab|K)(x) =(a,K(x)|K)(x) is an automorphism of Kab and the map ( , Kab|K) : K∗ →G(Kab|K) is a homomorphism.

To formulate the reciprocity law in terms of the map ( , Kab|K) we nowneed the Weil group.

Definition 1.18. The Weil group WK of a local field K is the inverse imageof Z under the map dK : G→ G(Knr|K) ∼= Z.

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Let IK denote the kernel of dK . This group is called the inertia group andconsists of all elements of G which act trivially on Knr. By definition WK =tn∈ZψnIK , where ψn is some element of d−1

K (ϕnK). This gives a topology onthe Weil group by defining a subset to be open if and only if the intersectionswith ψnIK are open for every n ∈ Z. This topology is finer than the subspacetopology inherited from G, because if U is open in G then U ∩ IK is open inIK so U ∩WK is open in WK .

Proposition 1.19. The map ( , Kab|K) gives an isomorphism of topologicalgroups K∗ → W ab

K and image of the unit group is the inertia group.

Proof. Let a = πnu then (a,Kab|K)|Knr = (a,Knr|K). Restricting to afinite subextension L gives (a,Knr|K)|L = (a, L|K). As L is unramified prop1.14 gives (a, L|K) = ϕnL|K = (ϕK)|nL where ϕL|K and ϕK are the Frobenius

elements. This implies (a,Knr|K) = ϕnK , thus the image of ( , Kab|K) isin W ab

K . The same argument shows that a is mapped to the inertia group ifand only if a is a unit.

If (x,Kab|K) is the identity map then for every abelian extension L wehave x ∈ NL|KL

∗, so by theorem 1.17 we have x = 1 as the open subgroupsform a neighborhood basis of 1.

If π ∈ K∗ is a prime element we have (π,Knr|K) = ϕK , so the imageof π in W ab

K , which we call ϕ′K , is in d−1K (ϕK). There are isomorphisms

K∗ → (π)× UK and WK → (ϕ′K)× IK , the first is given by proposition 1.5and the second is given by the fact that the exact sequence 0→ IK → WK →Z→ 0 splits and the topology on (ϕ′K) is discrete by definition. This gives asequence

(π)× UK( ,Kab|K)−→ (ϕ′K)× IK

d−→ Z

where the maps are the identity on the Z components and the last mapis projection on the first coordinate. It is thus sufficient to show that

UK( ,Kab|K)−→ IK is a isomorphism of topological groups. Theorem 1.17 shows

shows that the subgroups of finite index in UK correspond with those of IKand by the reciprocity theorem the quotients must be isomorphic. Thus forany open subgroup H of finite index in UK there is an exact sequence

0→ UK/H → IK/(H,Kab|K)→ 0.

Now proposition 2.7 of chapter IV of [5] gives lim← UK/H ∼= lim←K/(H,Kab|K).

As both UK and IK are profinite and H respective (H,Kab|K) range over allopen subgroups of finite index this is an isomorphism UK ∼= IK .

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1.3 L-functions and ε-factors

The reciprocity isomorphism can be used to get information about 1-dimensionalrepresentations of the Weil group as it is in principle simpler to analyze rep-resentations of K∗ than representations of the Weil group. This is becausethe group K∗ is the multiplicative group of a normed field which makes itpossible to apply techniques from analysis.

1.3.1 Characters

For technical reason we will also be interested in 1-dimensional representa-tions of K as additive group so the following definition is useful.

Definition 1.20. A locally profinite group is a topological group G suchthat every open neighborhood of 1 contains a compact open subgroup.

The following lemma shows that this concept generalizes K and K∗.

Lemma 1.21. For a local field K the additive group K and multiplicativegroup K∗ are locally profinite.

Proof. The valuation ring o is compact and open in K. For K note thatpn = x ∈ K : |x| < q−n, so the pn for n ∈ Z form a neighborhood basisof 0 of compact subgroups. For K∗ note that the 1 + pn for n ≥ 1 arecompact open in K and also in K∗ as they do not contain 0. As the pn forma neighborhood basis of 0 the 1 + pn form a basis of neighborhoods for 1 inK∗.

Now we define characters, which are essentially the same as 1-dimensionalrepresentations as C∗ = GL(1,C).

Definition 1.22. A character for a locally profinite group G is a continuoushomomorphism G → C∗. A character is unitary if its image is in S1. Thecharacters form a group for multiplication, denoted by G.

The following characterization is sometimes useful.

Proposition 1.23. For a homomorphism ψ : G→ C∗ equivalent are

1. ψ is a character;

2. ker(ψ) is open.

If ψ is a character and G is the union of its compact open subgroups theimage of ψ is contained in the unit circle.

13

Proof. If ψ−1(1) is open and g ∈ G then gψ−1(1) = ψ−1(ψ(g)), so all inverseimages are open, in particular those of the open sets. Conversely let ψ becontinuous and U an open neighborhood of 1 ∈ C∗, then ψ−1(U) is openand thus contains a compact open subgroup H. If U is small it contains nonontrivial subgroups C∗ so then H ⊂ ker(ψ), so ker(ψ) = ∪g∈ker(ψ)gH, whichis open. The second assertion follows as the image of a compact subgroup isin the unit circle.

If K is a local field then the subgroups pn for n ∈ N>0 form a neighborhoodbasis of 0 so the kernel of each character ψ of the additive group of K mustcontain such a group. The level of ψ is defined to be the smallest positiveinteger n such that pn ⊂ ker(ψ). For the multiplicative group the subgroups1 + pn form form a neighborhood basis of the unit element a basis so fora character χ the level is defined to be the smallest nonnegative integersuch that 1 + pn+1 ⊂ ker(χ). A character χ of K∗ is called unramified ifUK ⊂ ker(χ) or equivalently if it factors trough the valuation homomorphism.

The following proposition shows that the character group of K is fullydetermined by any nontrivial character. The construction of such an objectcan be found in [2].

Proposition 1.24. Let ψ ∈ K be of level d > 0 then:

1. For a ∈ K the map x 7→ ψ(ax), denoted by aψ, is in K. If a 6= 0 thenaψ ∈ K has level d− vK(a).

2. The map a 7→ aψ is a group isomorphism K ∼= K.

Proof. The map x 7→ ψ(ax) is a continuous homomorphism, so a character.If p is generated by π then pd ⊂ ker(ψ) is equivalent to πd ∈ ker(ψ) sothe level of aψ is d − vK(a). The map a 7→ aψ clearly is an injective grouphomomorphism. Let θ ∈ K be a character of level l. The character πd−lψ haslevel l so agrees with θ on pl. These characters define a character of pl−1/pl.For u, u′ ∈ UK the characters uπd−lψ and u′πd−lψ agree on pl−1/pl if and onlyif they agree mod p, thus there is a u1 ∈ UK such that if u1π

d−lψ = θ on pl−1.Continuing inductively there are uj ∈ UK such that ujπ

d−lψ = θ on pl−j. Asuj ≡ uk mod pk this defines an element u ∈ UK such that uπd−lψ = θ.

Now we give some examples of characters of Q∗p. Note that these can begeneralized to arbitrary local fields of characteristic 0.

Example 1.25. The map Q∗p → C∗ given by x 7→ |x|p is clearly a characterand is unramified.

14

To find a ramified character consider the maps Q∗p → UQp given by x 7→ x|x|

and the quotient map UQp → UQp/1 + p which are continuous homomor-phisms. As UQp → UQp/1 + p ∼= (Z/pZ)∗ any 1 dimensional representationof (Z/pZ)∗ gives by composition a character of Q∗p.

1.3.2 The Haar-integral

For simplicity from now on G is assumed to be abelian, as this is the onlycase needed. The proofs are slightly easier in the abelian case as there is nodistinction between right and left measures. The reader familiar with Haarmeasures can safely skip this section. What is defined here is not actually aHaar measure but something simpler which integrates only functions of thefollowing type.

Definition 1.26. For a locally profinite group G let C∞c (G) be the C-vectorspace of functions Φ : G → C which are locally constant and have compactsupport.

We have:

Lemma 1.27. For Φ : G→ C equivalent are:

1. Φ ∈ C∞c (G);

2. Φ is a finite linear combinations of characteristic functions of cosetsgH for some open compact subgroup H.

Proof. (2)⇒(1) is clear.(1)⇒(2). For every x in the support of Φ there is a compact open subgroupHx such that Φ is constant on xHx. By compactness finitely many of these,call them H1, ..., Hn, cover the support, so the support is a union of cosetsof H1 ∩ ... ∩ Hn. By compactness Φ can only be nonzero on finitely manycosets, so it is a linear combination of characteristic functions of the cosetsof H1 ∩ ... ∩Hn.

Lemma 1.28. The space C∞c (K) is spanned by the characteristic functionsof the sets a+ pn for a ∈ K and n ∈ Z.

Proof. As the subgroups pn for n > 0 form a neighborhood basis of 0 in Kevery compact open subgroup is a finite union of cosets of some pn, so thisfollows from lemma 1.27.

Definition 1.29. A Haar integral on G is a non-zero translation invariantlinear functional I : C∞c (G)→ C such that I(f) ∈ R≥0 if f(g) ∈ R≥0 for allg ∈ G.

15

A Haar integral only allows us to calculate integrals of functions in C∞c (G),but this is sufficient for our purposes.

Proposition 1.30. Up to multiplication with a constant G has a uniqueHaar integral.

Proof. Let Hii∈I be the directed system of compact open subgroups of Gand let X be the set of translates of the Hi and their finite unions.

Claim. There exists a finitely additive and translation invariant functionm : X → R. This function is unique up to a constant.

Proof of claim: Let m(Hi) := c for some i ∈ I and c ∈ R. By additivitythe function m has to take the value c

[Hi:Hi∩Hj ] on Hi ∩Hj for any j ∈ I and

by translation invariance the valuec·[Hj :Hi∩Hj ][Hi:Hi∩Hj ] on Hj. Defining m(Hj) to be

this value and defining the other values on X by additivity and translationinvariance it is clear that m is a finitely additive and translation invariantfunction m : X → R. As noted for given c the function m is determined onall compact open subgroups and therefore determined uniquely on X, so mis unique up to choice of c, which proves the claim.

Let f ∈ C∞c (G), then by lemma 1.27 there is a compact open subgroupH and elements gi ∈ G, for 1 ≤ i ≤ n such that the support of Φ is theunion of the cosets giH. Defining

∫Gfdm =

∑ni=0 f(gi) ·m(H) gives a Haar

integral. This integral is unique up to a constant as is gives a function mby integration the indicator functions and this function is unique up to aconstant.

1.3.3 L-functions and ε-factors

In this section we define two invariants attached to a character of K∗. Thefirst is the simple looking L-function.

Definition 1.31. Let χ be a character of K∗ and s a complex variable thenwe define the L-function L(χ, s) by

L(χ, s) =

11−χ(π)q−s

if χ is unramified

1 otherwise

Note that this is well defined as χ(π) does not depend on the choice ofπ and that an unramified character is determined by its L-function as itis defined by its value at π. For the moment this is all that is necessaryconcerning the L-function. We will see it again when we look at L-functionsin the global case.

16

The second invariant will be found by comparing a zeta functions withthe zeta function obtained by Fourier transforming the arguments. To definethe zeta function first some work needs to be done.

Definition 1.32. Let ψ be a nontrivial character of K and let Φ ∈ C∞c (F ).The Fourier transform Φ relative to ψ is the function defined by

Φ(x) =

∫F

Φ(y)ψ(xy)dµ(y).

Note that for fixed x the function y 7→ Φ(y)ψ(xy) has the same supportas Φ and is locally constant, so the integral is defined.

If we fix Haar measures µ on K and µ∗ on K∗ we have:

Proposition 1.33. Let Φ ∈ C∞c (K) then:

1. The function Φ is in C∞c (K).

2. There is a constant c(ψ, µ) ∈ R>0 such that for all Φ ∈ C∞c (K) andx ∈ K

ˆΦ(x) = cΦ(−x).

3. For a given ψ there is a unique measure µψ such that c(ψ, µψ) = 1.This measure satisfies µψ(o) = ql/2 where l is the level of ψ.

4. For a ∈ K∗ one has µaψ = |a|1/2µψ .

Proof. By linearity of the Fourier transform it suffices to check the statementsfor a basis of C∞c (K). Let l be the level of ψ and Φj the characteristic functionof pj. Suppose vK(x) < l− j, then ψ(xy), as a character of pj, factors troughpj/pl−vK(x) and is nontrivial on this group so∫

F

Φ(y)ψ(xy)dµ(y) = µ(pl−vK(x))∑

y∈pj/pl−vK (x)

ψ(xy).

This sum is the inner product of the character ψ with the trivial characterso by theorem 2.3 of [7] it is zero. If on the other hand vK(x) ≥ l − jthen ψ(xy) is the trivial character on pj, so for x ∈ pl−j the value of Φj isµ(pl−j) = µ(o)ql−j, where q is the characteristic of the residue field. Applying

this twice givesˆΦj(x) = µ(o)2q−lΦj(−x), thus points (1) and (2) hold for Φj.

For a translate of Φj, a function x 7→ Φj(x−a) the Fourier transform becomes∫F

Φj(y − a)ψ(xy)dµ(y) =

∫F

Φj(y)ψ(x(y + a))dµ(y + a) = aψ(x)Φ(x).

17

Again applying the Fourier transform gives∫F

ψ(ay)Φ(y)ψ(xy)dµ(y) =

∫F

Φ(y)ψ((x+ a)y)dµ(y) =ˆΦ(x+ a),

so the translates also satisfy the (2) and as these form a basis this proves (2)for C∞c (K) with c = µ(o)2q−j. Point (3) requires µ(o)2q−j and as c(ψ, bµ) =b2c(ψ, µ) putting µ(o) = ql/2 suffices. Point (4) follows directly from (3) andproposition 1.24

The unique measure in 3 is called the self dual measure for ψ.Let em be the characteristic function for pn − pn+1. Since p is compact

open and pn+1 is a compact open subgroup of finite index em is of compactsupport. It follows that for Φ ∈ C∞c (K) the function Φm = emΦ is in C∞c (K∗)as 0 is not in the support.

Definition 1.34. Let µ∗ be a Haar measure on K∗, χ a character of K∗,Φ ∈ C∞c (K∗) and q the order of the residue field. For a complex variable sdefine a function

ζ(Φ, χ, s) =∑m∈Z

zm(q−s)m,

where zm =∫F ∗

Φm(x)χ(x)dµ∗(x).

As a compact subset of K is bounded the support of Φ it is contained inpk for some k ∈ Z. For m < k the function emΦ is constant zero so thenzm = 0. This shows that ζ(Φ, χ, s) is a formal power series in q−s. Thefollowing lemma will show that it is also a rational function in q−s.

For a ∈ K∗ and Φ ∈ C∞c (K) denote by aΦ the function x 7→ Φ(a−1x).The relation for the coefficients of the zeta function is

zm(aΦ, χ) =

∫πmUK

Φ(a−1x)χ(x)dµ∗(x)

= χ(a)

∫πm−vK (a)UK

Φ(x)χ(x)dµ∗(x)

= χ(a)zm−vK(a)(Φ, χ),

so ζ(aΦ, χ, s) = χ(a)XvK(a)ζ(Φ, χ, s).Denote Z(χ, s) = ζ(Φ, χ, s) : Φ ∈ C∞c (K∗).

Lemma 1.35. If χ is a character of K∗ then

Z(χ, s) = L(χ, s)C[q−s, qs].

18

Proof. If Φ(0) = 0 the restriction of Φ to K∗ is in C∞c (K∗) so then onlyfinitely many terms are nonzero and ζ(Φ, χ, s) ∈ C[q−s, qs]. As χ(x)|x|−sis locally constant choosing Φ to be a sufficiently small neighborhood of 1implies ζ(Φ, χ, s) ∈ C. Thus by linearity of the map Φ 7→ ζ(Φ, χ, s) we haveζ(Φ, χ, s) : Φ ∈ C∞c (K∗) = C[q−s, qs], where C∞c (K∗) is identified witha subset of C∞c (K) by defining the value in 0 to be 0. Now let Φ0 be thecharacteristic function of o then

ζ(Φ0, χ, s) =∑m≥0

χ(πm)(q−s)m∫UK

χ(x)dµ∗(x).

If χ is unramified∫UK

χ(x)dµ∗(x) = µ∗(UK) so

ζ(Φ0, χ, s) = µ∗(UK)∑m≥0

χ(π)m(q−s)m = µ∗(UK)L(χ, s)

as the power series is a geometric series for sufficiently large s. If χ is notunramified χ|UK is a character of a compact group and so the kernel is openand χ factors through a finite abelian group G, so∫

UK

χ(x)dµ∗(x) = µ∗(ker(χ|UK ))∑g∈G

χ(g) = 0.

As Φ0 and C∞c (K)∗ span C∞c (K) this proves the lemma.

The following lemma is the main part of the proof of the next proposition.

Lemma 1.36. Let Λ be set of all maps λ : C∞c (F ) → C(q−s) such that forall Φ ∈ C∞c (F ) and a ∈ F ∗ we have

λ(aΦ) = χ(a)XvK(a)λ(Φ).

Then Λ is a vector space of dimension 1 over C(q−s).

Proof. As the map defined by Φ 7→ ζ(Φ, χ, s) is in Λ it is at least 1 di-mensional. Now choose n ≥ 0 such that Un

K ⊂ ker(χ) and let Φk be thecharacteristic function of Uk

K . The map Λ → C(q−s) given by λ 7→ λ(Φn)is linear. To prove the lemma it is sufficient to prove it is injective. Sup-pose λ(Φn) = 0 for some λ ∈ Λ. Then for k ≥ n and a ∈ Un

K we haveλ(Φk) = qn−kλ(Φn) = 0. Every element of C∞c (K∗) is a linear combinationof these elements so it is sufficient to prove this for Φ which are the character-istic function of a neighborhood of 0. For such a Φ the function Φ(ax)−Φ(x)is in C∞c (K∗) and λ(Φ(ax)−Φ(x)) = 0, so Φ(ax) = Φ(x) for all a ∈ K∗. Foran a such that χ(a) 6= 1 this implies Φ = 0.

19

Note that ζ depends up to a constant on the choice of µ∗ but the invariantdefined in the next proposition does not. If χ is a character let χ denote thedual character.

Proposition 1.37. Let ψ be a nontrivial character of K. For Φ ∈ C∞c (K)let Φ denote the Fourier transform of Φ for the Haar measure on K which isself dual with respect to ψ. If χ is a character of K∗ then there is a uniquec(χ, ψ, q−s) ∈ C(q−s) such that

ζ(Φ, χ, 1− s) = c(χ, ψ, q−s)ζ(Φ, χ, s)

for all Φ ∈ C∞c (K).

Proof. The map Φ 7→ ζ(Φ, χ, s) is in Λ so by lemma 1.36 it suffices to showthat the map Φ 7→ ζ(Φ, χ, 1− s) is in Λ. We have

aΦ(x) =

∫F

Φ(ay)ψ(xy)dµ(x)

=

∫F

Φ(y)ψ(a−1xy)dµ(a−1x)

= |a| · (a−1Φ)(x),

so by translation invariance of µ∗

ζ(Φ, χ, 1− s) =

∫F ∗|a|Φ(a−1y)χ−1(x)dµ∗(x)

= |a|χ−1(a)

∫F ∗

Φ(y)χ−1(ax)dµ∗(ax)

= |a|χ−1

∫F ∗

Φ(y)χ−1(x)dµ∗(x) = |a|χ−1ζ(Φ, χ, 1− s),

as required.

Definition 1.38. For a character ′χ of K∗ the ε-factor is defined by

ε(χ, s, ψ) = c(χ, ψ, q−s)L(χ, s)

L(χ, 1− s),

where ψ is a nontrivial character of K and s is a complex variable.

20

1.3.4 Calculation of the ε-factor

For this section character ψ of K is fixed and the measure on K is taken tobe self dual for ψ.

Equation 1.1 is called Tate’s local functional equation.

Theorem 1.39. The function ε(χ, s, ψ) satisfies the equation

ε(χ, s, ψ)ε(χ, 1− s, ψ) = χ(−1)

and it has the form

ε(χ, s, ψ) = q( 12−s)n(χ,ψ)ε(χ,

1

2, ψ),

for some n(χ, ψ) ∈ Z.

Proof. By proposition 1.33 we haveˆΦ(x) = Φ(−x), so the coefficients of the

function ζ(ˆΦ, χ, s) are

zm(ˆΦ, χ) =

∫F ∗

ˆΦ(x)χ(x)dµ∗(x) =

∫F ∗

Φ(−x)χ(−x)dµ∗(x) = χ(−1)zm(Φ, χ),

thus ζ(ˆΦ, χ, s) = χ(−1)ζ(Φ, χ, s). Twice applying prop 1.37 and the defini-

tion of the ε-factor gives

ζ(ˆΦ, χ, s) = ε(χ, s, ψ)ε(χ, 1− s, ψ)ζ(Φ, χ, s).

This proves the functional equation for the ε-factor.By proposition 1.37

ζ(Φ, χ−1, 1− s)L(χ−1, 1− s)

= ε(χ, s, ψ)ζ(Φ, χ, s)

L(χ, s)(1.1)

and by proposition 1.35 ζ(Φ,χ−1,1−s)L(χ−1,1−s) ∈ C[qs, q−s]. The quotient ζ(Φ,χ,s)

L(χ,s)can

be made constant by choice of Φ as in the proof of proposition 1.35: If χ isunramified choose Φ such that ζ(Φ, χ, s) = 1 and if χ is ramified choosing Φcharacteristic on o gives ζ(Φ, χ, s) = L(χ, s). This proves ε(χ, s, ψ),ε(χ, 1 −s, ψ) ∈ C[q−s, q−s]. By the functional equation the ε-factor is a unit in thering C[qs, q−s]. All units of this ring are of the form aqns for a ∈ C∗ and

n ∈ Z and thus ε(χ, 12, ψ) = aq

n2 and aqns = qn−( 1

2)nε(χ, 1

2, ψ), thus choosing

n(χ, ψ) = −n suffices.

21

It is also possible to give an explicit formula for the ε-factor, the rest ofthis section will be used to derive it.

Lemma 1.40. For a ∈ K∗ one has ε(χ, s, aϕ) = χ(a)|a|s− 12 ε(χ, s, ϕ).

Proof. The function ζ(Φ, χ, 1− s) change by a factor χ(a−1)|a| 12−s as for theFourier transform one has Φaψ(x) =

∫K

Φ(y)ψ(axy)dµaψ(y) = |a|−1/2a−1Φψ(x),

so by the formula ζ(aΦ, χ, s) = χ(a)XvK(a)ζ(Φ, χ, s) and the definition of εthe lemma holds.

Proposition 1.41. If χ is unramified and ψ has level one

ε(χ, s, ψ) = qs−12χ(π)−1.

Proof. The proof of lemma 1.35 shows that if Φ is the characteristic functionof o one has ζ(Φ, χ, s) = L(χ, s). On the other side Φ(x) = q1/2Φ(π−1x), asin the proof of proposition 1.33, so

ζ(Φ, χ, s) = q1/2

∫K∗

Φ(πx)χ(x)−1|x|sd∗x

= q1/2−sχ(π)−1

∫K∗

Φ(x)χ(x)−1|x|sd∗x

= q1/2−sχ(π)−1L(χ−1, s),

so ε(χ, s, ψ) = ζ(Φ,χ,s)L(χ−1,1−s) = qs−

12χ(π)−1.

Theorem 1.42. Let χ be a ramified character of level n ≥ 0, let ψ ∈ K beof level one, and a ∈ K∗ such that vK(a) = −n, then

ε(χ, s, ψ) = q−(ns− 12

)∑r∈R

χ(ax)ψ(ax),

where R ranges over a system of coset representatives of UK/Un+1K .

Proof. It is sufficient to calculate this for one Φ. Choosing the indicatorfunction on Un+1

K then χΦ = Φ so ζ(Φ, χ, s) =∫F ∗

Φ(x)χ(x)|x|dµ∗(x) =

µ∗(Un+1K ). As in proposition 1.33 Φ(y) = q

12−n−1ψ(y) for y ∈ p−n and zero

elsewhere. This shows zm(Φ, χ) = 0 for m < −n. For m > −n+ 1 choose anz ∈ pn such that χ(1 + z) 6= 1, this is possible as the level of χ is n. Now

zm(Φ, χ) = q12−n−1

∫pm−pm+1

ψ(y)χ(y)dµ∗(y).

22

by translation invariance this integral equals∫pm−pm+1

ψ(y(1 + z))χ(y(1 + z))dµ∗(y) = χ(1 + z)

∫pm−pm+1

ψ(y)χ(y)dµ∗(y)

as yz ∈ p. As χ(1 + z) 6= 1 here zm vanishes. Calculating z−n gives

q12−n−1

∫p−n−p−n+1 ψ(x)χ(x)dµ∗(x). Considering the levels of ψ and χ the

integrand is constant on cosets of Un+1K , so

ζ(Φ, χ, 1− s) = (q−s)−nµ∗(Un+1K )q

12−n−1

∑r∈R

χ(ax)ψ(ax).

The formula for ε(χ, s, ψ) follows from equation 1.1.

1.3.5 The local Langlands correspondence for GL(1)

The second theorem in this section is a simple consequence of the class fieldtheory of the previous chapter. First of all we like to know that that the in-variants defined so far completely determine the characters. The formulationof the theorem will, considering the proof, be strange, as not all assump-tions are used. It turn out that this form is the correct form for the higherdimensional cases, so it is formulated in that fashion.

Theorem 1.43 (converse theorem). If θ1 and θ2 are characters of K∗ suchthat for some nontrivial character ψ of K we have L(χθ1, s) = L(χθ2, s) andε(χθ1, s, ψ) = ε(χθ2, s, ψ) for all characters χ of K∗ then θ1 = θ2.

Proof. Let θ1 and θ2 satisfy the assumptions of the theorem, then L(θ1θ−12 , s) =

L(θ2θ−12 , s) = 1

1−q−s , so in particular θ1θ−12 is unramified. If for given a, b ∈ C

we have that 11−aq−s = 1

1−bq−s for all s ∈ C then a = b, so θ2θ−12 (π) = 1. As

unramified characters are determined by χ(π) this proves θ1θ−12 is the trivial

character, so θ1 = θ2.

Let WK be the Weil group of K and let r : WK → K∗ be the reciprocitymap. As the reciprocity map induces an isomorphism W ab

K∼= K∗ each char-

acter ρ of WK factors through K∗, so ρ = χ r for a unique character χ ofK∗. Thus for a character ρ = χ r of WK we define

L(ρ, s) = L(χ, s), ε(ρ, s, ϕ) = ε(χ, s, ϕ).

Let ch(G) be the set of characters of a group G. The preceding discussioncan be summarized by the following theorem.

23

Theorem 1.44. For ψ ∈ K nontrivial there is a unique bijection

β : ch(WK)→ ch(K∗)

such that L(χβ(ρ), s) = L(χ⊗ ρ, s) and ε(χβ(ρ), s, ψ) = ε(χ⊗ ρ, s, ψ) for allρ ∈ ch(WK) and χ ∈ ch(K∗).

If χ is a character of G(K|K) then χ|WKis a character of the Weil group

and χ is uniquely determined by χ|WKas WK is dense in G(K|K). The

characters of WK which are the restriction of a character of a character ofG(K|K) are those with finite image so we have:

Corollary. The map χ 7→ β(χ|WK) is a bijection between ch(G(K|K)) and

the unitary characters of K∗.

24

Chapter 2

Global theory

The purpose of this chapter is to give a bijection similar to the one givenin section 1.3.5, but now for number fields. It will again rely on class fieldtheory. In the number field case class field theory gives, for a number field F ,an isomorphism between G(F ab|F ) and a quotient of the idele class group,which is a topological group specifically constructed for this purpose. Thisisomorphism will give a bijection between the characters. On both sides of thebijection we can define L-functions which are infinite products of L-functionssimilar to those considered in section 1.3.3.

As the absolute Galois group of a number field is a profinite group anycontinuous representation factors through the Galois group of a finite exten-sion of F . Thus we only need to consider representations of Galois groups offinite extensions of number fields.

Some of the L-functions for the characters of the absolute Galois groupare familiar, namely the Dedekind zeta functions. Following [7] we give theproof of the Aramata-Brauer theorem which solves part of an old conjectureof Dedekind.

2.1 Number fields

We will of course needs some results from number theory, mainly concerningabsolute values and the relation between Galois groups and ramification.Throughout this chapter F will be a number field, meaning a finite extensionof the rational numbers.

Proofs of the results stated can be found in [5].

25

2.1.1 Absolute values on number fields

Recall the basics of absolute values as defined in section 1.1.1. We now haveto introduce some new terminology. Two absolute values | · |1 and | · |2 on Fare called equivalent if there is an r ∈ R>0 such that |x|r1 = |x|2 for all x ∈ F .An equivalence class of nontrivial absolute values is called a place of F . If Lis an extension of F and p′ is a place of L then we say p′ lies over a place p

of F if the restriction of the absolute values in p′ are in p.The equivalence classes of nonarchimedean absolute values are called fi-

nite, the others infinite. Write p|∞ if p is an infinite place and p -∞ if p isa finite place. If | · | is an absolute value which belongs to a place p then thecompletion of F with respect to | · | only depends on p and is denoted by Fp.

For the Archimedean places we have:

Theorem 2.1. If a field k is complete with respect to an archimedean valu-ation | · |k then there is an isomorphism σ from k to either R or C and ans ∈ (0, 1] satisfying

|x|k = |σ(x)|s

for all x ∈ k, where | · | is the standard absolute value on R or C.

This shows that there are only finitely many infinite places and Fp∼= R

or Fp∼= C if p is infinite. We call p real if Fp

∼= R and complex if Fp∼= C.

We denote the ring of integers of F by oF . The finite places correspondone on one with the prime ideals of oF and if p is a prime ideal the place itinduces will also be denoted by p. Also the completion of F with respect toa finite place is a local field.

A infinite place is called normalized if it is the composition of the standardabsolute value of C with an embedding. A finite place is called normalizedif in the completion the generator of the maximal ideal has norm 1

#κwhere

κ is the residue field. We will use | · |p for the normalized representative of p.We will need the following results on the behavior of the places of a number

field.

Theorem 2.2 (Approximation theorem). If p1, ..., pn are pairwise distinctplaces of F and a1, ..., an ∈ F then for every ε > 0 there exists an x ∈ Fsuch that |x− ai|pi < ε for all 1 ≤ i ≤ n.

Theorem 2.3 (Product formula). For a ∈ F ∗ the norm we have |a|p = 1 foralmost all places p and ∏

p

|a|p = 1.

26

2.1.2 Some Galois theory

We need the results of this section for the proofs later. The results on theGalois groups of completions are used to define an action of the Galois groupon the idele class group and the results on ramification are necessary to provethe basic properties of Artin L series.

For the rest of this section L|F is a Galois extension of number fields andG = G(L|F ). Also let p be a prime ideal of the ring of integers of F and letoL be the ring of integers of L. If P is a prime ideal of oL over p and σ ∈ Gthen σP also is a prime ideal of oL over p.

The decomposition group GP of P is defined by

GP = σ ∈ G : σP = P

and the inertia group of P is is the normal subgroup of GP defined by

IP = σ ∈ G : σx ≡ x mod P for all x ∈ oL.

As L is dense in LP and an element σ ∈ GP is continuous for the topologyinduced by the absolute value we get a map GP → G(LP|Fp) by definingfor σ ∈ GP a map σ : LP → LP. The map σ acts on x ∈ LP by writingx = limn→∞ xn and defining σ(x) = limn→∞ σ(x). Because of the followingproposition we will often identify σ with σ and GP with G(LP|Fp).

Proposition 2.4. The map GP → G(LP|Fp) is an isomorphism and theimage of IP is the inertia group as defined for G(LP|Fp).

The following proposition shows that it is visible in the Galois group ifL|F is unramified at p.

Proposition 2.5. The extension L|F is unramified at p if and only if IP = 1for some P above p

Let κ(P) and κ(p) be the residue fields of LP and Fp. We see that theGalois group behave as expected with respect to the residue fields.

Proposition 2.6. The extension κ(P)|κ(p) is normal and G(κ(P)|κ(p)) ∼=GP/IP.

The Frobenius ϕP ∈ GP/IP is defined as the element which acts on κ(P)by x 7→ x#κ(p). Not that for unramified extensions this corresponds to theFrobenius element of G(LP|Fp).

Lemma 2.7. If L|M |F are Galois extensions and P|P′|p are prime idealsthen the maps GP → GP′, IP → IP′ and GP/IP → GP′/IP′ induced by thequotient map G(L|F )→ G(M |F ) are surjective and the image of ϕP is ϕP′.

27

The next proposition shows that in a Galois extension all ideals over agiven ideal p are essentially equivalent.

Proposition 2.8. If P and P′ are prime ideals of oL lying over p thenthere is a σ ∈ G such that σP′ = P. For this σ we have GP = σGP′σ

−1

and IP = σIP′σ−1. Furthermore if τ is such that τ = ϕP′ mod IP′ then

στσ−1 = ϕP mod IP.

This proposition shows that if L|F is unramified at p then the elements ofG which occur as the Frobenius of a prime ideal P over p form a conjugacyclass of G. Thus if L|F is an abelian extension, then for a given prime idealp there is a uniquely determined Frobenius element in G(L|F ) for the primeideals over p.

The following propositions describe the splitting of p in the fixed fields ofthe decomposition and inertia groups of a prime ideal P over p. We fix eand f as the ramification index and inertia index of P over p.

Proposition 2.9. Let P′ be the prime ideal of LGP below P then

1. The ideal P is the only prime ideal of the ring of integers of L aboveP′.

2. The ramification index of P′ in L is e and the inertia index is f .

3. The inertia and ramification index of P′ over F are 1.

For the next proposition we make a further assumption on G. It couldalso be stated in terms of the extension L|LGP .

Proposition 2.10. Assume G = GP. There is one prime ideal P′ in LIP

which lies over p. This ideal has ramification index e and inertia index 1over p. Furthermore the ramification index of P over P′ is 1 and the inertiaindex is f .

Let M be a subextension of L|F with Galois group H ⊂ G and let q1, ...qkbe the prime ideals of the ring of integers of M which lie over p.

Lemma 2.11. Let τi ∈ G be an element such that τi(P ∩M) = qi. The thesystem τiki=1 forms a system of coset representatives of GP\G/H.

Denote the Frobenius element of HP/JP by γP.

Lemma 2.12. We have HP = GP ∩ H and JP = IP ∩ H where JP is the

inertia group. Furthermore we have γP = ϕ[G:H]P .

28

2.2 Galois representations

In this section we consider representations of Galois groups of finite exten-sions of number fields. To each representation we attach a complex function,called the Artin L-series. These representations are allowed to be of arbi-trary degree. This is useful as as the Artin L-series have nice properties withrespect to induction so by pure representation theory it is possible to trans-fer some results on Artin L-series of 1-dimensional representations to ArtinL-series of representations of arbitrary degree.

The reader is assumed to know the basics of representation theory.

2.2.1 Artin L-series

First we need a simple lemma from representation theory in order to be ableto pass from representations of the Galois group to a more manageable group.

If (ρ, V ) is a representation of some group G and I is a normal subgrouplet V I be the subspace of V of elements fixed by I, that is:

V I := v ∈ V : ρ(g)(v) = v for all g ∈ I).

Lemma 2.13. If (ρ, V ) is a representation of a group G and I is normal inG then ρ induces a representation of G/I on V I defined by gI(v) = g(v).

Proof. It suffices to show G maps V I onto itself. Let g ∈ G and v ∈ V I , andsuppose gv /∈ V I then there is some h ∈ I such that hgv 6= gv. This impliesg−1hgv 6= v, which is a contradiction as g−1hg ∈ I.

We denote the representation given by lemma 2.13 by (ρ′, V I). This con-struction is natural with respect to induction from subgroups as the followinglemma shows.

Lemma 2.14. If H is a subgroup of G and (ρ, V ) is a representation of Hthen (IndGH(ρ)′, IndGH(V )I) = (IndGH(ρ′), IndGH(V H∩I)).

Proof. Let W be the vector space of the representation IndGH(ρ) and let ρ′ bethe representation of H/H ∩ I given by lemma 2.13. Furthermore let R be asystem of coset representatives of H in G, then we have W = ⊕r∈RrV . Thusw = rv ∈ W is invariant under I if and only if r is is representative of a cosetof the subgroup H/H∩I of G/I and V is invariant under I ∩H. Thus W I =⊕r∈H/H∩IV I . This is the vector space of the representation IndGH/H∩I(ρ

′). Itis clear that the action of G/I is the same so the representations are thesame.

29

From now on we will again use the notation L|F for a Galois extension ofnumber fields and put G = G(L|F ). Also let p be a prime ideal of the ringof integers of F and P a prime ideal of the ring of integers of L lying over p.

Let (ρ, V ) be a complex finite dimensional representation of G. We define

Lp(L|F, ρ, s) := (det(1− ρ′(ϕP)N(p)−s))−1,

where 1 is the unit element of GL(V IP) and s is a complex variable.

Lemma 2.15. The function Lp(L|F, ρ, s) does not depend on the choice ofP over p.

Proof. Let P and P′ be prime ideals over p. By proposition 2.8 there existsa σ in G such that GP = σGP′σ

−1 and IP = σIP′σ−1 and ϕP = σϕP′σ

−1, so

det(1− ρ′(ϕP)N(p)−s) = det(ρ′(σ)−1(1− ρ′(ϕP)N(p))ρ′(σ)−1)

= det(1− ρ′(σ−1ϕPσ)N(p)−s)

= det(1− ρ′(ϕP′)N(p)−s).

Remark. By the same argument we can see that the Artin L-function onlydepends on the isomorphism class of a representation. Thus we could for-mulate everything in terms of characters of representations. However as theterm character is already used for a different object we will not do so.

By the last lemma the following definition is unambiguous.

Definition 2.16. The Artin L-series is defined by

L(L|F, ρ, s) :=∏

p

Lp(L|F, ρ, s),

where p ranges over all prime ideals of the ring of integers of F .

Observe that the Artin L-series of the trivial representation is the Dedekindzeta function of F .

We now show that the Artin L-functions are convergent on a half-plane.

Theorem 2.17. If ρ is a representation of G(L|F ) then L(L|F, ρ, s) con-verges uniformly and absolutely on the complex half-plane defined by Re(s) ≥1 + δ for any δ > 0 and is non-zero on this half-plane.

30

Proof. Because ρ′(ψP) has finite order all eigenvalues are roots of unity so

det(1− ρ′(ϕP)N(p)−s) =d∏i=1

(1− ξiN(p)−s),

where the ξi are roots of unity and d = dim(V IP). Thus we can write

L(L|F, ρ, s) =∏p

∏p|p

d∏i=1

(1− ξiN(p)−s).

By definition his product converges absolutely if∏p

∏p|p

d∏i=1

(1 + |ξiN(p)−s|). (2.1)

converges. We have

log(∏p

∏p|p

d∏i=1

(1 + |ξiN(p)−s|)) =∑p

∑p|p

d∑i=1

log(1 + |ξiN(p)−s|)

=∑p

∑p|p

d∑i=1

∞∑j=1

(−1)j+1 |ξiN(p)−s|j

j

≤∑p

∑p|p

d∑i=1

∞∑j=1

| 1

jpjs|

≤ d[F : Q]∑p

∞∑j=1

1

jpj(1+δ).

The last sum is d[F : Q] log(ζ(1 + δ)) so the product converges independentof s. As limx→∞ log(x) =∞ this shows product 2.1 converges. For any n > 0we have

log(|∏p≤n

∏p|p

d∏i=1

(1− ξiN(p)−s)|) ≤ log(∏p≤

∏p|p

d∏i=1

(1 + |ξiN(p)−s|))

so for any s with Re(s) > 1 + δ we have

|∏p

∏p|p

d∏i=1

(1− ξiN(p)−s)| = exp(log(|∏p

∏p|p

d∏i=1

(1− ξiN(p)−s)|))

which shows L(L|K, ρ, s) 6= 0 for any s with Re(s) > 1.

31

Now we are ready to prove the following result, which has several nicecorollaries. The third property is the most difficult to prove.

Proposition 2.18. Let L|F be a Galois extension and let M be a subexten-sion. Denote G = G(L|F ).

1. If (ρ, V ) and (γ,W ) are representations of G then

L(L|F, ρ⊕ γ, s) = L(L|F, ρ, s)L(L|F, γ, s).

2. If M is Galois over F and ρ is a representation of G(M |F ) then

L(L|F, ρ, s) = L(M |F, ρ, s),

where ρ is obtained by composing ρ with the quotient map G→ G(M |F ).

3. Let H = G(L|M). If ρ is a representation of H then

L(L|M,ρ, s) = L(M |F, IndGH(ρ), s).

Proof. For part 1 note that we have that (V ⊕W )IP = V IP ⊕W IP , so thecharacteristic polynomial of ρ(ϕP) ⊕ γ(ϕP) is the product of the charac-teristic polynomials of ρ(ϕP) and γ(ϕP), which proves Lp(L|F, ρ ⊕ γ, s) =Lp(L|F, ρ, s)Lp(L|F, γ, s) and thus L(L|F, ρ⊕γ, s) = L(L|F, ρ, s)L(L|F, γ, s).

If P|P′|p are prime ideals in the extensions L|M |F then lemma 2.7 showsthat V IP′ = V IP and that the action of ϕP′ is the same as the action of ϕP sothe characteristic polynomials are the same, so Lp(L|F, ρ, s) = Lp(M |F, ρ, s),which proves point 2.

Let (IndGH(ρ),W ) be the representation induced by (ρ, V ). We first provethe third point in case G = GP. In this case there is only one ideal P overp, so also a unique ideal q in M over p so we abbreviate I = IP. Now wereduce to the case I = 1. If we denote the inertia group of P over M byJ we have by proposition 2.12 that HP = H and J = I ∩H. We reduce tothe extension LI over F with the subextension LH·I = M ∩ LI . This givesGalois groups G(LI |F ) = G/I and G(LI |LH·I) = H · I/I ∼= H/J . We equipG(LI |LH·I) with the representation given by lemma 2.13 which is denoted by

(ρ′, V J). By lemma 2.14 we have IndG/IH/J(ρ′), Ind

G/IH/J(V J)) = (IndGH(ρ)′,W I),

so we have

Lp(LI |F, Ind

G/IH/J(ρ′), s) = Lp(L|F, IndGH(ρ), s). (2.2)

As P is the only prime ideal lying over p, the prime ideal q∩M I is the onlyprime ideal in the ring of integers of M I lying over p. By the same argument

32

as for equation 2.2 we have Lq(L|M,ρ, s) = Lqi∩MI (LI |M I , ρ′, s), so we canindeed assume I = 1.

If G = GP and I = 1 then G is generated by ϕP and H is generatedby some power of ϕP. By part (1) we can assume ρ is irreducible thus 1-dimensional. By proposition 2.10 we have that P is the only prime idealof the ring of integers of L over p and thus P′ = P ∩ LH is the uniqueprime ideal of the ring of integers of LH over p. Furthermore both idealsare unramified. Let j = [G : H] then ϕjP generates H and is the Frobenius.

We have W = ⊕j−1i=0ϕ

iPV so if we identify W with C and ρ(ϕjP) with the

appropriate element of GL(C) ∼= C∗ the matrix of ρ(ϕP) in GL(W ) is0 1 · · · 0 0

0 0. . . 0 0

......

. . . . . ....

0 0 · · · 0 1

ρ(ϕjP) 0 · · · 0 0

.

This matrix has characteristic polynomial Y j − ρ(ϕjP) and the characteristic

polynomial of ρ(ϕjP) as element of GL(C) is Y − ρ(ϕjP). By proposition

2.10 we have ϕp′ = ϕjp. This shows Lp(L|F, IndGH(ρ), s) = Lq(L|M,ρ, s) asN(p′) = N(p)j.

Now the general case. Let p be a prime ideal of the ring of integers of Fand let q1,...,qk be the prime ideals of the ring of integers of M over p. For1 ≤ i ≤ k choose prime ideals bi in L such that bi lies over qi and abbreviateP = b1. Also choose τi ∈ G(L|F ) such that τibi = P. By lemma 2.11 the thesystem T = τiki=1 forms a system of coset representatives of GP\G/H. Let(ρ,W ) be the representation induced by ρ on G. We will show by inductionon [L : F ] that

Lp(L|F, ρ, s) =r∏i=1

Lqi(L|M,ρ, s).

The statement is clear if L = F . Furthermore we can assume G 6= GP.For τ ∈ T let Hτ = τHτ−1 ∩ GP and let ρτ be the representation ofHτ defined by ρτ (x) = ρ(τ−1xτ). By proposition 22 of [7] we have that

ResGP(ρ) ∼= ⊕τ∈T Ind

GP

Hτ(ρτ ). If we put P′ = P ∩ LGP have Lp(L|F, ρ, s) =

LP′(L|LGP ,ResGP(ρ), s) as the representations are the same and N(p) =

33

N(P′). Thus by induction

Lp(L|F, ρ, s) = LP′(L|LGP ,ResGP(ρ), s)

=∏τ∈T

LP′(L|LGP , IndGP

Hτ(ρτ ), s)

=∏τ∈T

Lq′i(L|LHτ , ρτ , s),

where q′ is unique prime ideal of the ring of integers of LHτ which lies over P′.If τ is such that τP ∩ LH = qi for some 1 ≤ i ≤ k then Lq′(L|LHτ , ρτ , s) =Lqi(L|LH , ρ, s) by the same argument as for Lp(L|F, ρ, s) so by proposition

2.11 we have Lp(L|F, IndGH(ρ), s) =∏k

i=1 Lqi(L|M,ρ, s).

Proposition 2.18 makes it possible to translate theorems from representa-tion theory into results on L-functions.

For a finite group G let g be the order, rG the regular representation and 1the trivial representation. If ρ is a representation and k an integer we denoteby k · ρ the representation ⊕ki=1ρ.

Corollary. If ζL are ζF the Dedekind zeta functions of L and F then

ζL(s) = ζF (s)∏ρ

L(L|F, ρ, s)ρ(1), (2.3)

where the product is over a system of representatives of the equivalence classesof nontrivial irreducible representations of G(L|F ).

Proof. Let H = G(L|L) ⊂ G(L|F ) = G be the trivial group then we knowfrom representation theory that IndGH(rH) = rG =

∑ρ(1) · ρ where ρ ranges

over a system of representatives of the equivalence classes of irreducible rep-resentations of G. As H is the trivial group rH is also the trivial character sowe have ζL(s) = L(L|L, rH , s) = L(L|F, IndGH(rH), s) =

∏ρ L(L|F, ρ, s)ρ(1),

which proves the corollary as ζF is the L function of the trivial representationwhich occurs once in the product.

This formula is very elegant, however for applications it is not so usefulas it involves Artin L-series over field extensions with possibly non-solvableGalois groups. This is a problem as we can only prove some good propertiesin case the extension is solvable. We will use a lemma and a theorem fromrepresentation theory to express ζL(s)g

ζF (s)gas a product of Artin L functions and

L(L|F, ρ, s) as a quotient. For the proofs of the results from representationtheory see [7].

The characters in the next lemma can be given explicitly, however, thereis no need to do so here.

34

Lemma 2.19. If A ranges over all cyclic subgroups of a finite group G thereexist nontrivial representations λA such that⊕

A

IndGA(λA) = g(rG − 1).

By noting that L(L|F, g(rG − 1), s) = L(L|F, rG − 1, s)g = ζL(s)g

ζF (s)gwe get:

Corollary. If L|F is a Galois extension of number fields with Galois groupG(L|F ) and LA is the fixed field of L under the subgroup A of G(L|F ) then

ζL(s)g

ζF (s)g=∏A

L(L|LA, λA, s),

where the product runs over all cyclic subgroups of G(L|F ).

Now we cite:

Theorem 2.20 (Brauer). Every character of a finite group G is a Z-linearcombination of characters induced from characters of of degree 1 of subgroupsof G.

By part 3 of proposition 2.18 this implies:

Corollary. If L|F is a Galois extension of number fields with Galois groupG and ρ is a representation of G there is a finite collection H of subgroupsof G and for each H ∈ H a 1-dimensional representations ρH and a numbernH ∈ Z such that

L(L|F, ρ, s) =∏H∈H

L(L|LH , ρH , s)nH .

2.2.2 The conjectures of Dedekind and Artin

We now want to use the corollaries proved in the previous section. If werecall that the Dedekind zeta function ζL(s) of a number field L can bemeromorphically continued onto C and that a function on C is called entireif it is holomorphic in every point of C we can state the following conjecture:

Conjecture (Dedekind). For any finite extension L|F of number fields thefunction

ζL(s)

ζF (s)

is entire.

35

Equation 2.3 shows that for Galois extensions L|F Dedekinds conjecturewould follow from the following conjecture:

Conjecture (Artin). If χ is irreducible and nontrivial then L(L|K,χ, s) canbe analytically continued to an entire function.

This conjecture is still open, but the case of abelian extensions has beensettled and we will sketch a proof of that. This shows why we needed thefurther corollaries in the previous section, the corollary to lemma 2.19 showsthat the question of ζL(s)g

ζF (s)gbeing holomorphic reduces to the question of L-

functions of abelian extension being holomorphic, which we will see later.

Remark. Using Frobenius reciprocity it is not very difficult to show that evenfor extensions which are not Galois, Dedekind’s conjecture would follow fromArtin’s conjecture.

2.3 The idele class group

We will now construct the idele class group. This topological group makes itpossible to give an elegant formulation of class field theory for number fields.

2.3.1 Construction

The first step is the definition of the adeles and ideles which are a way tocombine information from all different completions of F . Denote by (ap)p anelement of

∏p Fp, where p ranges over all places of F . We define the adeles

as the subring of∏

p Fp given by the elements a = (ap)p for which ap is inthe valuation ring of Fp for all but finitely many of the finite places. Denotethe adeles of F by AF .

The ideles IF are the group of units of AF , so IF = A∗F . For any placep we can embed F ∗p in the ideles by mapping x to the idele which is x onthe p-th coordinate and 1 elsewhere. When no confusion is possible we willidentify F ∗p with its image in IF .

If p is a finite place then UFp was defined in subsection 1.1.1. Here weabbreviate this as Up. For a finite set S of places containing the infiniteplaces we define the S-ideles as the subgroup

ISF :=∏p∈S

F ∗p ×∏p/∈S

Up

of the ideles. We equip the S-ideles with the product topology, so the S idelesare a locally compact Hausdorff topological group. It is clear that IF is the

36

union of the ISF where S ranges over all finite subsets of places containing theinfinite ones.

We now define a topology on the ideles. This will be a slightly alteredversion of the subspace topology in such a way that the ideles become atopological group. As a basis we give the sets

∏p Vp, where Vp is open in K∗p

for all p and Vp = Up for all but finitely many of the finite places p. Thefollowing statement is immediate.

Lemma 2.21. For a finite set S of places containing the infinite places theinclusion ISF → IF is a homeomorphsim onto its image and the image is openin IF .

From this lemma we get:

Lemma 2.22. The group IF is a locally compact Hausdorff topological group.

Proof. The space IF is locally compact Hausdorff as it is a union of opensubspaces which are locally compact Hausdorff. Now let S range over allfinite sets of places which contain the infinite places. If U ⊂ IF is openthen U = ∪S(U ∩ ISF ) and the inverse image of U under the multiplicationmap IF × IF → IF is the union of the inverse images of ISF ∩ U under themultiplication maps ISF×ISF → ISF and so it is open. This shows multiplicationis continuous. The continuity of inversion follows by the same argument.

If x ∈ F ∗ then |x|p = 1 for almost all p so the group F ∗ can be embeddedin IF by x 7→ (xp)p where xp is the image of x under the standard embeddingF → Fp. We will identify F ∗ with its embedding in IF .

Definition 2.23. The group CF := IF/F∗ is called the idele class group.

Proposition 2.24. The subgroup F ∗ is discrete and closed in IF , thereforeCF is a locally compact and Hausdorff topological group.

Proof. The subset U = α ∈ IF : |αp|p = 1 for p -∞, |αp − 1|p < 1 for p|∞is an open neighborhood of 1 ∈ IF . If x is a principal idele other than 1 inU then

1 =∏

p

|x− 1|p

=∏p-∞

|x− 1|p ·∏p|∞

|x− 1|p

<∏p-∞

|x− 1|p ≤∏p-∞

max|x|p, 1 = 1,

37

so 1 is the only principal idele in U . By continuity of multiplication andinversion there is a open neighborhood V of 1 such that V V −1 ⊂ U . Ify ∈ IF then yV contains at most one principal idele. Suppose yv1, yv2 ∈ yVand yv1, yv2 ∈ F ∗ then yv1(yv2)−1 is a principal idele which is in V V −1 ⊂ U ,so v1 = v2. As IF is Hausdorff it follows that F ∗ is discrete and closed. Thequotient of a topological group by a closed subgroup is a topological groupand the quotient of a locally compact Hausdorff space by a closed equivalencerelation is locally compact Hausdorff, so this finishes the proof.

2.3.2 Galois module structure

Let L|F be a Galois extension of number fields and let IL and IF be the idelegroups. The action of the Galois group can be transferred to the ideles asfollows. We can consider IF as a subgroup of IL by identifying (ap)p ∈ IFwith the element (aP)P ∈ IL which is given by aP = ap where p is theideal over which P lies. As L is dense in LP for any σ ∈ G(L|F ) we getan isomorphism LP → (σL)σP, which we again denote by σ. We get anautomorphism IL → IL by mapping α = (αP)P to the idele with P-thcoefficient σ(ασ−1P). Again by abuse of notation we denote this map by σ.

The action of G = G(L|F ) defined this way has the following property.

Proposition 2.25. If L|F is a Galois extension then IGL = IF .

Proof. The isomorphism induced by σ on LP is the identity on Fp if P liesover p, so IF ⊆ IGL . Now suppose (αP)P ∈ IGL . If αP /∈ F ∗p the there is someτ ∈ G(LP|Fp) ∼= GP such that τ(αP) 6= αP. As τ ∈ GP we have τP = P, sowe must have αP ∈ F ∗p . If P and P′ are above a places p then the is a σ inG such that σP = P′. The P′-th coefficient of σ(α) is σ(ασ−1P′) = σ(αP).As αP is in Fp this shows αP = αP′ . This proves that α is in IF .

Definition 2.26. If L|F is a finite extension then the norm of an ideleα = (αP) ∈ IL is the element of NL|K(α) of IF defined by

(NL|K(α))p =∏P|p

NLP|Fp(αP).

Proposition 2.27. Then idele norm has the following properties:

1. If M |L|F are extensions of number fields then NM |F = NL|F NML.

2. If G = G(M |F ) and H = G(M |L) and IL is considered as a subgroupof IM then the norm of α ∈ IL is given by

∏σ∈G/H σ(α).

3. NL|F = α[L:F ] for α ∈ IF .

38

4. If α is the principal idele of IL defined by x ∈ L then the idele normof α is the idele defined by N(x) in IF , where N : L → F is the normmap defined in subsection 1.1.2.

Proof. Part 1 and 2 follow directly from the definition and the properties ofthe norm of fields. Part 3 follows from the fact that [L : F ] =

∑P|p[LP : Fp]

for any place p and part 4 follows from part 2.

The discussion translates as follows to the idele class group.

Proposition 2.28. The map CF → CL defined by αF ∗ 7→ α′L∗ is injective,where α′ is α, considered as an elemnt of IL.

Proof. The map is well defined as the inclusion IF → IL maps F ∗ into L∗. Forinjectivity it is enough to show that if αF ∗ is mapped to L∗ then αF ∗ = F ∗

or equivalently, if α ∈ L∗ ∩ IF then α ∈ F ∗. Choosing a Galois extension Mof F containing L gives

IF ∩ L∗ ⊂ IF ∩M∗ ⊂ (IK ∩M∗)G = IF ∩ (M∗)G = IF ∩ F ∗ = F ∗.

Proposition 2.29. For a Galois extension L|F the Galois group G = G(L|F )acts on CL and CG

L = CF .

Proof. From the definition it is clear that the map IL → IL induced byσ ∈ G maps L∗ to L∗, so it induces a well defined homomorphism of CL byαL∗ 7→ σ(α)L∗. By definition there is an exact sequence

1→ L∗ → IL → CL → 1.

As by assumption (L∗)G = K∗ and by proposition 2.25 IGL = IK it suffices toshow that the sequence

1→ (L∗)G → IGL → CGL → 1

is exact. The map (L∗)G → IGL is the restriction of an injective map soinjective. The kernel of IGL → CG

L is IGL ∩ L∗ = IK ∩ L∗ = K∗ = (L∗)G. Forthe surjectivity of IGL → CG

L let αL∗ ∈ CGL . The G invariance means that

for ever σ ∈ G there exist an xσ such that σα = αxσ. The map σ 7→ xσ isa crossed homomorphism as xστ = σ(τ(α))α−1 = σ(τ(α))σ(α)−1σ(α)α−1 =σ(xτ )xσ, so by Hilbert 90 there is an y ∈ L∗ such that xσ = σ(y)y−1. Nowαy−1L∗ = αL∗ and σ(αy−1) = σ(α)σ(y−1) = ασ(y)y−1σ(y−1) = σ(αy−1), soσ(αy−1) ∈ IGL = IF .

39

2.4 Global class field theory

We give the following formulation of class field theory for number fields. Thisversion is a short summary of what can be found in [4].

Recall that the local norm residue symbol was discussed in section 1.2.3.

Theorem 2.30. There is a unique continuous homomorphism ψF : CF →G(F ab|F ) such that for any finite extension L|F and every place P over aplace p the diagram

F ∗p( ,LP|Fp)

//

G(LP|Fp)

CF

α 7→ψF (α|L)// G(L|F )

is commutative, where ( , LP|Fp) is the local norm residue symbol.

The map ψF has the following property:

Proposition 2.31. Let L be an abelian extension of F unramified at p andα an idele with as p-th coefficient a prime element of Fp and 1 on the othercoordinates. Then ψF (α) = ϕP, where P is a prime ideal over p and ϕP isthe Frobenius.

The following theorem is the analog for number fields of theorem 1.19.

Theorem 2.32. If DF is the connected component of 1 ∈ CF then ψF givesan isomorphism of topological groups CF/DF

∼= G(F ab|F ).

2.5 Hecke characters

Now we will discuss characters of the idele class group and their L-functions,omitting the proofs. The L-functions will be defined by a product very similarto the definition of the Artin L-function and will be convergent on a half-plane. By analytic methods it is possible to show that the Hecke L-functionhave an analytic continuation to the whole complex plane and to give asimple condition under which they have no poles. There are two approachesto these theorems. The first one is by Hecke and involves integration overcomplex functions called theta series. This approach is similar to the proofof the analogous properties of the Riemann-zeta function and can be foundin [5]. The other approach is by Tate and is known as Tate’s thesis, whichcan be found in [2].

We will confine ourselves to stating the results which we will need later.

40

Definition 2.33. A Hecke character is a continuous homomorphism CF →C∗ such that the image is in S1.

Note that as CF is the quotient of IF by F ∗ we can also view Heckecharacters as characters of IF such that χ(F ∗) = 1. This point of viewprevents some cumbersome notation.

For k > 0 define subgroups of F ∗p by

U(k)p =

1 + pk if p -∞,

R∗>0 for p real,C∗ for p complex,

and put U(0)p = Up.

Let m =∏

p pnp be an ideal the ring of integers of F Define Imf =∏

p-∞ U(np)p and define an embedding in IF by embedding every factor of

the product using the the standard embedding F ∗p → IF .

Definition 2.34. A integral ideal m is called a module of definition of aHecke character χ if χ(Im

f ) = 1 and ordp(m) = 0 if χ(Up) = 1.

Lemma 2.35. For every Hecke character there is a module of definition.

Proof. If p is finite then Up is locally profinite so Imf is a product of locally

profinite groups so it is locally profinite. Proposition 1.23 shows the kernelis open and by definition must contain a subset of the form

(∏p∈S

Wp ×∏p/∈S

Up) ∩ Imf ,

where S is a finite set of places and the Wp are closed and open. Because

the U(k)p for k ∈ N>0 form a neighborhood basis of 1 in Up for every p ∈ S

there is some np such that U(np)p ⊂ Up ∩Wp. Now m =

∏p∈S U

(np)p ×

∏p/∈S Up

satisfies χ(Imf ) = 1. If χ(Up) = 1 then m′ = m · p−np also satisfies χ(Im′

f ) = 1,thus we can assume ordp(m) = 0.

We now assume χ is a Hecke character with module of definition m. Wewill only use m as a list of p such that χ(πp) does not depend on the choiceof prime element πp, so the choice of m is not important.

Definition 2.36. The Hecke L-series of χ is defined by

L(χ, s) =∏

p

1

1− χ(πp)N(p)−s,

where p varies over all prime ideals of the ring of integers of F which arerelatively prime to m and πp is a prime element of Fp.

41

Note that this is well defined as χ(πp) does not depend on the choice ofπp because two prime elements can only differ by a unit.

If a is an integral ideal of F relatively prime to m then by abuse of notation

we denote by χ(a) the value of χ on an idele (πordp(a)p )p where πp is a prime

element of Kp for local fields and we put πp = 1 and ordp(a) = 0 for infiniteplaces p.

It is also possible to define the Hecke L-series by the sum in the nexttheorem. The proof of the first part of this theorem is similar to the proofof the analogous property of Artin L series and the second part is similar tothe analogous product formula of the Riemann zeta function.

Theorem 2.37. The product L(χ, s) converges uniformly and absolutely onthe complex half-plane defined by Re(s) ≥ 1 + δ for any δ > 0. on thishalf-plane we have

L(χ, s) =∑

a

χ(a)

N(a)s,

where a ranges over all integral ideals of F relatively prime to m.

Now we have:

Theorem 2.38. If χ is a Hecke character with module of definition m 6= 1then L(χ, s) has an analytic continuation to an entire function.

2.6 The global correspondence

Now we will use the isomorphism of theorem 2.32 to give a correspondencebetween characters of G(F |F ) and characters of CF which are trivial on theconnected component of 1. This bijection is the analog for number fieldsof corollary 1.3.5. There also is an analog of theorem 1.44, however thattheorem is more complicated as it involves the Weil group for a number field,for which there is no natural definition.

Let φ be a character of G(F |F ). As G(F |F ) is profinite the kernel is anopen subgroup by proposition 1.23. The open subgroups correspond to thefinite extensions of F so φ factors trough some G(L|F ) where L is a finiteabelian extension of F and ker(φ) = G(F |L). As the field L is uniquelydetermined by the subgroup ker(φ) we can define the L-function of φ as theL-function of φ, considered as a character of G(L|F ). We will denote thisfunction by L(φ, s).

Let ch(G) be the set of characters of a group G.

42

Theorem 2.39. The map

ch(G(F |F ))→ ch(CF )

given by φ 7→ φ ψF is a bijection between ch(G(F |F )) and the characters ofCF which are trivial on the connected component of 1. If φ is a character ofG(F |F ) we have

L(φ, s) = L(φ ψF , s).

Proof. The map is a bijection because by theorem 2.32 the kernel of the mapψF is the connected component of 1. Let φ be a character of G(F |F ). Wehave L(φ, s) = L(L|F, φ, s) where L is the fixed field of the kernel of φ. Wenote that L is a finite abelian extension of F and that φ is injective as acharacter of G(L|F ).

For a given prime ideal p we have the diagram

F ∗p( ,F ab

p |Fp)//

W abFp

F ∗p

( ,LP|Fp)//

G(LP|Fp)

CF

α 7→ψF (α)|L // G(L|F ),

where WFp is the Weil group as defined in section 1.2.4 and the unnamedmaps are standard. The diagram is commutative by definition of the Weilgroup and by theorem 2.30. Furthermore, by theorem 1.19 the image of thegroup of units of F ∗p by ( , F ab

p |Fp) is the inertia group of W abFp

and the

image of the inertia group by the quotient map W abFp→ G(LP|Fp) is IP. By

commutativity of the diagram we conclude that the image of ψF (Up) underthe quotient map G(F |F ) → G(L|F ) is the inertia group IP. We concludeφ(IP) 6= 1 is equivalent to φ ψF (Up) 6= 1.

Let (ρ, V ) be a representation of G(L|F ) with character φ then V is bydefinition 1-dimensional so we have

Lp(L|F, φ, s) =

(1− φ(ϕP)N(p)−1)−1 if φ(IP) = 1,

1 if φ(IP) 6= 1.

If p is ramified in L we have that IP 6= 1, so φ(IP) 6= 1. This showsLp(L|F, φ, s) = 1 and there is no factor for p occurring in the product defi-nition of L(φ, s). If p is unramified in L we have that IP = 1, so φ(IP) = 1.Therefore Lp(L|F, φ, s) = (1 − φ(ϕP)N(p)−1)−1 and as φ ψF (Up) = 1 the

43

factor at p in the Hecke L-series is (1−φψF (πp)N(p)−s)−1. As p is unrami-fied theorem 2.31 gives that ψF (π) = ϕP, so the factors occurring at p in theArtin and Hecke L-series are equal. This shows that L(φ, s) and L(φ, s) aredefined by the same product so for any δ > 0 they agree on the half-planedefined by Re(s) > 1 + δ.

Now we can deduce a several of corollaries.As all irreducible nontrivial characters of abelian groups are 1-dimensional

they correspond to Hecke L series with module of definition m 6= 1, so bytheorem 2.38 we have:

Corollary. Artins conjecture holds for abelian Galois groups.

The following result follows directly from the corollary to theorem 2.20.

Corollary. If L|F is a Galois extension of number fields with Galois groupG and ρ is a representation of G then the Artin L-function attached to ρ hasa meromorphic continuation to C.

Also we have the promised result on quotients of Dedekind zeta functions,which follows from the corollary to lemma 2.19.

Theorem 2.40 (Aramata-Brauer). For a finite Galois extension L|F ofnumber fields the function

ζL(s)

ζF (s)

is entire.

44

Populaire samenvatting

Ieder geheel getal is te schrijven als product van priemgetallen. Stel dat ditniet waar zou zijn dan is er een kleinste getal n waarvoor dit niet geldt, maardan moet gelden n = n1n2 met n1, n2 < n, omdat n in het bijzonder geenpriemgetal mag zijn. Omdat n1 en n2 kleiner zijn dan n moeten n1 en n2 welte schrijven zijn als product van priemgetallen, maar dan is n dat ook.

We kunnen makkelijk zien dat de eerste priemgetallen

2, 3, 5, 7, 11, 13...

zijn. De Griek Euclides bewees dat er oneindig veel priemgetallen zijn. Steldat p1, ...pk een lijstje met alle priemgetalen is dan is

p1 · p2 · · · pk + 1

een getal dat niet niet deelbaar is door de getallen p1 tot pk. We weten echterdat er een priemgetal moet zijn dat p1 · p2 · · · pk + 1 deelt dus ons lijstje wasniet volledig.

Omdat er oneindig veel priemgetallen zijn kunnen we ons afvragen hoe zeverdeeld zijn over de gehele getallen. Na veel rekenen vermoedden Legendreen Gauss al in de 18e eeuw dat voor een getal n er ongeveer

n

log(n)

priemgetellen kleiner dan n zijn. Om over de verdeling van de priemgetallenna te denken is het nuttig om van de rij priemgetallen een functie op hetcomplexe vlak te maken. De functie van s gedefineerd door

ζ(s) =∞∑n=1

1

ns=∏p

1

1− p−s

heet de Riemann zeta functie. De gelijkheid van de som en het product kanbewezen worden door gebruik te maken van de ontbinding in priemfactoren.

45

Omdat geldt

1+1

2+

1

3+

1

4+

1

5+

1

6+... > (

1

2+

1

2)+(

1

4+

1

4)+(

1

6+

1

6)+... = 1+

1

2+

1

3+

1

4+

1

5+

1

6+...

is de zeta functie niet gedefineerd voor s = 1. Het gedrag van de functie ronds = 1 is echter zeer van belang, hoe snel de functie ”naar oneindig gaat” issterk gerelateerd aan de verdeling van de priemgetallen. Het vermoeden vanLagrange en Gauss werd bewezen door Hadamard en de la Vallee-Poussindoor het gedrag van de zeta functie rond s = 1 te bekijken. In figuur 2.1 iste zien dat het punt s = 1 inderdaad bijzonder is.

Figure 2.1: Riemann zeta functie rond s = 1.

We kunnen ook de gehele getallen opdelen in de rijen

0, 3, 6, 9, 12, ...

1, 4, 7, 10, 13, ...

en2, 5, 8, 11, 14, ...

We zien meteen dat is de eerste rij maar een priemgetal zit, al deze getallenzijn deelbaar door 3. In minstens een van de andere rijen moeten dus oneindigveel priemgetallen zitten. Om te bewijzen dat er in beide rijtjes oneindig veel

46

priemgetallen zitten en zelfs ongeveer evenveel is een generalistatie van deRiemann zeta functie nodig, bedacht door Dirichlet. Deze functie op hetcomplexe vlak wordt gegeven door een product van de vorm

L(χ, s) =∏p

1

1− χ(p)p−s,

waar χ een eenvoudige functie op de priemgetallen is. Met deze functiesbewees Dirichlet dat de priemgetallen ongeveer gelijk verdeeld zijn over het2e en 3e rijtje.

Als χ vervangen wordt door een minder eenvoudige functie krijgt men eenzogenaamde Artin L-reeks. Met deze functies is het lastiger rekenen danmet de concretere functies van Dirichlet. Dat de Artin L-reeksen toch mooieeigenschappen hebben is het vermoeden van Artin. Dit is nog niet in hetalgemene geval bewezen. In deze scriptie bekijken we het bewijs van hetvermoeden van Artin voor een speciale klasse van Artin L-reeksen.

47

Bibliography

[1] Colin J. Bushnell, Guy Henniart, The local Langlands conjecture forGL(2), Springer-Verlag, 2006.

[2] John W.S. Cassels, Albrecht Frolich, Algebraic number theory, Academicpress, 1967.

[3] Serge Lang, Algebra, revised third edition, Springer-Verlag, 2002.

[4] James S. Milne, Class field theory, version 4.00, http://www.jmilne.

org/math/, 2008.

[5] Jurgen Neukirch, Algebraic number theory, Springer-Verlag, 1999.

[6] Jean-Pierre Serre, Local fields, Springer-Verlag, 1979.

[7] Jean-Pierre Serre, Linear Representations of Finite Groups, Springer-Verlag, 1977.

[8] John Tate, Number-theoretic background, Proc. Sympos. Pure Math.Vol. 33 - Part 2, Amer. Math. Society, 1979.

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