TATE’S THESIS

BAPTISTE DEJEAN

Abstract. L-functions are of great interest to number theorists. Key to their

study are their meromorphic extensions and functional equations. Hecke de-

fined a class of L-functions analogous to Dirichlet’s and used unpleasant meth-ods to give their meromorphic extensions and functional equations [4]. In his

thesis, Tate bypassed Hecke’s methods by using simple Fourier analysis to

derive the same [9]. We give Tate’s derivation, but following Kudla [5] (whofollows Weil [10],) we reinterpret the pervasive proportionality of distributions.

To illustrate the theory, we compute a few concrete examples.

Contents

1. Introduction 22. Motivation, characters 22.1. Characters 32.2. Restricted direct products 42.3. Adeles and ideles 42.4. Abelian Fourier analysis 63. Local theory 63.1. Setting the stage 63.2. Lemma 3.4’s proof 73.3. Recap 93.4. Local Fourier analysis and the local functional equation 104. Global theory 124.1. From local analysis to global analysis 124.2. Remarks about characters of A∗K 134.3. From local eigendistributions to global eigendistributions 134.4. Global Fourier analysis 144.5. Global functional equation 155. Computations 175.1. Warm-up: Riemann zeta function 175.2. Dirichlet L-functions 185.3. Hecke characters 195.4. Example: Hecke characters of Q

(√5)

mod(√

5)

205.5. Final remarks 21References 22

Date: October 26, 2017.

1

2 BAPTISTE DEJEAN

1. Introduction

L-functions are of great classical interest, for they contain detailed informationabout distribution of primes. Obtaining their meromorphic extensions to the com-plex plane and their functional equations are key tools for studying them. In histhesis, Tate [9] used fairly simple Fourier analysis to obtain these extensions andfunctional equations for a specific class of L-functions, a significant improvementover the techniques of the time [4]. We will closely follow the exposition in Kudla[5], which follows Weil [10] in reinterpreting the local functional equation.

In Section 2, we will introduce the main players.In Section 3, we will work out the situation in local contexts. A space of distri-

butions will turn out to be one-dimensional, forcing important distributions to berelated by scale-factors. One of these scale factors is, conveniently, the local factorof a classical L-function.

In Section 4, adeles and ideles will be useful global contexts, for we can producetheir pictures by multiplying together all local pictures. Therefore, the entire L-function whose factors appeared locally will appear globally. There will be a scalefactor which we can compute in terms of this L-function; however, we can alsouse Poisson summation to compute this scale factor as 1. This gives a functionalequation for our L-function.

The reader should note that there only two crucial theoretical points in Sections3 and 4, namely Lemma 3.4 and Theorem 4.12.

Finally, we will show the relationship to classical pictures in Section 5, computingthe functional equations for Dirichlet L-functions and certain Hecke L-functionsover Q

(√5).

2. Motivation, characters

By “valuation” we mean a norm; that is, we write codomains multiplicativelyand allow archimedean valuations. By a finite valuation we mean a nonarchimedeanvaluation, and by an infinite valuation we mean an archimedean valuation.

Notations 2.1. We will use the following notation.

• K is a global field; that is, a finite extension of Q or Fp(t).• If K is a number field, OK is its ring of integers.• Kv is a completion of K with respect to a valuation v.• OKv is the ring of integers in Kv for finite v, mv is its maximal ideal, andπv is a uniformizer.• Uv is the group of elements of K∗v of norm 1.• dx is an additive Haar measure on Kv, and d∗x is a multiplicative Haar

measure on K∗v .• πv is a uniformizer of Kv for finite v.• For a finite valuation v, pv is the prime of OK inducing it, and for a primep of OK , vp is the valuation induced by it.• If v is finite, fov is the characteristic function of OKv .

We will tacitly normalize all valuations v so that multiplication by any α ∈ Kv

scales dx by |α|; that is, d(ax) = |a|vdx. Thus, d∗x and dx|x|v agree up to a constant.

If Kv = C, this means ||v is the square of the usual Euclidean norm.

TATE’S THESIS 3

2.1. Characters.

Definition 2.2. If G is a locally compact abelian group, a quasicharacter of G isa continuous homomorphism ω : G→ C∗, and a character of G is a quasicharacterwhose image lands in S1.

Remarks 2.3. If G is compact, any quasicharacter is a character. Quasicharactersand characters form groups under pointwise multiplication.

We write down the following lemma for future reference.

Lemma 2.4. If G is a compact abelian group, ω is a nontrivial character of G,and dx is a Haar measure on G,

∫Gω(x) dx = 0.

Proof. Let a be any element of G on which ω is nontrivial. If dx is a Haar measure,

ω(a)

∫G

ω(x) dx =

∫G

ω(ax) dx =

∫G

ω(x) dx,

which can only hold if∫Gω(x) dx = 0. �

Recall that a Dirichlet character χ mod m is a character of (Z/m)∗, and χ is

primitive if it is not the pullback of a Dirichlet character mod n for any n properlydividing m. By pulling back, we can identify primitive Dirichlet characters with

characters of Z∗ = lim←− (Z/mZ)∗. We will further refine this group to the group of

ideles and then treat characters on these, though the benefit of this will not becomeclear until Section 4.

Construction 2.5. For a given character ω : Kv → S1, we can consider all qua-sicharacters of the form ω(x) · |x|sv for s ∈ C. Parameterized by s, these form aconnected Riemann surface, and the analytic structure we assign this surface isindependent of our choice of ω. These partition all quasicharacters.

The picture is this: if v is infinite, these Riemann surfaces are copies of C, andif v is finite, they are copies of the cylinder given by taking C mod 2πi/ log |πv|−1v .The characters on this surface are the imaginary axis Re(s) = 0 (a circle if v isfinite).

We will now begin to classify characters of K∗v for an arbitrary completion Kv

of K.

Definition 2.6. A quasicharacter ω of K∗v is unramified if it is trivial on Uv;that is, if some complex s exists such that ω(x) = |x|sv for all x. For finite v, aquasicharacter of Kv is unramified if it is trivial on OKv .

We can classify quasicharacters ω of K∗v : by compactness of Uv, ω can be writtenin the form ω′(x) · |x|sv for some character ω′ and some s ∈ C. ω′ is not well-defined,though its restriction to Uv is, and s is not well-defined, though its real part is.

Definition 2.7. The exponent of ω is Re(s).

We can now see the following. Our Riemann surfaces are the cosets of theunramified quasicharacters; that is, they correspond to restrictions to Uv. Thetranslates on each Riemann surface of the characters correspond to exponents.

We can classify these even further with the following observation. If v is fi-nite, a character of Uv = lim←−n (OKv/mnv )

∗is the pullback of a character of some

(OKv/mnv )∗.

4 BAPTISTE DEJEAN

Definition 2.8. The conductor of a character ω of Uv is the smallest c such thatω is the pullback of a character of (OKv/mcv)

∗. If ω is a character of Kv, let the

conductor of ω be the conductor of its restriction to Uv.

Thus, we can specify a quasicharacter by specifying a conductor c, a characterof (OKv/mnv )

∗, and the image of a uniformizer πv; the first two are finite amounts

of data.The classification in the infinite case is even simpler: if Kv = R, a character of

Uv is of the form x 7→ xn, where n is a congruence class mod 2, and if Kv = C, acharacter of Uv is of the form x 7→ xn, where n ∈ Z.

2.2. Restricted direct products. Let (Gv)v∈V be an indexed family of locallycompact groups, cofinitely many of which have an open compact subgroup Hv.Their restricted direct product

∏′v (Gv, Hv) is the group of all elements of

∏v Gv

with cofinitely many components in Hv. Topologize this by taking as an open basesets of the form

∏v Uv intersected with the restricted direct product, where every

Uv ⊆ Gv is open and cofinitely many Uvs contain Hv.We will usually suppress the Hv, writing the product instead as

∏′v Gv.

Now, for any finite collection v1, . . . , vn of indices,∏n

1 Gvi ×∏v 6=v1,...,vn Hv is

an open subset of∏′v Gv, and the subspace and product topologies coincide. The

subspace topology is locally compact by Tychonoff’s theorem, and subsets of thisform cover

∏′v Gv. Therefore

∏′v Gv is again locally compact.

Every Gv embeds into∏′v Gv by sending gv to the tuple whose vth coordinate

is gv and whose other coordinates are 1; call this embedding ιv.For v ∈ V let ωv be a (quasi)character of Gv, and assume cofinitely many

ωvs are unramified. We can now define a (quasi)character of∏′v Gv by the rule

(xv) 7→∏v ωv (xv).

Proposition 2.9. This gives a bijection between (quasi)characters of∏′v Gv and

such collections (ωv) of (quasi)characters.

The inverse map is obvious: define our ωvs by ωv (xv) = ω (xv), where we identifyxv ∈ Gv with its image under ιv. All that remains to be checked is that these areindeed inverse bijections; for a proof, see Lemmas 3.2.1 and 3.2.2 of [9].

Definition 2.10. If ω is a quasicharacter of∏′v Gv, let ωv be its factor at v, and

call (ωv) the local factors of ω.

2.3. Adeles and ideles.

Examples 2.11. We will only consider two examples of restricted direct products.

• The restricted direct product∏′v (Kv,OKv ) of the completions of K is

called the ring of adeles of K and denoted AK . As the name suggests,this is a ring, with product inherited from the unrestricted product

∏vKv.

(Its topology is not inherited from the unrestricted product.) Further, wecan regard K as a discrete subring of AK with the diagonal embeddingx 7→ (x)v.

• The restricted direct product∏′v (K∗v ,Uv) is the group of ideles of K and

denoted A∗K . Just like in the adelic case, we can regard K∗ as a discretesubgroup.

This is the group of units in AK , but with a finer topology than the sub-space topology. It is, however, the coarsest topology refining the subspace

TATE’S THESIS 5

topology under which inversion is continuous; equivalently, the topologyinherited from the embedding

α 7→(α, α−1

): A∗K → AK × AK .

This construction has two points. First, AK and A∗K are locally compact, so onecan do harmonic analysis on them. Second, K sits discretely in AK and K∗ sitsdiscretely in A∗K ; see [2, § 14] and [2, § 16]. However, the following results showthat this is a precarious situation.

Theorem 2.12 (Weak approximation theorem). If V is any finite set of valuationsof K, K is dense in

∏v∈V Kv.

Theorem 2.13 (Strong approximation theorem). If we omit any one of K’s com-pletios, K is dense in the restricted direct product

∏′vKv.

See [2, § 6] and [2, § 15] for proofs.Let K be a number field. If (xv) is an idele, for every finite v let ev be the unique

integer satisfying xv/πevv ∈ Uv. Cofinitely many evs are 0, so

∏v nonarch. p

evv is a

fractional ideal of OK .

Definition 2.14.∏v nonarch. p

evv is the fractional ideal generated by (xv).

(xv) 7→∏v nonarch. p

evv is an epimorphism.

Define a projection πv from A∗Q to Z∗ as follows. If (xv) ∈ A∗Q, cofinitely manyxvs are in their respective Zpv . Therefore a unique rational number α exists suchthat α (xv) = (αxv) has all finite components in their respective Zpv and αx∞ is

nonnegative, namely sgn (x∞) ·∏v 6=∞ |xv|

−1v . Now (α · xv)v 6=∞ is an element of∏

v 6=∞ Z∗pv = Z∗. Let this be πv ((xv)).This is a homomorphism whose kernel contains Q∗.

Construction 2.15. If χ : (Z/m)∗ → S1 is a Dirichlet character, we can pull back

χ−1 under the projection A∗Q → Z∗ → (Z/m)∗

and obtain an idelic character ωtrivial on Q∗.

The reason for using χ−1 is as follows. If p is a prime, consider the idele ιvp(p)whose pth component is p and whose other components are 1. If p is coprime to theconductor of χ, ωp(p) = χ(p) holds, where p is the residue of p mod the conductorof χ.

For a Dirichlet character χ, define its Dirichlet L-function

(2.16) L(s, χ) =∑n≥1

χ(n)

ns=

∏p prime

1

1− χ(p) · p−s

for Re(s) > 1. This will turn out to be (up to a meromorphic factor) a constantof proportionality between two adelic distributions, and the factors of the Eulerproduct will turn out to be constants of proportionality between p-adic distribu-tions. We will also extend L(s, χ) meromorphically to the entire plane and useadelic Fourier analysis to derive a functional equation.

The same theory will apply similarly for Hecke characters and zeta-functions, butto avoid overcomplicating things we will explain this after completing our discussionof Dirichlet L-functions.

6 BAPTISTE DEJEAN

2.4. Abelian Fourier analysis. Here we will give, but not prove, the main resultsof abelian Fourier analysis. The reader wishing to know more may consult [8, § 3].

Let G be a locally compact abelian group with a Haar measure dx.

Definition 2.17. The Pontryagin dual G of G is defined as the group of charactersof G, with the compact-open topology.

Theorem 2.18 (Pontryagin duality). There is a natural isomorphism G ∼= ˆG.

That is, Homcts

(−, S1

)is an involution of the category of locally compact abelian

groups.Pontryagin duality forms the correct context for abstract abelian Fourier analy-

sis. In particular, we will need the following fact.

Definition 2.19. If f ∈ L2(G) ∩ L1(G), define the Fourier transform f : G → Cof f by f(ξ) =

∫Gf(y)ξ(y) dy.

Theorem 2.20. The Fourier transform extends by continuity to give an isomor-phism of L2(G) with L2(G). There is a unique Haar measure on G so that the

Fourier transform is unitary andˆf(x) = f(−x) holds; this measure is said to be

dual to dx.

3. Local theory

3.1. Setting the stage. We will define the topological vector space S (Kv) ofSchwartz-Bruhat functions on Kv as follows. These constructions are largely unim-portant; we just need a well-behaved space of test functions. For example, [9] usesa suitable subspace of L1. That being said, S (Kv) is particularly convenient.

If v is finite, define S (Kv) to be the C-vector space of compactly-supportedlocally-constant functions Kv → C, frivolously topologized as the direct limit of itsfinite-dimensional subspaces. Thus any linear map out of S (Kv) is automaticallycontinuous.

If Kv = R, for f ∈ C∞(R) and N, i ∈ N define

‖f‖(N,i) = supx

((1 + |x|)N ·

∣∣∣f (n)(x)∣∣∣) .

Let S(R) be the space of f such that every ‖f‖(N,i) is finite; that is, f and all itsderivatives are dominated by any power of x. The ‖‖(N,i)s are now seminorms onS(R); let these generate our topology on S(R).

If Kv = C, define S(C) as in the case K = R, but with the seminorms given by

‖f‖(N,i,j) = supx

((1 + |x|)N ·

∣∣∣∂ix∂jxf(x)∣∣∣) .

Both of these are Frechet spaces; see [3, Prop 8.2].

Definition 3.1. A tempered distribution on Kv is a continuous linear functionalon S (Kv); that is, a continuous linear map S (Kv)→ C.

Examples 3.2. The point mass δ0 defined by 〈δ0, f〉 = f(0) is a distribution. Moregenerally, if dx′ is a Radon measure on Kv and v is finite, integration against dx′ isa distribution. This is also true for infinite v if there is some compact K0 and reala, c > 0 such that, on compact sets containing K0, dx′ is dominated by c|x|av dx.

TATE’S THESIS 7

Denote the space of tempered distributions by S (Kv)′. For λ ∈ S (Kv)

′and

f ∈ S (Kv) we will write λ(f) as 〈λ, f〉.We can define an action of K∗v on S (Kv)

′with 〈aλ, f〉 = 〈λ, f

(xa−1

)〉. Given a

quasicharacter ω of K∗v , we can define another action of K∗v on S (Kv)′

by multi-plying by ω(a).

Definition 3.3. If λ is a distribution on which these two actions coincide, we sayλ is an ω-eigendistribution, and denote the space of ω-eigendistributions by S′(ω).

The uncomfortable reader may be convinced this terminology is sound as follows.If λ is an eigendistribution of every λ 7→ aλ, as defined by the first action, triviallythe eigenvalues ω(a) form a quasicharacter, therefore λ is an ω-eigendistribution.

The crux of the local theory is the following.

Lemma 3.4. For any quasicharacter ω, S′(ω) is one-dimensional.

On our quest to prove this, we will meet the factors of the Euler products ofclassical L-function as scale factors. Passing to the global (idelic) picture willmultiply all of these together, yielding the entire L-function as a scale factor.

3.2. Lemma 3.4’s proof. Lemma 3.4 is proven as (coincidentally) Theorem 3.4 of[5]. The arguments we will use here are essentially the same, though we will use lessfancy language at the cost of our setup being less canonical. Our topologies aren’twell-behaved enough to make such grandiose claims as “a distribution decomposesuniquely as a distribution supported at 0 plus a distribution supported away from0,” but we will set up a slightly more careful version of this claim.

If v is finite, let f be any Schwartz-Bruhat function with f(0) 6= 0. There is ashort exact sequence

0 C∞c (K∗v ) S (Kv) C 0,ev0

where C∞c (K∗v ) is the space of functions in S (Kv) supported away from 0. We cansplit this, writing S (Kv) = C∞c (K∗v ) ⊕ Cf . A distribution is thus defined by itsrestriction to C∞c (K∗v ) and its value on f .

Define ω-eigendistributions on K∗v in the obvious way, namely as linear λ :C∞c (K∗v ) → C for which

⟨λ, f

(xa−1

)⟩= 〈ω(a)λ, f〉 for all a ∈ K∗v . The following

facts are clear.

• ω-eigendistributions on K∗v are spanned by ω(x) d∗x.• Any ω-eigendistribution in S′(ω) restricts to an ω-eigendistribution on K∗v .• δ0 spans all distributions supported at 0, and is an ω-eigendistribution if

and only if ω is trivial.

We can now verify Lemma 3.4 by assuming λ is an ω-eigendistribution and askingwhat this forces about 〈λ, f〉. This will essentially reduce to a formal game.

Case 1: v is finite and ω is ramified.Let λ be an ω-eigendistribution. Then some c ∈ C must exist such that λ

coincides with cω(x) d∗x on C∞c (K∗v ). We will check that this determines λ on allremaining functions.

Let a be any element of Uv that ω is nontrivial on. That λ is an ω-eigendistributionforces

〈λ, fov 〉 = ω(a) 〈λ, fov 〉

8 BAPTISTE DEJEAN

to hold, so 〈λ, fov 〉 = 0. It follows that 〈λ, fov (xπ−nv )〉 = 0 for all n ∈ Z; that is, λ is0 on the characteristic function of any mnv .

However, as any f ∈ S (Kv) is constant on a neighborhood mnv of 0, f differs byone of these characteristic functions from a function supported away from 0; thisimplies

〈λ, f〉 =⟨λ, f − f(0)fov

(xπ−nv

)⟩= c

∫K∗v−mnv

f(x)ω(x) d∗x.

This shows that S′(ω) is at most one-dimensional, for λ is determined by c. Fur-ther, we can now check S′(ω) is populated by defining a nonzero ω-eigendistribution

〈z0(ω), f〉 :=

∫K∗v−mnv

f(x)ω(x) d∗x.

This is independent of our choice of n, so long as it’s large enough for f to beconstant on mnv . We can reinterpret z0(ω) in a more natural way as the principalvalue integral

〈z0(ω), f〉 = PV

∫K∗v

f(x)ω(x) d∗x = limn→∞

∫K∗v−mnv

f(x)ω(x) d∗x.

Case 2: v is finite and ω is unramified and nontrivial.We proceed similarly. Let λ be an ω-eigendistribution. Again, some c ∈ C

exists such that λ coincides with cω(x) d∗x on C∞c (K∗v ). As f is Schwartz-Bruhat,f(x)− f

(xπ−1v

)is supported away from 0, implying

⟨λ, f(x)− f

(xπ−1v

)⟩= c

∫ (f(x)− f

(xπ−1v

))ω(x) d∗x.

As λ is an ω-eigendistribution, we also have⟨λ, f(x)− f

(xπ−1v

)⟩= 〈λ, f〉 −

⟨λ, f

(xπ−1v

)⟩= (1− ω (πv)) 〈λ, f〉 .

This gives us

〈λ, f〉 = c (1− ω (πv))−1∫ (

f(x)− f(xπ−1v

))ω(x) d∗x,

showing that S′(ω) is at most one-dimensional. By defining

〈z0(ω), f〉 =

∫ (f(x)− f

(xπ−1v

))ω(x) d∗x,

we obtain a nonzero ω-eigendistribution.Case 3: ω is trivial.Let λ be an ω-eigendistribution, i.e. let λ be K∗v -invariant, and let c be as usual.

We have

〈λ, fov 〉 =⟨λ, fov

(xπ−1v

)⟩,

and

0 =⟨λ, fov (x)− fov

(xπ−1v

)⟩= c

∫Uvd∗x

TATE’S THESIS 9

follows. This forces c to be 0; that is, λ is trivial on functions supported away from0. Therefore λ is a scalar multiple of δ0. Conversely, any scalar multiple of δ0 is anω-eigendistribution.

Notice that, if we define z0(1) as in Case 2, z0(1) = δ0∫Uv d

∗x.

Case 4: v is infinite.We summarize, but do not prove, the results in this case. If we define

Lv(ω) =

π−s/2Γ

(s2

)if Kv = R and ω(x) = |x|s

π−s+12 Γ

(s+12

)if Kv = R and ω(x) = sgn(x) · |x|s

(2π)1−s+|n|2 Γ(s+ |n|

2

)if Kv = C and ω(x) =

(x/|x|1/2

)n |x|s ,then we can holomorphically choose a generator z0(ω) for S′(ω) whose restriction toK∗v is Lv(ω)−1ω(x) d∗x. We refer the reader to [1, Prop 3.1.8], [5, Pages 121-122],and [10] for more details.

3.3. Recap. For all ω, we have just defined a distribution z0(ω), which gives anonzero generator for the one-dimensional vector space S′(ω). Another key point isthat we have parameterized z0(ω) holomorphically in ω. This allows us to concludethat certain constants of proportionality are holomorphic.

Unless ω is the trivial character in the finite case, xn in the real case, or(x/|x|1/2

)n1xn2 in the complex case, the restriction of z0(ω) to K∗v is nonzero.

This means there is another generator, call it z(ω), whose restriction to K∗v isω(x) d∗x. By our calculations in proving Lemma 3.4, we obtain

(3.5) z(ω) = z0(ω) · Lv(ω),

where

(3.6) Lv(ω) =

11−ω(πv) if v is finite and ω is unramified

1 if v is finite and v is ramified

π−s/2Γ(s2

)if Kv = R and ω(x) = |x|s

π−s+12 Γ

(s+12

)if Kv = R and ω(x) = sgn(x) · |x|s

(2π)1−s+|n|2 Γ(s+ |n|

2

)if Kv = C and ω(x) =

(x/|x|1/2

)n |x|s.

Remark 3.7. Notice z(ω) is integration against ω(x) d∗x if ω has positive exponent;that is, if this integral converges. Therefore z(ω) can be viewed as a meromorphicextension of ω(x) d∗x. Lv(ω)−1 is the factor necessary to eliminate the zeroes andpoles of z(ω), yielding a nonzero holomorphic family of distributions z0(ω). Thatz0(ω) be scaled exactly as we have defined it is also critical; see Remark 4.1.

Remark 3.8. If χ is a Dirichlet character, we have already described (Construction2.15) how to pull back χ−1 to obtain a character of A∗Q. If ω is this character’s

factor at a prime p, note Lvp

(ω · ||svp

)= 1

1−χ(p)·p−s is the factor at p of the Euler

product L(s, χ). In Section 5, this will be the connection to Dirichlet (and Hecke)L-functions.

10 BAPTISTE DEJEAN

3.4. Local Fourier analysis and the local functional equation. Much likein the real case, if Kv is a local field and ψ : Kv → S1 is a nontrivial character,y 7→ ψ(y · −) gives an isomorphism of Kv with its character group; that is, Kv

is self-dual. Identifying Kv with its dual in this way, the Fourier transform of a

function f ∈ L1 (Kv) ∩ L2 (Kv) is given by f(ξ) =∫Kv

f(x)ψ(xξ) dx. The Fourier

transform is not only an automorphism of L2 (Kv), but also of S (Kv). The choice

of dx such thatˆf = f(−x) holds is called self-dual with respect to ψ. See [8,

Sec 3.3] and [8, Prop 7-1] for more details.For the rest of this subsection, fix a character ψ : Kv → S1, and define the

Fourier transform using the corresponding self-dual measure dx.

Definitions 3.9. If λ is a distribution, its Fourier transform λ is defined by⟨λ, f

⟩=⟨λ, f

⟩. If ω is a character of K∗v , its shifted dual is ω = ||v · ω−1.

If λ is an ω-eigendistribution, λ is a ω-eigendistribution, for if a ∈ K∗v ,

⟨λ, f

(xa−1

)⟩=⟨λ, f (xa−1)

⟩=⟨λ, |a|v f(xa)

⟩= |a|v

⟨λ, f(xa)

⟩= |a|v · ω

(a−1

) ⟨λ, f

⟩= |a|v · ω−1(a)

⟨λ, f

⟩.

Combining this with Lemma 3.4 and the fact that z0 is holomorphic in ω, weobtain the following.

Theorem 3.10 (Local functional equation). Some holomorphic nonzero factor

εv(ω, ψ) exists such that εv(ω, ψ) · z0(ω) = z0 (ω), and if ω is nontrivial some

meromorphic γv(ω, ψ) exists such that γv(ω, ψ) · z(ω) = z (ω). These are related by

the equation γv(ω, ψ) = εv(ω, ψ) · Lv(ω)Lv(ω).1

Proposition 3.11. For y ∈ Kv, εv(ω, ψ(− · y)) = |y|1/2v ω(y)εv(ω, ψ).

Proof. Work through definitions. �

Therefore it suffices to compute εv(ω, ψ) for a single ψ.To compute εv(ω, ψ), we need only choose a test function at which to evaluate

z0(ω) and z0 (ω). First, we will make some necessary calculations about dy and fov .If v is finite, define the conductor ν of ψ by the condition that mνv is the kernel

of ψ. By definition,

fov (x) =

∫OKv

ψ(xy) dy.

By Lemma 2.4, this works out to be fov (x) = fov (xπ−νv )∫OKv

dy. Similarly,fov (x) =

fov · |πv|νv

(∫OKv

dy)2

; for Fourier inversion to hold, dy must assign OKv the measure

|πv|−ν/2v , so fov (x) = |πv|−ν/2v fov (xπ−νv ).Now we actually compute εv(ω, ψ). If v is finite and ω is unramified, we see that

1Tate proved this differently, using the formula 〈z(ω), f〉 〈z(ω), g〉 =⟨z(ω), f

⟩〈z(ω), g〉 to

establish this functional equation on the strip 0 < Re(s) < 1 and using it to extend z. See [9,Thm 2.4.1].

TATE’S THESIS 11

(3.12) 〈z0(ω), fov 〉 =

∫K∗v

(fov (x)− fov

(xπ−1v

))ω(x) d∗x

=

∫Uvd∗x+

∫K∗v−Uv

0 d∗x =

∫Uvd∗x.

As z0(ω) is an ω-eigendistribution,⟨z0(ω), fov

⟩=⟨z0(ω), |πv|ν/2v fov

(xπ−νv

)⟩= ω (πνv ) |πv|ν/2v 〈z0(ω), fov 〉 ,

and 〈z0(ω), fov 〉 = 〈z0(ω), fov 〉 directly from the definition of z0; therefore εv(ω, ψ) =

ω (π−νv ) · |πv|−ν/2v .The other cases are similar; we briefly describe them. If v is finite and ω is

ramified, let c be the conductor of ω. We use the test function

gω(x) =

{ψ(x) if x ∈ mν−cv

0 else,

whose Fourier transform is given by

gω(ξ) =

{|πv|ν/2−c if ξ ≡ 1 (mod mcv)

0 else.

For the case Kv = R, we will only give εv(ω, ψ) for the character ψ(x) = e−2πi·x.

If ω(−1) = 1, use the test function e−π·x2

, and if ω(−1) = −1, use the test function

x · e−π·x2

.For the case Kv = C, we will only give εv(ω, ψ) for the character ψ(x) =

e−4πi·Re(x) and let n ∈ Z be the integer such that ω(x) = xn on S1. If n ≥ 0,

use the test function x|n|e−2π·|x|, and if n ≤ 0 use x|n|e−2π·|x|.For more details on the computation of εv, see [5, Prop 3.8] or [9, Sec 2.5]. (Note

that Tate’s factor ρ is the inverse of our factor γ.) We now list the results:

(3.13) εv(ω, ψ) =

ω (π−νv ) · |πv|−ν/2v if v is finite and ω is unramified∫πν−cv Uv ω

−1(x)ψ(x) dx if v is finite and ω is ramified

with conductor c

1 if Kv = R, ω(−1) = 1,

and ψ(x) = e−2πi·x

−i if Kv = R, ω(−1) = −1,

and ψ(x) = e−2πi·x

(−i)|n| if Kv = C, ω(x) = xn on S1,

and ψ(x) = e−4πi·Re(x)

.

The remaining cases follow from Proposition 3.11. We can rewrite the case wherev is finite and ω is unramified more conveniently as the Gauss sum

(3.14) εv(ω, ψ) = ω(πc−νv

)|πv|ν/2v

∑x∈(OKv/mcv)

∗

ω−1(x)ψ(πν−cv x

).

12 BAPTISTE DEJEAN

4. Global theory

4.1. From local analysis to global analysis. We have already seen how localcharacters multiply to give global characters; here we will do the same for Schwartz-Bruhat functions, distributions, and measures. We will point out that most of thisis to work out a framework in which we can multiply the local instances of (3.5)and Theorem 3.10. For more details and proofs, we refer the reader to [9, § 3] and[5, Sec 4].

Define a topological vector space S (AK) of Schwartz-Bruhat functions on AK asthe “restricted tensor product” of all of our S (Kv)s; that is, as the space spannedby all formal products

⊗v fv, where every fv is in S (Kv), cofinitely many fvs

are fov , and linearity in every fv is imposed. Topologize this as the direct limit ofthe tensor products of finitely many S (Kv)s. Notice every element

∑i

⊗v fi,v of

S (AK) defines a real function (xv) 7→∑i

∏v fi,v (xv) on AK . This assignment is

easily seen to be injective, so we will identify S (AK) with a space of functions ofAK .

As before, define a tempered distribution as a continuous linear functional onS (AK). A tempered distribution is of the form

⊗v fv 7→

∏v 〈λv, fv〉, where every

λv is in S (Kv)′

and cofinitely many λvs are trivial on fov ; see [5, Lem 4.1]. Denotethis distribution

∏v λv.

To multiply Haar measures, let∏′v Gv be a restricted direct product, and choose

Haar measures dxv on every Gv of K such that cofinitely many OKvs are assignedmeasure 1. Define a measure “dx =

∏v dxv” on

∏′v Gv as follows. Say we have

Borel Xv ⊆ Kv, cofinitely many of which are OKv . Give∏vXv the measure∏

v

∫Xvdxv. Extend by declaring dx to be a Radon measure; dx is now a Haar

measure. For a more rigorous construction, see [9, Sec 3.3].

Remark 4.1. By (3.12), the normalizations required to multiply local Haar measureson K∗v and to multiply our local z0s coincide: we need cofinitely many Uvs to beassigned measure 1. This is not a coincidence. For the surface of unramifiedω, the holomorphic family z0(ω) of distributions is determined up to a nonzeroholomorphic factor by the requirement that it be a basis element of S′(ω). Thisholomorphic factor is fixed by the requirement that 〈z0(ω), fov 〉 = 1 for all unramifiedω. That is, we deliberately scaled z0(ω) to make these conditions coincide. Fromnow on, we will fix measures d∗xv assigning cofinitely many Uvs measure 1.

Notation 4.2. Let dx be a Haar measure on AK , and let d∗x be a Haar measureon A∗K constructed as a product of the local measures we just fixed.

Integrating Schwartz-Bruhat functions on AK is easy: if⊗

v fv is a generator ofS (AK),

(4.3)

∫AK

⊗v

fv dx =∏v

∫Kv

fv dxv.

We have a global “absolute value” that serves the same purpose as our local ones.Define || : A∗K → R∗ by |(xv)| =

∏v |xv|v; that is, let || be the quasicharacter whose

local factors are ||v. As a corollary of our construction of a global Haar measurefrom local ones, we see that d(ax) = |a|dx for all ideles a, therefore d∗x and dx

|x|agree up to a constant.

TATE’S THESIS 13

Notations 4.4. We collect all our global notation in one place so we don’t losethe reader.

• AK is the restricted direct product∏′vKv.

• A∗K is the restricted direct product∏′vK∗v .

• dx is a Haar measure on AK .• d∗x is a Haar measure on A∗K .• || is the global absolute value we just defined.

We will also need the following.

• UAK is group of ideles of absolute value 1.

4.2. Remarks about characters of A∗K .

Proposition 4.5. || splits after restricting to its image.

Proof. If K is a number field, K has an infinite valuation v, from which one canconstruct a splitting of ||: send x ∈ (0,∞) to ιv(x) if Kv = R or ιv (

√x) if Kv = C.

If K has positive characteristic, the image of || is isomorphic to Z, so the conclusionfollows by abstract nonsense. �

Rephrasing our characterization of local quasicharacters slightly, pick a splittingA∗K ∼= UAK ⊕ im ||, and if x ∈ A∗K , let x ∈ UAK be the first component of x in thissplitting. For a quasicharacter ω of A∗K , we might hope to write ω = ω′ (x) · |x|s forsome character ω′ of UAK and let the exponent of ω be Re(s). Unfortunately, UAK isnot compact, so we cannot always do this. However, UAK/K∗ is compact. One seesthis by constructing a fundamental domain for K∗ in UAK whose closure is compactby Tychonoff’s theorem; this closure now surjects to UAK/K∗ under the quotientmap, forcing UAK/K∗ to be compact. The construction of this fundamental domain,while beautiful, would lead us too far astray; the reader may consult [9, Sec 4.3].

Therefore, if ω is a character which is trivial on K∗, the restriction of ω to UAKis in fact a character, so we can write ω = ω′ (x) · |x|s and define the exponent of ωto be Re(s). This is independent of our decomposition A∗K ∼= UAK ⊕ im ||.

We will restrict our attention to these characters. (The actual reason for doingso is to make Theorem 4.12 hold).

Convention 4.6. A character of A∗K is assumed to be trivial on K∗.

As an important note, || is still a quasicharacter; that is, |x| = 1 for x ∈ K∗. Wecan show this by using the norm to reduce this to the cases K = Q and K = Fp(t),which we can directly check, or can show this by constructing a fundamental domainD for K in AK and showing that D and xD have the same measure.

We can now construct a picture of global quasicharacters much like the one forlocal quasicharacters. Given a global character ω, the space of all quasicharacters ofthe form ω(x)|x|s for s ∈ C forms a Riemann surface under this parameterization;this is a cylinder if K has positive characteristic and a plane if K is a numberfield. This Riemann surface structure is independent of our choice of representativecharacter ω; the characters correspond to the imaginary axis Re(s) = 0. Thetranslates of this imaginary axis (a circle if K has positive characteristic) correspondto the different exponents of quasicharacters.

4.3. From local eigendistributions to global eigendistributions. For a char-acter ω of A∗K , define two actions of A∗K on the space S (AK)

′of distributions by

defining aλ as ω(a)λ or by 〈aλ, f〉 =⟨λ, f

(xa−1

)⟩, and define an ω-eigendistribution

14 BAPTISTE DEJEAN

as a distribution on which these two actions coincide. By Lemma 3.4, we obtainthe following global dimension result.

Corollary 4.7. The space of ω-eigendistributions is one-dimensional and spannedby the distribution

z0(ω) =∏v

z0 (ωv) .

We have just defined a global version of z0; we will now do the same to z.

Definition 4.8.

〈z(ω), f〉 =

∫A∗K

f(x)ω(x) d∗x,

provided this integral converges for all f .

By Remark 3.7, (4.3), and appropriate convergence theorems, this converges forω of exponent greater than 1, and 〈z(ω),

⊗v fv〉 =

∏v 〈z (ωv) , fv〉. Therefore, for

such ω, multiplying the local instances of (3.5) yields

(4.9) z(ω) = z0(ω) · Λ(ω),

where Λ(ω) =∏v Lv (ωv).

Note that z and Λ are meromorphic in ω where they are defined, and furtherthat z0 is holomorphic in ω and nonzero.

4.4. Global Fourier analysis. Let (ψv)v be nontrivial characters of the comple-tions Kv of K, cofinitely many of which have kernel OKv . By Proposition 2.9,∏v ψv is a character of AK , and y 7→ ψ(y · −) gives an isomorphism of AK with its

character group; that is, AK is self-dual.If we further require ψ to be trivial on K, this restricts to an isomorphism of K

with the character group of AK/K. It is enough to take on faith for now that sucha ψ exists; we will explicitly construct one at the beginning of Section 5, and thatthis ψ works is [9, Thm 4.1.4] and [9, Lem 4.1.5].

Let ψ be a character of AK , trivial on K. If f ∈ S (AK), define its Fourier

transform by f(ξ) =∫AK f(x)ψ(xξ) dx. By (4.3), the Fourier transform of

⊗v fv

is⊗

v fv, defined by the local Fourier transforms given by the local ψvs. Thereforethe Fourier transform is an automorphism of S (AK), and we can normalize dx soˆf(x) = f(−x) holds. In fact, this self-dual dx is just the product

∏v dxv of the

local self-dual measures with respect to the local ψvs.

We define the Fourier transform of a distribution λ, again, by⟨λ, f

⟩=⟨λ, f

⟩.

We can now multiply the local instances of Theorem 3.10 and obtain a globalanalogue.

Theorem 4.10. There is a nonzero holomorphic factor ε(ω) such that ε(ω)·z0(ω) =

z0 (ω); ε(ω) is given by∏v εv (ωv, ψv). (This is in fact a finite product.)

Notice ε(ω) doesn’t depend on ψ by Proposition 3.11.

TATE’S THESIS 15

4.5. Global functional equation. We can try to extend z(ω) by (4.9), but thisrequires a meromorphic extension of Λ. Instead, we reverse the situation, extendingz(ω) meromorphically and defining Λ(ω) as the factor making (4.9) true. Once thisis done, (4.9) and Theorem 4.10 combine to yield the equation

(4.11) z(ω) =Λ(ω)

Λ (ω) · ε(ω)· z (ω).

We may be tempted to call this the global functional equation. Instead, we willgive that name to the following equation, which is the critical computation of Tate’sthesis.

Theorem 4.12 (Global functional equation). z has a meromorphic extension toall quasicharacters satisfying

z(ω) = z (ω).

Corollary 4.13. Λ has a meromorphic extension satisfying Λ (ω) = ε(ω) · Λ(ω).

Proof. Combine Theorems 4.11 and 4.12. �

Before proving Theorem 4.12, we need a prerequisite.

Lemma 4.14 (Poisson summation). For f ∈ S (AK),∑x∈K

f(x) =∑ξ∈K

f(ξ).

Proof sketch. Say∑s∈K f(x + y) converges absolutely and uniformly on compact

sets for any y ∈ AK , and the same holds for f . Then this formula follows fromrepeating the classical proof; see [9, Lem 4.2.4] or [8, Thm 7-7]. Further, anyf ∈ S (AK) satisfies these conditions by [8, Lem 7-6].

�

Remark 4.15. The triviality of ψ on K forces the self-dual measure dx to assignthe measure 1 to a fundamental domain for K. Poisson summation, in this form, isequivalent to this fact; Tate [9] uses it to prove Poisson summation. Ramakrishnanand Valenza [8] avoid using this fact, instead taking the scaling of dx into accountvia Fourier inversion.

Theorem 4.16 (Riemann-Roch). For f ∈ S (AK) and a ∈ A∗K ,

|a|−1∑x∈K

f(xa−1

)=∑ξ∈K

f(ξa).

Proof. Apply Lemma 4.14 to f(xa−1

). �

If K has positive characteristic, the case f =⊗

v fov recovers the classical

Riemann-Roch theorem for curves over finite fields; see [8, Thm 7.12]. This cantherefore can be viewed as a restatement and number-theoretic analogue of theclassical Riemann-Roch theorem.

16 BAPTISTE DEJEAN

Proof of Theorem 4.12. We treat only the case that K is a number field. Thepositive-characteristic case is similar; see [8, Thm 7-16].

Choose a splitting A∗K = UAK ⊕ im ||, identifying im || = (0,∞) with a subgroupof A∗K . Let dx be Lebesgue measure, and let dxr = dx/x. There is now a uniquemeasure dxu on UAK such that d∗x on A∗K is the product dxu × dxr.

If ω has exponent greater than 1, we see

〈z(ω), f〉 =

∫A∗K

f(x)ω(x) d∗x =

∫ ∞0

∫UAK

f(xt)ω(xt) dxu dtr

=

∫ ∞1

∫UAK

f(xt)ω(xt) dxu dtr +

∫ 1

0

∫UAK

f(xt)ω(xt) dxu dtr

=

∫ ∞1

∫UAK

f(xt)ω(xt) dxu dtr +

∫ ∞1

∫UAK

f(xt−1

)ω(xt−1

)dxu dtr.

We treat the second term. Let E be a fundamental domain for K∗ in UK ; weremind the reader that [9, Sec 4.3] has a construction. Notice∫

UAKf(xt−1

)ω(xt−1

)dxu =

∫E

∑y∈K∗

f(xyt−1

)ω(xt−1

)dxu.

We can almost apply Riemann-Roch 4.16; we first need to add the final term∫Ef(0)ω

(xt−1

)dxu. Once we have, Riemann-Roch gives

∫UAK

f(xt−1

)ω(xt−1

)dxu +

∫E

f(0)ω(xt−1

)dxu

=

∫E

∣∣x−1t∣∣ ∑ξ∈K

f(ξx−1t

)ω(xt−1

)dx =

∫E

∑ξ∈K

f(ξx−1t

)ω(x−1t

)dxu.

Reversing steps yields that this is∫UAK

f(xt)ω(xt) dxu +

∫E

f(0)ω(xt) dxu.

We have, therefore,

〈z(ω), f〉 =

∫ ∞1

∫UAK

f(xt)ω(xt) dxu dtr +

∫ ∞1

∫UAK

f(xt)ω(xt) dxu

− f(0)

∫ ∞1

∫E

ω(xt−1

)dxu dtr + f(0)

∫ ∞1

∫E

ω(xt) dxu dtr

If ω is unramified, that is if ω is nontrivial on UAK , the last two terms are 0.Otherwise, ω(x) = |x|s for some s ∈ C, so these terms are

− f(0)

∫ ∞1

|t|−s dtr∫E

dxu + f(0)

∫ ∞1

|t|1−s dtr∫E

dxu

= −

(f(0)

s+f(0)

1− s

)∫E

dxu.

TATE’S THESIS 17

In the notation of [9], we obtain

(4.17) 〈z(ω), f〉 =

∫ ∞1

∫UAK

f(xt)ω(xt) dxu dtr +

∫ ∞1

∫UAK

f(xt)ω(xt) dxu

−

{{(f(0)

s+f(0)

1− s

)∫E

dxu

}},

where the expression {{g(s)}} is 0 if ω is unramified and g(s) if ω = ||s. (4.17)now converges for any ω other than 1 and ||, not just ω of exponent greater than 1.

We can therefore define z(ω) by (4.17), and the functional equation z(ω) = z (ω) isimmediate.

�

As a bonus, z has simple poles exactly at 1 and ||. With the parameterization ||sof the Riemann surface containing these, we can immediately read off the residues

as −δ0∫Edxu at 1 and δ0

∫Edxu at ||.

5. Computations

Now we will compute specific examples. We will identify prime integers with thecorresponding finite valuations of Q.

If K = Q, we can choose additive characters as follows. Let ψ∞(x) = e−2πi·x.For a finite prime p, Qp/Zp is (algebraically) isomorphic to Z [1/p] /Q, which (alge-braically) embeds into R/Z. Composing these gives a homomorphism λ : Qp/Zp →R/Z. Let ψp(x) = e2πi·λ(x); this is an additive character of conductor 0. Every ψpis unramified, hence ψ =

∏p ψp gives a character of AQ trivial on Q.

For an arbitrary number field K and a valuation v of K lying over a valuationp of Q, we can define ψv by composing ψp with the trace map Kv → Qp. If p is

finite, this has kernel diff−1 (Kv/Qp). Cofinitely many of these differents are 1, soψ =

∏v ψv gives a character of AK , which is also trivial on K.

For finite v, let ν be the conductor of ψv; that is, m−νv = diff (OKv/Zp). As wechecked in Subsection 3.4, for finite v, the measure on Kv which is self-dual with

respect to ψv assigns OKv the measure |πv|−ν/2v .For more details, see [9, Sec 2.2] and [9, Lem 4.1.5]. We will now fix this choice

of ψ for computation of ε.

5.1. Warm-up: Riemann zeta function. Let K be Q. For s ∈ C, let ωs be||s, so ωs = ||1−s. The local factors of ωs are then ||sv. Λ (ωs) is the Riemann zetafunction ζ(s) times the extra factor L∞ (ωs) = π−s/2Γ

(s2

). Our extension of Λ thus

gives an extension of ζ. For our functional equation, every εp is 1, so we obtain

Λ (ωs) = Λ (ω1−s) ;

that is,

(5.1) π−s/2Γ(s

2

)ζ(s) = π

s−12 Γ

(1− s

2

)ζ(1− s).

This is the classical functional equation for ζ.

18 BAPTISTE DEJEAN

5.2. Dirichlet L-functions. Let χ be a primitive Dirichlet character mod m. Aswe have already described (Construction 2.15), there is an idelic character ω, trivialon Q∗, such that Lp (ω · ||s) = 1

1−χ(p)·p−s for finite primes p. Therefore

Λ (ω · ||s) = L(s, χ) · L∞ (ω · ||s)

for s with Re(s) > 1. This equation gives a meromorphic extension of L(s, χ), andby Corollary 4.13 it satisfies

L(s, χ)L∞(ω · ||s) = ε(ω · ||s)L(1− s, χ−1)L∞(ω−1 · ||1−s).

We will now compute our local ε factors. For finite p relatively prime to m, ωpis unramified and ψp has conductor ν = 0, so by (3.13) εp(ωp · ||sp, ψp) = 1.

As ω∞ is a character of R∗, we only have two choices for ω∞, determinedby the image of −1. ω∞(−1) = χ(−1)−1 = χ(−1), giving ε∞ (ω∞ · ||s∞, ψ∞) ={

1 if χ(−1) = 1

i if χ(−1) = −1.

We now need only to deal with the remaining ε factors. The conductor cp of ωpis the multiplicity with which p divides m. Computing, we obtain

∏p|m

εp (ωp · ||p, ψp) =∏p|m

∫p−cpZ∗p

ω−1p (x)|x|−sp ψp(x) dx

= m−s∫m−1·

∏p|m Z∗p

ω−1

∏p|m

ιp (xp)

·∏p|m

ψp (xp) dx

= m−s∑

x∈(Z/m)∗

χ (x) ·∏p|m

e2πi·xp/m = m−s∑

x∈(Z/m)∗

χ(x) · e2πi·x/m,

where xp denotes the image of x under the composition Z/m → Z/pcp → Z/mof maps given by the Chinese remainder theorem. We collect our final functionalequation: it is

(5.2)

L(s, χ)L∞ (ω · ||s) = ε∞L(1− s, χ−1)L∞(ω−1 · ||1−s

)m−s

∑x∈(Z/m)∗

χ(x) · e2πi·x/m,

where

ε∞ =

{1 if χ(−1) = 1

−i if χ(−1) = −1,

L∞ (ω · ||s) =

{π−s/2Γ

(s2

)if χ(−1) = 1

π−s+12 Γ

(s+12

)if χ(−1) = −1

,

and L∞(ω−1 · ||1−s

)=

{πs−12 Γ

(1−s2

)if χ(−1) = 1

πs−22 Γ

(2−s2

)if χ(−1) = −1

.

TATE’S THESIS 19

5.3. Hecke characters. Let K be a number field. Fix an ideal f of OK , and letf =

∏p p

cp be its factorization into prime ideals. For x ∈ OK , let x denote theresidue of x residue mod f.

Definition 5.3. A Hecke character mod f is a character χ of the group of fractionalideals relatively prime to f satisfying the following condition: a character ω∞ of∏v arch.K

∗v exists such that, for all x ≡ 1 (mod f), χ((x)) = ω−1∞ (x). Notice χ

uniquely specifies ω∞.

Definition 5.4. A Hecke character mod f is called primitive if it is not induced(by restriction) by a Hecke character mod f′ for any f′ properly dividing f.

If χ is a Hecke character mod f, χf (x) = χ((x)) · ω∞(x) gives a well-definedcharacter of (OK/f)∗. Hence we can define Hecke characters as characters whichfactor as χf (x) · ω−1∞ (x), for some χf and ω∞, on principal ideals coprime to f.

Given a Hecke character χ mod f, we can define an L-function

(5.5) L(s, χ) =∑I

χ(I)

‖I‖s=∏p

1

1− χ(p)‖p‖−s,

where the sum is taken over ideals of OK relatively prime to f, the product istaken over primes of OK not dividing f, and ‖I‖ = |OK/I|. As in the Dirichletcase, we want to interpret this as a product of most of the factors of Λ (ω · ||s) foran appropriate ω. We give a very explicit and computational, yet not very clean,construction of ω.

• If p is a prime not dividing f, the requirement Lvp

(ωvp · ||svp

)= 1

1−χ(p)‖p‖−s

forces ωvp to be unramified with ωvp (πv) = χ−1(p).• For infinite v, let ωv be the factor of ω∞ at v.• For every remaining valuation v, let ωv be the factor of ωram at v, whereωram :

∏p|fK

∗vp → S1 is defined as follows. First, define ωram on

∏p|f Uvp

as the pullback of χ−1f under the quotient∏

p|f Uvp → (OK/f)∗. The factor-

ization χ((x)) = χf (x)·ω−1∞ (x) now guarantees that, for whatever extensionωram we choose, ω(x) =

∏v ωv(xv) is trivial on every α ∈ K∗ coprime to f.

We now need to extend ωram to the rest of∏v|fK

∗v → S1 to make ω

trivial on all of K∗. This is obtained by the weak approximation theorem2.12. (Continuity follows by translating to the case x ∈

∏v|f Uv, where we

have already explicitly constructed a continuous ωram.)

This gives a map from Hecke characters χ to characters ω of A∗K .We can go the other way: let ω be a character of A∗K and f an ideal such that, for

every prime p, pcp |f (where cp is the conductor of ωv). Then we can define a Heckecharacter χ by letting χ(p) = ωvp

(πvp)

and extending by linearity. The associatedω∞ is then

∏v arch. ωv.

These are inverse bijections between primitive characters and characters of A∗K .

Hence, the classical and idelic pictures are equivalent. χ−1f is (with some obvious

identifications) the product of the ramified ωvs, restricted to their Uvs. ω∞ is ω∞.For the unramified ωvs, ωv (πv) encode χ itself. The remaining ωv (πv)s, thoughnot seen in the classical picture, are there to make ω trivial on K∗.

As a corollary of the equivalence between the classical and idelic pictures, forK a number field, the content of Corollary 4.13 is exactly the functional equation

20 BAPTISTE DEJEAN

for L(s, χ) for primitive Hecke characters χ. In particular, if we define Λ(s, χ) =Λ (ω · ||s), where χ and ω are corresponding Hecke and idelic characters, we obtain

(5.6) Λ(s, χ) = L(s, χ) ·∏

v arch.

Lv (ωv · ||sv)

for Re(s) > 1. The meromorphic extension and functional equation of Λ yields thesame for L.

Finally, we point out that Dirichlet characters are the case K = Q; ω∞ =1 corresponds to χ(−1) = 1 and ω∞ = sgn corresopnds to χ(−1) = −1. Thedistinction between these two kinds of Dirichlet characters comes (in this view)from their having different infinite parts.

5.4. Example: Hecke characters of Q(√

5)

mod(√

5). To illustrate the situ-

ation just described, we will work out the exercise on [5, Page 130]: compute the

functional equations of L(s, χ) for Hecke characters of Q(√

5)

mod f =(√

5).

First, we will determine which ω∞s are allowed. There are two infinite places ofQ(√

5), namely

∣∣a+ b√

5∣∣∞1

=∣∣a+ b

√5∣∣ and

∣∣a+ b√

5∣∣∞2

=∣∣a− b√5

∣∣, and both

of these are real. ω∞ is a character of R∗ ×R∗; therefore we may write it uniquelyas ω∞ (x1, x2) = |x1|a1 |x2|a2 sgn (x1)

s1 sgn (x2)s2 for a1, a2 ∈ R · i and s1, s2 ∈ Z/2.

For ω∞ to correspond to a Hecke character χ, ω−1∞ (x) = χ((x)) for x ≡ 1

(mod(√

5)) forces ω∞ to be trivial on units of OQ(

√5) which are congruent to

1 mod(√

5). As −1 and 1+

√5

2 generate the units of OQ(√5), the group of units

congruent to 1 mod(√

5)

is principal and generated by −(

1+√5

2

)2. ω∞ is trivial

on −(

1+√5

2

)2if and only if

∣∣∣ 1+√52

∣∣∣2(a1−a2) · (−1)s1+s2 = 1; that is, some n ∈ Z

exists such that a1 − a2 = πin/2 log(

1+√5

2

)and s1 + s2 ≡ n (mod 2).

Say we have such an ω∞. As OQ(√5) is a PID and its units surject to those of

OQ(√5)/(√

5) ∼= F5, we can retrieve the χ with this ω∞ by defining χ((x/y)) =

ω−1∞ (x) ·ω∞(y) for x, y ≡ 1 (mod(√

5)). This gives a well-defined character on all

ideals coprime to(√

5). Computing the corresponding χf, we see

χf(3) = χf

1 +√

5

2

= ω∞

(1 +√

5

2

)χ

((1 +√

5

2

))= ω∞

(1 +√

5

2

),

and 3 generates OQ(√5)/(√

5) ∼= F∗5.

Let ω be the idelic character corresponding to χ. At every finite place v otherthan v√5,

∣∣√5∣∣v

= 1 and ωv is unramified, so ωv(√

5)

= 1. As ω(√

5)

= 1,

ωv√5

(√5)

= ω−1∞(√

5). This, that ωv√5

has conductor 1, and ωv√5(3) = χ−1f

(3)

=

ω−1∞

(1+√5

2

)completely determine ωv√5

.

The description of ωv for other finite v is straightforward. Pick a generator xof pv which is congruent to 1 mod

(√5). ωv is unramified, and ω(x) = 1 forces

ωv (πv) = ωv(x) = ω−1∞ (x) to hold.

TATE’S THESIS 21

We can now finally compute all our local ε factors. diff(OQ(

√5)/Z

)=(√

5), so

εv (ωv · ||sv, ψv) = 1 for every finite v other than v√5. From (3.14), we obtain

εv√5

(ωv√5

· ||sv√5, ψv√5

)= 51/2−2sω−1∞ (5)

∑x∈F∗5

χf(x)e4πi·x/5.

Our infinite ε-factors are

ε∞i

(ω∞i

· ||s∞i, ψ∞i

)=

{1 if ai = 0

i if ai = 1.

Our functional equation is therefore

Λ(s, χ) = Λ(1− s, χ−1

)ε∞ · 51/2−2sω−1∞ (5)

∑x∈F∗5

χf(x)e4πi·x/5,

where

Λ(s, χ) = L(s, χ) · L∞1

(ω∞1

· ||s∞1

)· L∞2

(ω∞2

· ||s∞2

),

Λ(1− s, χ−1

)= L

(1− s, χ−1

)· L∞1

(ω−1∞1

· ||1−s∞1

)· L∞2

(ω−1∞2

· ||1−s∞2

),

ε∞ =

1 if a1 = a2 = 0

−i if a1 6= a2

−1 if a1 = a2 = 1

,

and L∞1 and L∞2 are as in 5.2.

5.5. Final remarks. We refer the reader to [8] for classical applications; amongthese are the Tchebotarev density theorem.

If K is a field of positive characteristic containing Fq, then ζK(s) = Λ (||s) is infact a rational function in q−s! See Exercise 22 in Chapter 7 of [8] for a proof. Theproperties of this rational function are generalized by Weil’s celebrated conjectures,now resolved.

22 BAPTISTE DEJEAN

References

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[2] Cassels, J. W. S. “Global Fields.” In Algebraic Number Theory, edited by J. W. S. Cassels

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John Wiley & Sons, Inc., 1999.

[4] Hecke, E. “Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung derPrimzahlen.” Mathematische Zeitschrift, Issue 1-2, Volume 6 (March 1920): 11-51.

[5] Kudla, Stephen S. “Tate’s Thesis.” In An Introduction to the Langlands Program, edited byJoseph Bernstein and Stephen Gelbart, 109-131. New York: Springer Science+Business Media,

2004.

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[8] Ramakrishnan, Dinakar, and Valenza, Robert J. Fourier Analysis on Number Fields. London:Academic Press Inc., 1967.

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