TATE's THESIS Contents 1. Introduction 2 2. Motivation

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 methods 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] (who follows Weil [10],) we reinterpret the pervasive proportionality of distributions. To illustrate the theory, we compute a few concrete examples.

Contents 1. Introduction 2 2. Motivation, characters 2 2.1. Characters 3 2.2. Restricted direct products 4 2.3. Ad`elesand id`eles 4 2.4. Abelian Fourier analysis 6 3. Local theory 6 3.1. Setting the stage 6 3.2. Lemma 3.4’s proof 7 3.3. Recap 9 3.4. Local Fourier analysis and the local functional equation 10 4. Global theory 12 4.1. From local analysis to global analysis 12 ∗ 4.2. Remarks about characters of AK 13 4.3. From local eigendistributions to global eigendistributions 13 4.4. Global Fourier analysis 14 4.5. Global functional equation 15 5. Computations 17 5.1. Warm-up: Riemann zeta function 17 5.2. Dirichlet L-functions 18 5.3. Hecke characters 19 √ √ 5.4. Example: Hecke characters of Q 5 mod 5 20 5.5. Final remarks 21 References 22

Date: October 26, 2017. 1 2 BAPTISTE DEJEAN

1. Introduction L-functions are of great classical interest, for they contain detailed information about distribution of primes. Obtaining their meromorphic extensions to the complex plane and their functional equations are key tools for studying them. In his thesis, Tate [9] used fairly simple Fourier analysis to obtain these extensions and functional equations for a specific class of L-functions, a significant improvement over 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 distributions will turn out to be one-dimensional, forcing important distributions to be related by scale-factors. One of these scale factors is, conveniently, the local factor of a classical L-function. In Section 4, adèlesand idèles will be useful global contexts, for we can produce their pictures by multiplying together all local pictures. Therefore, the entire L- function whose factors appeared locally will appear globally. There will be a scale factor which we can compute in terms of this L-function; however, we can also use Poisson summation to compute this scale factor as 1. This gives a functional equation for our L-function. The reader should note that there only two crucial theoretical points in Sections 3 and 4, namely Lemma 3.4 and Theorem 4.12. Finally, we will show the relationship to classical pictures in Section 5, computing the functional equations for Dirichlet L-functions and certain Hecke L-functions √ over Q 5 .

2. Motivation, characters By “valuation” we mean a norm; that is, we write codomains multiplicatively and allow archimedean valuations. By a ﬁnite valuation we mean a nonarchimedean valuation, 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.

•O Kv is the ring of integers in Kv for finite v, mv is its maximal ideal, and πv is a uniformizer. ∗ •U v is the group of elements of Kv of norm 1. ∗ • dx is an additive Haar measure on Kv, and d x is a multiplicative Haar ∗ measure on Kv . • 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 prime p of OK , vp is the valuation induced by it. o • If v is finite, fv is the characteristic function of OKv .

We will tacitly normalize all valuations v so that multiplication by any α ∈ Kv ∗ dx scales dx by |α|; that is, d(ax) = |a|vdx. Thus, d x and agree up to a constant. |x|v 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 is a continuous homomorphism ω : G → C∗, and a character of G is a quasicharacter whose image lands in S1. Remarks 2.3. If G is compact, any quasicharacter is a character. Quasicharacters and 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, R and dx is a Haar measure on G, G ω(x) dx = 0. Proof. Let a be any element of G on which ω is nontrivial. If dx is a Haar measure, Z Z Z ω(a) ω(x) dx = ω(ax) dx = ω(x) dx, G G G R which can only hold if G ω(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 properly dividing m. By pulling back, we can identify primitive Dirichlet characters with characters of ∗ = lim ( /m )∗. We will further refine this group to the group of Zb ←− Z Z idèlesand then treat characters on these, though the benefit of this will not become clear until Section 4. 1 Construction 2.5. For a given character ω : Kv → S , we can consider all qua- s sicharacters of the form ω(x) · |x|v for s ∈ C. Parameterized by s, these form a connected Riemann surface, and the analytic structure we assign this surface is independent of our choice of ω. These partition all quasicharacters. The picture is this: if v is infinite, these Riemann surfaces are copies of C, and −1 if v is finite, they are copies of the cylinder given by taking C mod 2πi/ log |πv|v . The characters on this surface are the imaginary axis Re(s) = 0 (a circle if v is finite). ∗ We will now begin to classify characters of Kv for an arbitrary completion Kv of K. ∗ Definition 2.6. A quasicharacter ω of Kv is unramified if it is trivial on Uv; s that is, if some complex s exists such that ω(x) = |x|v for all x. For finite v, a quasicharacter of Kv is unramified if it is trivial on OKv . ∗ We can classify quasicharacters ω of Kv : by compactness of Uv, ω can be written 0 s 0 0 in the form ω (x)·|x|v 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 the unramified quasicharacters; that is, they correspond to restrictions to Uv. The translates on each Riemann surface of the characters correspond to exponents. We can classify these even further with the following observation. If v is finite, a character of U = lim (O /mn)∗ is the pullback of a character of some v ←−n Kv v n ∗ (OKv /mv ) . 4 BAPTISTE DEJEAN

Definition 2.8. The conductor of a character ω of Uv is the smallest c such that c ∗ ω is the pullback of a character of (OKv /mv) . 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 character n ∗ of (OKv /mv ) , 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 n Uv is of the form x 7→ x , where n is a congruence class mod 2, and if Kv = C, a n character of Uv is of the form x 7→ x , where n ∈ Z.

2.2. Restricted direct products. Let (Gv)v∈V be an indexed family of locally compact groups, cofinitely many of which have an open compact subgroup Hv. Q0 Q 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 base Q sets of the form v Uv intersected with the restricted direct product, where every Uv ⊆ Gv is open and cofinitely many Uvs contain Hv. Q0 We will usually suppress the Hv, writing the product instead as v Gv. Now, for any finite collection v , . . . , v of indices, Qn G × Q H is 1 n 1 vi v6=v1,...,vn v Q0 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 this Q0 Q0 form cover v Gv. Therefore v Gv is again locally compact. Q0 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 Q0 ωvs are unramified. We can now define a (quasi)character of Gv by the rule Q v (xv) 7→ v ωv (xv). Q0 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 identify xv ∈ Gv with its image under ιv. All that remains to be checked is that these are indeed inverse bijections; for a proof, see Lemmas 3.2.1 and 3.2.2 of [9]. Q0 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. Adèlesand idèles. Examples 2.11. We will only consider two examples of restricted direct products. Q0 • The restricted direct product v (Kv, OKv ) of the completions of K is called the ring of adèles of K and denoted AK . As the name suggests, Q this is a ring, with product inherited from the unrestricted product v Kv. (Its topology is not inherited from the unrestricted product.) Further, we can regard K as a discrete subring of AK with the diagonal embedding x 7→ (x)v. Q0 ∗ • The restricted direct product v (Kv , Uv) is the group of idèles of K and ∗ ∗ denoted AK . Just like in the adèliccase, we can regard K as a discrete subgroup. This is the group of units in AK , but with a finer topology than the subspace topology. It is, however, the coarsest topology refining the subspace TATE’S THESIS 5

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

−1 ∗ α 7→ α, α : AK → AK × AK . ∗ This construction has two points. First, AK and AK are locally compact, so one ∗ can do harmonic analysis on them. Second, K sits discretely in AK and K sits ∗ discretely in AK ; see [2, § 14] and [2, § 16]. However, the following results show that this is a precarious situation. Theorem 2.12 (Weak approximation theorem). If V is any finite set of valuations Q of K, K is dense in v∈V Kv. Theorem 2.13 (Strong approximation theorem). If we omit any one of K’s com- Q0 pletios, K is dense in the restricted direct product v Kv. See [2, § 6] and [2, § 15] for proofs. Let K be a number field. If (xv) is an idèle,for every finite v let ev be the unique ev Q ev integer satisfying xv/πv ∈ Uv. Cofinitely many evs are 0, so v nonarch. pv is a fractional ideal of OK .

Q ev Deﬁnition 2.14. v nonarch. pv is the fractional ideal generated by (xv).

Q ev (xv) 7→ v nonarch. pv is an epimorphism. Define a projection π from ∗ to ∗ as follows. If (x ) ∈ ∗ , cofinitely many v AQ Zb v AQ xvs are in their respective Zpv . Therefore a unique rational number α exists such that α (xv) = (αxv) has all finite components in their respective Zpv and αx∞ is Q −1 nonnegative, namely sgn (x∞) · v6=∞ |xv|v . Now (α · xv)v6=∞ is an element of Q ∗ = ∗. Let this be π ((x )). v6=∞ Zpv Zb v v 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 ∗ → ∗ → ( /m)∗ and obtain an idèliccharacter ω AQ Zb Z trivial on Q∗. −1 The reason for using χ is as follows. If p is a prime, consider the idèle ιvp (p) whose pth component is p and whose other components are 1. If p is coprime to the conductor of χ, ωp(p) = χ(pe) holds, where pe is the residue of p mod the conductor of χ. For a Dirichlet character χ, define its Dirichlet L-function

X χ(n) Y 1 (2.16) L(s, χ) = = ns 1 − χ(p) · p−s n≥1 p prime for Re(s) > 1. This will turn out to be (up to a meromorphic factor) a constant of proportionality between two ad`elicdistributions, and the factors of the Euler product will turn out to be constants of proportionality between p-adic distributions. We will also extend L(s, χ) meromorphically to the entire plane and use ad`elicFourier analysis to derive a functional equation. The same theory will apply similarly for Hecke characters and zeta-functions, but to avoid overcomplicating things we will explain this after completing our discussion of Dirichlet L-functions. 6 BAPTISTE DEJEAN

2.4. Abelian Fourier analysis. Here we will give, but not prove, the main results of 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 Gˆ of G is deﬁned as the group of characters of G, with the compact-open topology. ˆ Theorem 2.18 (Pontryagin duality). There is a natural isomorphism G =∼ Gˆ. 1 That is, Homcts −,S is an involution of the category of locally compact abelian groups. Pontryagin duality forms the correct context for abstract abelian Fourier analysis. In particular, we will need the following fact.

Deﬁnition 2.19. If f ∈ L2(G) ∩ L1(G), deﬁne the Fourier transform fˆ : Gˆ → C ˆ R of f by f(ξ) = G f(y)ξ(y) dy. Theorem 2.20. The Fourier transform extends by continuity to give an isomorphism of L2(G) with L2(Gˆ). There is a unique Haar measure on Gˆ so that the ˆ Fourier transform is unitary and fˆ(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) of Schwartz-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] uses 1 a suitable subspace of L . That being said, S (Kv) is particularly convenient. If v is finite, define S (Kv) to be the C-vector space of compactly-supported locally-constant functions Kv → C, frivolously topologized as the direct limit of its finite-dimensional subspaces. Thus any linear map out of S (Kv) is automatically continuous. ∞ If Kv = R, for f ∈ C (R) and N, i ∈ N define

N (n) kfk(N,i) = sup (1 + |x|) · f (x) . x Let S(R) be the space of f such that every kfk(N,i) is finite; that is, f and all its derivatives are dominated by any power of x. The kk(N,i)s are now seminorms on S(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 N i j kfk(N,i,j) = sup (1 + |x|) · ∂x∂xf(x) . x Both of these are Frèchet spaces; see [3, Prop 8.2].

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

Examples 3.2. The point mass δ0 defined by hδ0, fi = f(0) is a distribution. More 0 0 generally, if dx is a Radon measure on Kv and v is finite, integration against dx is a distribution. This is also true for infinite v if there is some compact K0 and real 0 a a, c > 0 such that, on compact sets containing K0, dx is dominated by c|x|v dx. TATE’S THESIS 7

0 0 Denote the space of tempered distributions by S (Kv) . For λ ∈ S (Kv) and f ∈ S (Kv) we will write λ(f) as hλ, fi. ∗ 0 −1 We can define an action of Kv on S (Kv) with haλ, fi = hλ, f xa i. Given a ∗ ∗ 0 quasicharacter ω of Kv , we can define another action of Kv on S (Kv) by multiplying by ω(a). Definition 3.3. If λ is a distribution on which these two actions coincide, we say λ is an ω-eigendistribution, and denote the space of ω-eigendistributions by S0(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, trivially the 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, S0(ω) is one-dimensional. On our quest to prove this, we will meet the factors of the Euler products of classical L-function as scale factors. Passing to the global (idèlic)picture will multiply 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 less fancy language at the cost of our setup being less canonical. Our topologies aren’t well-behaved enough to make such grandiose claims as “a distribution decomposes uniquely as a distribution supported at 0 plus a distribution supported away from 0,” 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 a short exact sequence

∞ ∗ ev0 0 Cc (Kv ) S (Kv) C 0, ∞ ∗ where Cc (Kv ) is the space of functions in S (Kv) supported away from 0. We can ∞ ∗ split this, writing S (Kv) = Cc (Kv ) ⊕ Cf. A distribution is thus defined by its ∞ ∗ restriction to Cc (Kv ) and its value on f. ∗ Define ω-eigendistributions on Kv in the obvious way, namely as linear λ : ∞ ∗ −1 ∗ Cc (Kv ) → C for which λ, f xa = hω(a)λ, fi for all a ∈ Kv . The following facts are clear. ∗ ∗ • ω-eigendistributions on Kv are spanned by ω(x) d x. 0 ∗ • Any ω-eigendistribution in S (ω) restricts to an ω-eigendistribution on Kv . • 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 asking what this forces about hλ, fi. 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 Cc (Kv ). We will check that this determines λ on all remaining functions. Let a be any element of Uv that ω is nontrivial on. That λ is an ω-eigendistribution forces

o o hλ, fv i = ω(a) hλ, fv i 8 BAPTISTE DEJEAN

o o −n to hold, so hλ, fv i = 0. It follows that hλ, fv (xπv )i = 0 for all n ∈ Z; that is, λ is n 0 on the characteristic function of any mv . n However, as any f ∈ S (Kv) is constant on a neighborhood mv of 0, f differs by one of these characteristic functions from a function supported away from 0; this implies Z o −n ∗ hλ, fi = λ, f − f(0)fv xπv = c f(x)ω(x) d x. ∗ n Kv −mv This shows that S0(ω) is at most one-dimensional, for λ is determined by c. Fur- ther, we can now check S0(ω) is populated by defining a nonzero ω-eigendistribution Z ∗ hz0(ω), fi := f(x)ω(x) d x. ∗ n Kv −mv This is independent of our choice of n, so long as it’s large enough for f to be n constant on mv . We can reinterpret z0(ω) in a more natural way as the principal value integral Z Z ∗ ∗ hz0(ω), fi = PV f(x)ω(x) d x = lim f(x)ω(x) d x. ∗ n→∞ ∗ n Kv Kv −mv 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 Cc (Kv ). As f is Schwartz-Bruhat, −1 f(x) − f xπv is supported away from 0, implying Z −1 −1 ∗ λ, f(x) − f xπv = c f(x) − f xπv ω(x) d x. As λ is an ω-eigendistribution, we also have

−1 −1 λ, f(x) − f xπv = hλ, fi − λ, f xπv = (1 − ω (πv)) hλ, fi . This gives us Z −1 −1 ∗ hλ, fi = c (1 − ω (πv)) f(x) − f xπv ω(x) d x, showing that S0(ω) is at most one-dimensional. By deﬁning Z −1 ∗ hz0(ω), fi = f(x) − f xπv 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 Kv -invariant, and let c be as usual. We have

o o −1 hλ, fv i = λ, fv xπv , and Z o o −1 ∗ 0 = λ, fv (x) − fv xπv = c d x Uv TATE’S THESIS 9 follows. This forces c to be 0; that is, λ is trivial on functions supported away from 0. Therefore λ is a scalar multiple of δ0. Conversely, any scalar multiple of δ0 is an ω-eigendistribution. R ∗ Notice that, if we define z0(1) as in Case 2, z0(1) = δ0 d x. Uv Case 4: v is infinite. We summarize, but do not prove, the results in this case. If we define

 −s/2 s s π Γ 2 if Kv = R and ω(x) = |x|  s+1 − 2 s+1 s Lv(ω) = π Γ 2 if Kv = R and ω(x) = sgn(x) · |x| , 1−s+ |n| |n| 1/2n s  2 (2π) Γ s + 2 if Kv = C and ω(x) = x/|x| |x|

0 then we can holomorphically choose a generator z0(ω) for S (ω) whose restriction to ∗ −1 ∗ Kv is Lv(ω) ω(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 a nonzero generator for the one-dimensional vector space S0(ω). Another key point is that we have parameterized z0(ω) holomorphically in ω. This allows us to conclude that certain constants of proportionality are holomorphic. Unless ω is the trivial character in the ﬁnite case, xn in the real case, or n1 1/2 n2 ∗ x/|x| x in the complex case, the restriction of z0(ω) to Kv is nonzero. ∗ This means there is another generator, call it z(ω), whose restriction to Kv is ω(x) d∗x. By our calculations in proving Lemma 3.4, we obtain

(3.5) z(ω) = z0(ω) · Lv(ω), where

 1 if v is finite and ω is unramified  1−ω(πv )  1 if v is finite and v is ramified  −s/2 s s (3.6) Lv(ω) = π Γ 2 if Kv = R and ω(x) = |x| . − s+1 s+1 s π 2 Γ if K = and ω(x) = sgn(x) · |x|  2 v R  1−s+ |n| |n| 1/2n s  2 (2π) Γ s + 2 if Kv = C and ω(x) = x/|x| |x|

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 meromorphic ∗ −1 extension of ω(x) d x. Lv(ω) is the factor necessary to eliminate the zeroes and poles of z(ω), yielding a nonzero holomorphic family of distributions z0(ω). That z0(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 (Construction 2.15) how to pull back χ−1 to obtain a character of A∗ . If ω is this character’s Q factor at a prime p, note L ω · ||s = 1 is the factor at p of the Euler vp vp 1−χ(p)·p−s 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 like 1 in the real case, if Kv is a local field and ψ : Kv → S 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 1 2 ˆ R function f ∈ L (Kv) ∩ L (Kv) is given by f(ξ) = f(x)ψ(xξ) dx. The Fourier Kv 2 transform is not only an automorphism of L (Kv), but also of S (Kv). The choice ˆ of dx such that fˆ = f(−x) holds is called self-dual with respect to ψ. See [8, Sec 3.3] and [8, Prop 7-1] for more details. 1 For the rest of this subsection, fix a character ψ : Kv → S , and define the Fourier transform using the corresponding self-dual measure dx. Definitions 3.9. If λ is a distribution, its Fourier transform λˆ is defined by D E D E ˆ ˆ ∗ −1 λ, f = λ, f . If ω is a character of Kv , its shifted dual isω ˆ = ||v · ω .

ˆ ∗ If λ is an ω-eigendistribution, λ is aω ˆ-eigendistribution, for if a ∈ Kv ,

D E D E D E ˆ −1 −1 ˆ λ, f xa = λ, f \(xa ) = λ, |a|vf(xa)

D ˆ E −1 D ˆE −1 Dˆ E = |a|v λ, f(xa) = |a|v · ω a λ, f = |a|v · ω (a) λ, f .

Combining this with Lemma 3.4 and the fact that z0 is holomorphic in ω, we obtain the following. Theorem 3.10 (Local functional equation). Some holomorphic nonzero factor

εv(ω, ψ) exists such that εv(ω, ψ) · z0(ω) = z\0 (ˆω), and if ω is nontrivial some meromorphic γv(ω, ψ) exists such that γv(ω, ψ) · z(ω) = z[(ˆω). These are related by Lv (ˆω) 1 the equation γv(ω, ψ) = εv(ω, ψ) · . Lv (ω) 1/2 Proposition 3.11. For y ∈ Kv, εv(ω, ψ(− · y)) = |y|v 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 o z0(ω) and z\0 (ˆω). First, we will make some necessary calculations about dy and fv . ν If v is finite, define the conductor ν of ψ by the condition that mv is the kernel of ψ. By definition, Z o fcv (x) = ψ(xy) dy. OKv o o −ν R co By Lemma 2.4, this works out to be fcv (x) = fv (xπv ) dy. Similarly, fcv (x) = OKv 2 o ν R fv ·|πv| dy ; for Fourier inversion to hold, dy must assign OKv the measure v OKv −ν/2 o −ν/2 o −ν |πv|v , so fcv (x) = |πv|v fv (xπv ). Now we actually compute εv(ω, ψ). If v is finite and ω is unramified, we see that

D E 1Tate proved this diﬀerently, using the formula hz(ω), fi hz(ˆω), gˆi = z(ˆω), fˆ hz(ω), gi 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

Z o o o −1 ∗ (3.12) hz0(ω), fv i = fv (x) − fv xπv ω(x) d x ∗ Kv Z Z Z = d∗x + 0 d∗x = d∗x. ∗ Uv Kv −Uv Uv

As z0(ˆω) is anω ˆ-eigendistribution,

D oE D ν/2 o −ν E ν ν/2 o z\0(ˆω), fv = z0(ˆω), |πv|v fv xπv =ω ˆ (πv ) |πv|v hz0(ˆω), fv i , o o and hz0(ω), fv i = hz0(ˆω), fv i directly from the definition of z0; therefore εv(ω, ψ) = −ν −ν/2 ω (πv ) · |πv|v . 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

( ν−c ψ(x) if x ∈ mv gω(x) = , 0 else whose Fourier transform is given by

( ν/2−c c |πv| if ξ ≡ 1 (mod mv) gˆω(ξ) = . 0 else −2πi·x For the case Kv = R, we will only give εv(ω, ψ) for the character ψ(x) = e . 2 If ω(−1) = 1, use the test function e−π·x , and if ω(−1) = −1, use the test function 2 x · e−π·x . 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:

ω (π−ν ) · |π |−ν/2 if v is finite and ω is unramified  v v v R −1  ν−c ω (x)ψ(x) dx if v is finite and ω is ramified  πv Uv   with conductor c  1 if K = , ω(−1) = 1,  v R −2πi·x (3.13) εv(ω, ψ) = and ψ(x) = e .  −i if Kv = R, ω(−1) = −1,   and ψ(x) = e−2πi·x   |n| n 1 (−i) if Kv = C, ω(x) = x on S ,   and ψ(x) = e−4πi·Re(x) The remaining cases follow from Proposition 3.11. We can rewrite the case where v is finite and ω is unramified more conveniently as the Gauss sum

c−ν ν/2 X −1 ν−c (3.14) εv(ω, ψ) = ω πv |πv|v ω (x)ψ πv x . c ∗ x∈(OKv /mv ) 12 BAPTISTE DEJEAN

4. Global theory 4.1. From local analysis to global analysis. We have already seen how local characters 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 this is 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 as the “restricted tensor product” of all of our S (Kv)s; that is, as the space spanned N by all formal products v fv, where every fv is in S (Kv), cofinitely many fvs o are fv , and linearity in every fv is imposed. Topologize this as the direct limit of P N the tensor products of finitely many S (Kv)s. Notice every element fi,v of P Q i v 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 of AK . As before, define a tempered distribution as a continuous linear functional on N Q S (AK ). A tempered distribution is of the form v fv 7→ v hλv, fvi, where every 0 o λv is in S (Kv) and cofinitely many λvs are trivial on fv ; see [5, Lem 4.1]. Denote Q this distribution v λv. Q0 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 OKv s are assigned Q Q0 measure 1. Define a measure “dx = dxv” on Gv as follows. Say we have v v Q Borel Xv ⊆ Kv, cofinitely many of which are OK . Give Xv the measure Q R v v dxv. Extend by declaring dx to be a Radon measure; dx is now a Haar v Xv measure. For a more rigorous construction, see [9, Sec 3.3]. Remark 4.1. By (3.12), the normalizations required to multiply local Haar measures ∗ on Kv and to multiply our local z0s coincide: we need cofinitely many Uvs to be assigned measure 1. This is not a coincidence. For the surface of unramified ω, the holomorphic family z0(ω) of distributions is determined up to a nonzero holomorphic factor by the requirement that it be a basis element of S0(ω). This o holomorphic factor is fixed by the requirement that hz0(ω), fv i = 1 for all unramified ω. That is, we deliberately scaled z0(ω) to make these conditions coincide. From ∗ now 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 measure ∗ on AK constructed as a product of the local measures we just ﬁxed. N Integrating Schwartz-Bruhat functions on AK is easy: if v fv is a generator of S (AK ),

Z O Y Z (4.3) fv dx = fv dxv. AK v v Kv We have a global “absolute value” that serves the same purpose as our local ones. ∗ ∗ Q Deﬁne || : AK → 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 measure ∗ dx from local ones, we see that d(ax) = |a|dx for all id`eles a, therefore d x and |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 lose the reader. Q0 • AK is the restricted direct product v Kv. ∗ Q0 ∗ • AK is the restricted direct product v Kv . • dx is a Haar measure on AK . ∗ ∗ • d x is a Haar measure on AK . • || is the global absolute value we just deﬁned. We will also need the following.

•U AK is group of id`elesof absolute value 1. ∗ 4.2. Remarks about characters of AK . 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 can construct 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 conclusion follows by abstract nonsense. Rephrasing our characterization of local quasicharacters slightly, pick a splitting ∗ ∼ U ⊕ im ||, and if x ∈ ∗ , let x ∈ U be the first component of x in this AK = AK AK e AK ∗ 0 s splitting. For a quasicharacter ω of AK , we might hope to write ω = ω (xe) · |x| for 0 some character ω of UAK and let the exponent of ω be Re(s). Unfortunately, UAK is ∗ not compact, so we cannot always do this. However, UAK /K is compact. One sees ∗ this by constructing a fundamental domain for K in UAK whose closure is compact ∗ by Tychonoff’s theorem; this closure now surjects to UAK /K under the quotient ∗ map, 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 UAK 0 s is in fact a character, so we can write ω = ω (xe) · |x| and define the exponent of ω ∗ ∼ to be Re(s). This is independent of our decomposition AK = UAK ⊕ im ||. We will restrict our attention to these characters. (The actual reason for doing so is to make Theorem 4.12 hold). ∗ ∗ Convention 4.6. A character of AK is assumed to be trivial on K . As an important note, || is still a quasicharacter; that is, |x| = 1 for x ∈ K∗. We can 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 domain D 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 for local quasicharacters. Given a global character ω, the space of all quasicharacters of the 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 number field. This Riemann surface structure is independent of our choice of representative character ω; the characters correspond to the imaginary axis Re(s) = 0. The translates of this imaginary axis (a circle if K has positive characteristic) correspond to the different exponents of quasicharacters. 4.3. From local eigendistributions to global eigendistributions. For a char- ∗ ∗ 0 acter ω of AK , define two actions of AK on the space S (AK ) of distributions by defining aλ as ω(a)λ or by haλ, fi = 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 obtain the following global dimension result.

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

Y z0(ω) = z0 (ωv) . v

We have just deﬁned a global version of z0; we will now do the same to z. Deﬁnition 4.8. Z hz(ω), fi = f(x)ω(x) d∗x, ∗ AK provided this integral converges for all f.

By Remark 3.7, (4.3), and appropriate convergence theorems, this converges for N Q ω of exponent greater than 1, and hz(ω), v fvi = v hz (ωv) , fvi. Therefore, for such ω, multiplying the local instances of (3.5) yields

(4.9) z(ω) = z0(ω) · Λ(ω), Q where Λ(ω) = v Lv (ωv). Note that z and Λ are meromorphic in ω where they are deﬁned, and further that z0 is holomorphic in ω and nonzero.

4.4. Global Fourier analysis. Let (ψv)v be nontrivial characters of the completions Kv of K, cofinitely many of which have kernel OK . By Proposition 2.9, Q v 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 such a ψ exists; we will explicitly construct one at the beginning of Section 5, and that this 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 ˆ R N transform by f(ξ) = f(x)ψ(xξ) dx. By (4.3), the Fourier transform of fv AK v N ˆ is v fv, defined by the local Fourier transforms given by the local ψvs. Therefore the Fourier transform is an automorphism of S (AK ), and we can normalize dx so ˆ Q f(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 ψ s. v D E D E We define the Fourier transform of a distribution λ, again, by λ,ˆ f = λ, fˆ . We can now multiply the local instances of Theorem 3.10 and obtain a global analogue.

Theorem 4.10. There is a nonzero holomorphic factor ε(ω) such that ε(ω)·z0(ω) = Q z\0 (ˆω); 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 this requires a meromorphic extension of Λ. Instead, we reverse the situation, extending z(ω) meromorphically and deﬁning Λ(ω) as the factor making (4.9) true. Once this is done, (4.9) and Theorem 4.10 combine to yield the equation

Λ(ω) (4.11) z(ω) = · z[(ˆω). Λ (ˆω) · ε(ω) We may be tempted to call this the global functional equation. Instead, we will give that name to the following equation, which is the critical computation of Tate’s thesis. Theorem 4.12 (Global functional equation). z has a meromorphic extension to all quasicharacters satisfying

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

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 X f(x) = fˆ(ξ). x∈K ξ∈K P Proof sketch. Say s∈K f(x + y) converges absolutely and uniformly on compact ˆ sets for any y ∈ AK , and the same holds for f. Then this formula follows from repeating the classical proof; see [9, Lem 4.2.4] or [8, Thm 7-7]. Further, any f ∈ 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 assign the measure 1 to a fundamental domain for K. Poisson summation, in this form, is equivalent to this fact; Tate [9] uses it to prove Poisson summation. Ramakrishnan and Valenza [8] avoid using this fact, instead taking the scaling of dx into account via Fourier inversion.

∗ Theorem 4.16 (Riemann-Roch). For f ∈ S (AK ) and a ∈ AK , X X |a|−1 f xa−1 = fˆ(ξa). x∈K ξ∈K

−1 Proof. Apply Lemma 4.14 to f xa .

N o If K has positive characteristic, the case f = v fv recovers the classical Riemann-Roch theorem for curves over ﬁnite ﬁelds; see [8, Thm 7.12]. This can therefore can be viewed as a restatement and number-theoretic analogue of the classical Riemann-Roch theorem. 16 BAPTISTE DEJEAN

Proof of Theorem 4.12. We treat only the case that K is a number ﬁeld. The positive-characteristic case is similar; see [8, Thm 7-16]. ∗ Choose a splitting AK = UAK ⊕ im ||, identifying im || = (0, ∞) with a subgroup ∗ of AK . Let dx be Lebesgue measure, and let dxr = dx/x. There is now a unique ∗ ∗ measure dxu on UAK such that d x on AK is the product dxu × dxr. If ω has exponent greater than 1, we see

Z Z ∞ Z ∗ hz(ω), fi = f(x)ω(x) d x = f(xt)ω(xt) dxu dtr ∗ 0 U AK AK Z ∞ Z Z 1 Z = f(xt)ω(xt) dxu dtr + f(xt)ω(xt) dxu dtr 1 UAK 0 UAK Z ∞ Z Z ∞ Z −1 −1 = f(xt)ω(xt) dxu dtr + f xt ω xt dxu dtr. 1 UAK 1 UAK ∗ We treat the second term. Let E be a fundamental domain for K in UK ; we remind the reader that [9, Sec 4.3] has a construction. Notice Z Z −1 −1 X −1 −1 f xt ω xt dxu = f xyt ω xt dxu. U E AK y∈K∗ We can almost apply Riemann-Roch 4.16; we ﬁrst need to add the ﬁnal term R −1 E f(0)ω xt dxu. Once we have, Riemann-Roch gives

Z Z −1 −1 −1 f xt ω xt dxu + f(0)ω xt dxu U E AK Z Z −1 X ˆ −1 −1 X ˆ −1 −1 = x t f ξx t ω xt dx = f ξx t ωˆ x t dxu. E ξ∈K E ξ∈K Reversing steps yields that this is Z Z ˆ ˆ f(xt)ˆω(xt) dxu + f(0)ˆω(xt) dxu. U E AK We have, therefore,

Z ∞ Z Z ∞ Z ˆ hz(ω), fi = f(xt)ω(xt) dxu dtr + f(xt)ˆω(xt) dxu 1 U 1 U AK AK Z ∞ Z Z ∞ Z −1 ˆ − f(0) ω xt dxu dtr + f(0) ωˆ(xt) dxu dtr 1 E 1 E

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

Z ∞ Z Z ∞ Z −s ˆ 1−s − f(0) |t| dtr dxu + f(0) |t| dtr dxu 1 E 1 E ! f(0) fˆ(0) Z = − + dxu. s 1 − s E TATE’S THESIS 17

In the notation of [9], we obtain

Z ∞ Z Z ∞ Z ˆ (4.17) hz(ω), fi = f(xt)ω(xt) dxu dtr + f(xt)ˆω(xt) dxu 1 U 1 U AK AK (( ! )) f(0) fˆ(0) Z − + dxu , s 1 − s E 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[(ˆω) is immediate. As a bonus, z has simple poles exactly at 1 and ||. With the parameterization ||s of the Riemann surface containing these, we can immediately read off the residues R ˆ R as −δ0 E dxu at 1 and δ0 E dxu at ||.

5. Computations Now we will compute specific examples. We will identify prime integers with the corresponding finite valuations of Q. −2πi·x If K = Q, we can choose additive characters as follows. Let ψ∞(x) = e . For a finite prime p, Qp/Zp is (algebraically) isomorphic to Z [1/p] /Q, which (algebraically) embeds into R/Z. Composing these gives a homomorphism λ : Qp/Zp → 2πi·λ(x) R/Z. Let ψp(x) = e ; this is an additive character of conductor 0. Every ψp Q is 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 valuation p of Q, we can define ψv by composing ψp with the trace map Kv → Qp. If p is −1 finite, this has kernel diff (Kv/Qp). Cofinitely many of these differents are 1, so Q ψ = v ψv gives a character of AK , which is also trivial on K. −ν For finite v, let ν be the conductor of ψv; that is, mv = diff (OKv /Zp). As we checked in Subsection 3.4, for finite v, the measure on Kv which is self-dual with −ν/2 respect to ψv assigns OKv the measure |πv|v . 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 1−s s || , so ωcs = || . The local factors of ωs are then ||v.Λ(ωs) is the Riemann zeta −s/2 s function ζ(s) times the extra factor L∞ (ωs) = π Γ 2 . 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,

−s/2 s s−1 1 − s (5.1) π Γ ζ(s) = π 2 Γ ζ(1 − s). 2 2 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. As we have already described (Construction 2.15), there is an id`eliccharacter ω, trivial ∗ s 1 on Q , such that Lp (ω · || ) = 1−χ(p)·p−s for ﬁnite primes p. Therefore

s s Λ(ω · || ) = L(s, χ) · L∞ (ω · || ) for s with Re(s) > 1. This equation gives a meromorphic extension of L(s, χ), and by Corollary 4.13 it satisﬁes

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

We will now compute our local ε factors. For ﬁnite p relatively prime to m, ωp s is unramiﬁed and ψp has conductor ν = 0, so by (3.13) εp(ωp · ||p, ψp) = 1. ∗ As ω∞ is a character of R , we only have two choices for ω∞, determined −1 s by the image of −1. ω∞(−1) = χ(−1) = χ(−1), giving ε∞ (ω∞ · || , 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 ωp is the multiplicity with which p divides m. Computing, we obtain

Z Y Y −1 −s εp (ωp · ||p, ψp) = ωp (x)|x|p ψp(x) dx −cp ∗ p|m p|m p Zp   Z −s −1 Y Y = m ω  ιp (xp) · ψp (xp) dx m−1·Q ∗ p|m Zp p|m p|m X Y X = m−s χ (x) · e2πi·xp/m = m−s χ(x) · e2πi·x/m, ∗ ∗ x∈(Z/m) p|m x∈(Z/m)

cp where xp denotes the image of x under the composition Z/m → Z/p → Z/m of maps given by the Chinese remainder theorem. We collect our ﬁnal functional equation: it is

(5.2) s −1 −1 1−s −s X 2πi·x/m L(s, χ)L∞ (ω · || ) = ε∞L(1 − s, χ )L∞ ω · || m χ(x) · e , ∗ x∈(Z/m) where

( 1 if χ(−1) = 1 ε∞ = , −i if χ(−1) = −1

( −s/2 s s π Γ 2 if χ(−1) = 1 L∞ (ω · || ) = s+1 , − 2 s+1 π Γ 2 if χ(−1) = −1 ( s−1 2 1−s −1 1−s π Γ 2 if χ(−1) = 1 and L∞ ω · || = s−2 . 2 2−s π Γ 2 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 let Q cp f = p p be its factorization into prime ideals. For x ∈ OK , let xe denote the residue of x residue mod f. Definition 5.3. A Hecke character mod f is a character χ of the group of fractional ideals relatively prime to f satisfying the following condition: a character ω∞ of Q ∗ −1 v arch. Kv exists such that, for all x ≡ 1 (mod f), χ((x)) = ω∞ (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 f0 for any f0 properly dividing f.

If χ is a Hecke character mod f, χf (x) = χ((x)) · ω∞(x) gives a well-defined ∗ e character of (OK /f) . Hence we can define Hecke characters as characters which −1 factor as χf (xe) · ω∞ (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

X χ(I) Y 1 (5.5) L(s, χ) = = , kIks 1 − χ(p)kpk−s I p where the sum is taken over ideals of OK relatively prime to f, the product is taken over primes of OK not dividing f, and kIk = |OK /I|. As in the Dirichlet case, we want to interpret this as a product of most of the factors of Λ (ω · ||s) for an 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 L ω · ||s = 1 vp vp vp 1−χ(p)kpk−s −1 forces ωvp to be unramified with ωvp (πv) = χ (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, where Q ∗ 1 Q ωram : p|f Kvp → S is defined as follows. First, define ωram on p|f Uvp −1 Q ∗ as the pullback of χf under the quotient p|f Uvp → (OK /f) . The factor- −1 ization χ((x)) = χf (xe)·ω∞ (x) now guarantees that, for whatever extension Q ∗ ωram we choose, ω(x) = v ωv(xv) is trivial on every α ∈ K coprime to f. Q ∗ 1 We now need to extend ωram to the rest of v|f Kv → S to make ω trivial on all of K∗. This is obtained by the weak approximation theorem Q 2.12. (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 AK . ∗ We can go the other way: let ω be a character of AK and f an ideal such that, for cp every prime p, p |f (where cp is the conductor of ωv). Then we can define a Hecke character χ by letting χ(p) = ωv πv and extending by linearity. The associated Q p p ω∞ is then v arch. ωv. ∗ These are inverse bijections between primitive characters and characters of AK . −1 Hence, the classical and idèlicpictures are equivalent. χf 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, though not seen in the classical picture, are there to make ω trivial on K∗. As a corollary of the equivalence between the classical and idèlicpictures, for K 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 idèliccharacters, we obtain

Y s (5.6) Λ(s, χ) = L(s, χ) · Lv (ωv · ||v) v arch. for Re(s) > 1. The meromorphic extension and functional equation of Λ yields the same 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. The distinction 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 situation 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 of 5, namely a + b 5 = a + b 5 and a + b 5 = a − b 5 , and both Q ∞1 ∞2 ∗ ∗ of these are real. ω∞ is a character of R × R ; therefore we may write it uniquely a1 a2 s1 s2 as ω∞ (x1, x2) = |x1| |x2| sgn (x1) sgn (x2) for a1, a2 ∈ R · i and s1, s2 ∈ Z/2. −1 For ω√∞ to correspond to a Hecke character χ, ω∞ (x) = χ((x)) for x ≡ 1 √ (mod 5 ) forces ω∞ to be trivial on units of O ( 5) which are congruent to √ √ Q 1 mod 5. As −1 and 1+ 5 generate the units of O √ , the group of units 2 Q( 5) √ √ 2 1+ 5 congruent to 1 mod 5 is principal and generated by − 2 . ω∞ is trivial √ 2 √ 2(a1−a2) on − 1+ 5 if and only if 1+ 5 · (−1)s1+s2 = 1; that is, some n ∈ 2 2 Z √ 1+ 5 exists such that a1 − a2 = πin/2 log 2 and s1 + s2 ≡ n (mod 2). √ Say we have such an ω∞. As O is a PID and its units surject to those of √ Q( 5) √ ∼ O / 5 = F5, we can retrieve the χ with this ω∞ by defining χ((x/y)) = Q( 5) √ −1 ω∞ (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

 √  √ ! √ !! √ ! 1^ + 5 1 + 5 1 + 5 1 + 5 χ (3) = χ = ω χ = ω , f e f  2  ∞ 2 2 ∞ 2 √ and 3 generates O √ / 5 =∼ ∗. e Q( 5) F5 Let ω be√ the idèliccharacter corresponding to χ.√ At every finite place√ v other than v√ , 5 = 1 and ω is unramified, so ω 5 = 1. As ω 5 = 1, √ 5 v √ v v −1 −1 ωv√ 5 = ω∞ 5 . This, that ωv√ has conductor 1, and ωv√ (3) = χf 3 = 5 √ 5 5 −1 1+ 5 ω completely determine ω √ . ∞ 2 v 5 The description of ωv for other finite v is straightforward. Pick a generator x √ of pv which is congruent to 1 mod 5 . ωv is unramified, and ω(x) = 1 forces −1 ωv (πv) = ωv(x) = ω∞ (x) to hold. TATE’S THESIS 21 √ We can now finally compute all our local ε factors. diff O √ / = 5, so Q( 5) Z s √ εv (ωv · ||v, ψv) = 1 for every finite v other than v 5. From (3.14), we obtain

s 1/2−2s −1 X 4πi·x/5 εv√ ωv√ · || √ , ψv√ = 5 ω (5) χf(x)e . 5 5 v 5 5 ∞ ∗ x∈F5 Our inﬁnite ε-factors are ( 1 if a = 0 ε ω · ||s , ψ = i . ∞i ∞i ∞i ∞i i if ai = 1 Our functional equation is therefore

−1 1/2−2s −1 X 4πi·x/5 Λ(s, χ) = Λ 1 − s, χ ε∞ · 5 ω∞ (5) χf(x)e , ∗ x∈F5 where

Λ(s, χ) = L(s, χ) · L ω · ||s · L ω · ||s , ∞1 ∞1 ∞1 ∞2 ∞2 ∞2 Λ 1 − s, χ−1 = L 1 − s, χ−1 · L ω−1 · ||1−s · L ω−1 · ||1−s , ∞1 ∞1 ∞1 ∞2 ∞2 ∞2  1 if a = a = 0  1 2 ε∞ = −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; among these are the Tchebotarev density theorem. s If K is a ﬁeld of positive characteristic containing Fq, then ζK (s) = Λ (|| ) is in fact a rational function in q−s! See Exercise 22 in Chapter 7 of [8] for a proof. The properties of this rational function are generalized by Weil’s celebrated conjectures, now resolved. 22 BAPTISTE DEJEAN

References [1] Bump, Daniel A. Automorphic Forms and Representations. Cambridge: Cambridge University Press, 1998. [2] Cassels, J. W. S. “Global Fields.” In Algebraic Number Theory, edited by J. W. S. Cassels and A. Fröhlich, 42-84. London: Academic Press Inc., 1967. [3] Folland, Gerald B. Real Analysis: Modern Techniques and Their Applications. New York: John Wiley & Sons, Inc., 1999. [4] Hecke, E. “Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen.” 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 by Joseph Bernstein and Stephen Gelbart, 109-131. New York: Springer Science+Business Media, 2004. [6] Lang, Serge. Algebraic Number Theory. New York: Springer Science+Business Media, 1986. [7] Neukirch, Jürgen.“Zeta Functions and L-series.” In Algebraic Number Theory, 419-549. Trans- lated by Norbert Schappacher. New York: Springer Science+Business Media, 1999. [8] Ramakrishnan, Dinakar, and Valenza, Robert J. Fourier Analysis on Number Fields. London: Academic Press Inc., 1967. [9] Tate, J. T. “Fourier Analysis in Number Fields and Hecke’s Zeta-Functions.” In Algebraic Number Theory, edited by J. W. S. Cassels and A. Fröhlich, 305-347. London: Academic Press Inc., 1967. [10] Weil, André.“Fonction zêtaet distributions.” Séminare Bourbaki, Volume 9 (1964-1966), Talk no. 312, 523-531.