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] (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 p p 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 com- plex 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 distri- butions 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, ad`elesand id`eles 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 p 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 2 Kv ∗ dx scales dx by jαj; that is, d(ax) = jajvdx. Thus, d x and agree up to a constant. jxjv If Kv = C, this means jjv 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 ! : 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, ! is a nontrivial character of G, R and dx is a Haar measure on G, G !(x) dx = 0. Proof. Let a be any element of G on which ! is nontrivial. If dx is a Haar measure, Z Z Z !(a) !(x) dx = !(ax) dx = !(x) dx; G G G R which can only hold if G !(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 id`elesand 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 ! : Kv ! S , we can consider all qua- s sicharacters of the form !(x) · jxjv for s 2 C. Parameterized by s, these form a connected Riemann surface, and the analytic structure we assign this surface is independent of our choice of !. 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 jπvjv . 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 ! of Kv is unramified if it is trivial on Uv; s that is, if some complex s exists such that !(x) = jxjv for all x. For finite v, a quasicharacter of Kv is unramified if it is trivial on OKv . ∗ We can classify quasicharacters ! of Kv : by compactness of Uv, ! can be written 0 s 0 0 in the form ! (x)·jxjv for some character ! and some s 2 C. ! 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 ! 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 fi- nite, 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 ! of Uv is the smallest c such that c ∗ ! is the pullback of a character of (OKv =mv) . If ! is a character of Kv, let the conductor of ! 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 2 Z. 2.2. Restricted direct products. Let (Gv)v2V 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.