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 defined a class of L-functions analogous to Dirichlet s and used unpleasant methods to gie their meromorphic extensions and functional equations [4]. In his thesis, Tate bypassed Hecke s methods by using simple Fourier analysis to derie the same [9]. We gie Tate s deriation, but following Kudla [] who follows Weil [0],) we reinterpret the perasie proportionality of distributions. To illustrate the theory, we compute a few concrete examples. Contents. Introduction. Motiation, characters.. Characters 3.. Restricted direct products 4.3. Adèles and idèles 4.4. Abelian Fourier analysis 6 3. Local theory 6 3.. Setting the stage 6 3.. Lemma 3.4 s proof 7 3.3. Recap 9 3.4. Local Fourier analysis and the local functional equation 0 4. Global theory 4.. From local analysis to global analysis 4.. Remarks about characters of A K 3 4.3. From local eigendistributions to global eigendistributions 3 4.4. Global Fourier analysis 4 4.. Global functional equation. Computations 7.. Warm-up: Riemann zeta function 7.. Dirichlet L-functions 8.3. Hecke characters 9.4. Example: Hecke characters of Q ) mod ) 0.. Final remarks References Date: October 6, 07.
BAPTISTE DEJEAN. 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 improement oer the techniques of the time [4]. We will closely follow the exposition in Kudla [], which follows Weil [0] in reinterpreting the local functional equation. In Section, 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, coneniently, the local factor of a classical L-function. In Section 4, adèles and 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; howeer, we can also use Poisson summation to compute this scale factor as. This gies 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.. Finally, we will show the relationship to classical pictures in Section, computing the functional equations for Dirichlet L-functions and certain Hecke L-functions oer Q ).. Motiation, characters By aluation we mean a norm; that is, we write codomains multiplicatiely and allow archimedean aluations. By a finite aluation we mean a nonarchimedean aluation, and by an infinite aluation we mean an archimedean aluation. Notations.. We will use the following notation. K is a global field; that is, a finite extension of Q or F p t). If K is a number field, O K is its ring of integers. K is a completion of K with respect to a aluation. O K is the ring of integers in K for finite, m is its maximal ideal, and π is a uniformizer. U is the group of elements of K of norm. dx is an additie Haar measure on K, and d x is a multiplicatie Haar measure on K. π is a uniformizer of K for finite. For a finite aluation, p is the prime of O K inducing it, and for a prime p of O K, p is the aluation induced by it. If is finite, f o is the characteristic function of O K. We will tacitly normalize all aluations so that multiplication by any α K scales dx by α ; that is, dax) = a dx. Thus, d x and dx x agree up to a constant. If K = C, this means is the square of the usual Euclidean norm.
TATE S THESIS 3.. Characters. Definition.. 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 S. Remarks.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.4. If G is a compact abelian group, ω is a nontriial character of G, and dx is a Haar measure on G, ωx) dx = 0. G Proof. Let a be any element of G on which ω is nontriial. If dx is a Haar measure, ωa) ωx) dx = ωax) dx = ωx) dx, which can only hold if ωx) dx = 0. G G G Recall that a Dirichlet character χ mod m is a character of Z/m), and χ is primitie if it is not the pullback of a Dirichlet character mod n for any n properly diiding m. By pulling back, we can identify primitie Dirichlet characters with characters of Ẑ = lim Z/mZ). We will further refine this group to the group of idèles and then treat characters on these, though the benefit of this will not become clear until Section 4. Construction.. For a gien character ω : K S, we can consider all quasicharacters of the form ωx) x s 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 is infinite, these Riemann surfaces are copies of C, and if is finite, they are copies of the cylinder gien by taking C mod πi/ log π. The characters on this surface are the imaginary axis Res) = 0 a circle if is finite). We will now begin to classify characters of K for an arbitrary completion K of K. Definition.6. A quasicharacter ω of K is unramified if it is triial on U ; that is, if some complex s exists such that ωx) = x s for all x. For finite, a quasicharacter of K is unramified if it is triial on O K. We can classify quasicharacters ω of K : by compactness of U, ω can be written in the form ω x) x s for some character ω and some s C. ω is not well-defined, though its restriction to U is, and s is not well-defined, though its real part is. Definition.7. The exponent of ω is Res). We can now see the following. Our Riemann surfaces are the cosets of the unramified quasicharacters; that is, they correspond to restrictions to U. The translates on each Riemann surface of the characters correspond to exponents. We can classify these een further with the following obseration. If is finite, a character of U = lim O K /m n n ) is the pullback of a character of some O K /m n ). G
4 BAPTISTE DEJEAN Definition.8. The conductor of a character ω of U is the smallest c such that ω is the pullback of a character of O K /m c ). If ω is a character of K, let the conductor of ω be the conductor of its restriction to U. Thus, we can specify a quasicharacter by specifying a conductor c, a character of O K /m n ), and the image of a uniformizer π ; the first two are finite amounts of data. The classification in the infinite case is een simpler: if K = R, a character of U is of the form x x n, where n is a congruence class mod, and if K = C, a character of U is of the form x x n, where n Z... Restricted direct products. Let G ) V be an indexed family of locally compact groups, cofinitely many of which hae an open compact subgroup H. Their restricted direct product G, H ) is the group of all elements of G with cofinitely many components in H. Topologize this by taking as an open base sets of the form U intersected with the restricted direct product, where eery U G is open and cofinitely many U s contain H. We will usually suppress the H, writing the product instead as G. Now, for any finite collection,..., n of indices, n G i,..., n H is an open subset of G, and the subspace and product topologies coincide. The subspace topology is locally compact by Tychonoff s theorem, and subsets of this form coer G. Therefore G is again locally compact. Eery G embeds into G by sending g to the tuple whose th coordinate is g and whose other coordinates are ; call this embedding ι. For V let ω be a quasi)character of G, and assume cofinitely many ω s are unramified. We can now define a quasi)character of G by the rule x ) ω x ). Proposition.9. This gies a bijection between quasi)characters of G and such collections ω ) of quasi)characters. The inerse map is obious: define our ω s by ω x ) = ω x ), where we identify x G with its image under ι. All that remains to be checked is that these are indeed inerse bijections; for a proof, see Lemmas 3.. and 3.. of [9]. Definition.0. If ω is a quasicharacter of G, let ω be its factor at, and call ω ) the local factors of ω..3. Adèles and idèles. Examples.. We will only consider two examples of restricted direct products. The restricted direct product K, O K ) of the completions of K is called the ring of adèles of K and denoted A K. As the name suggests, this is a ring, with product inherited from the unrestricted product K. Its topology is not inherited from the unrestricted product.) Further, we can regard K as a discrete subring of A K with the diagonal embedding x x). The restricted direct product K, U ) is the group of idèles of K and denoted A K. Just like in the adèlic case, we can regard K as a discrete subgroup. This is the group of units in A K, but with a finer topology than the subspace topology. It is, howeer, the coarsest topology refining the subspace
TATE S THESIS topology under which inersion is continuous; equialently, the topology inherited from the embedding α α, α ) : A K A K A K. This construction has two points. First, A K and A K are locally compact, so one can do harmonic analysis on them. Second, K sits discretely in A K and K sits discretely in A K ; see [, 4] and [, 6]. Howeer, the following results show that this is a precarious situation. Theorem. Weak approximation theorem). If V is any finite set of aluations of K, K is dense in V K. Theorem.3 Strong approximation theorem). If we omit any one of K s completios, K is dense in the restricted direct product K. See [, 6] and [, ] for proofs. Let K be a number field. If x ) is an idèle, for eery finite let e be the unique integer satisfying x /π e U. Cofinitely many e s are 0, so nonarch. pe is a fractional ideal of O K. Definition.4. nonarch. pe is the fractional ideal generated by x ). x ) nonarch. pe is an epimorphism. Define a projection π from A Q to Ẑ as follows. If x ) A Q, cofinitely many x s are in their respectie Z p. Therefore a unique rational number α exists such that α x ) = αx ) has all finite components in their respectie Z p and αx is nonnegatie, namely sgn x ) x. Now α x ) is an element of Z p = Ẑ. Let this be π x )). This is a homomorphism whose kernel contains Q. Construction.. If χ : Z/m) S is a Dirichlet character, we can pull back χ under the projection A Q Ẑ Z/m) and obtain an idèlic character ω triial on Q. The reason for using χ is as follows. If p is a prime, consider the idèle ι p p) whose pth component is p and whose other components are. If p is coprime to the conductor of χ, ω p p) = χ p) holds, where p is the residue of p mod the conductor of χ. For a Dirichlet character χ, define its Dirichlet L-function.6) Ls, χ) = n χn) n s = p prime χp) p s for Res) >. This will turn out to be up to a meromorphic factor) a constant of proportionality between two adèlic distributions, and the factors of the Euler product will turn out to be constants of proportionality between p-adic distributions. We will also extend Ls, χ) meromorphically to the entire plane and use adèlic Fourier analysis to derie a functional equation. The same theory will apply similarly for Hecke characters and zeta-functions, but to aoid oercomplicating things we will explain this after completing our discussion of Dirichlet L-functions.
6 BAPTISTE DEJEAN.4. Abelian Fourier analysis. Here we will gie, but not proe, 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.7. The Pontryagin dual Ĝ of G is defined as the group of characters of G, with the compact-open topology. Theorem.8 Pontryagin duality). There is a natural isomorphism G = Ĝ. That is, Hom cts, S ) is an inolution 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.9. If f L G) L G), define the Fourier transform ˆf : Ĝ C of f by ˆfξ) = fy)ξy) dy. G Theorem.0. The Fourier transform extends by continuity to gie an isomorphism of L G) with L Ĝ). There is a unique Haar measure on Ĝ so that the Fourier transform is unitary and ˆfx) = f x) holds; this measure is said to be dual to dx. 3. Local theory 3.. Setting the stage. We will define the topological ector space S K ) of Schwartz-Bruhat functions on K as follows. These constructions are largely unimportant; we just need a well-behaed space of test functions. For example, [9] uses a suitable subspace of L. That being said, S K ) is particularly conenient. If is finite, define S K ) to be the C-ector space of compactly-supported locally-constant functions K C, friolously topologized as the direct limit of its finite-dimensional subspaces. Thus any linear map out of S K ) is automatically continuous. If K = R, for f C R) and N, i N define ) f N,i) = sup + x ) N f n) x). x Let SR) be the space of f such that eery f N,i) is finite; that is, f and all its deriaties are dominated by any power of x. The N,i) s are now seminorms on SR); let these generate our topology on SR). If K = C, define SC) as in the case K = R, but with the seminorms gien by ) f N,i,j) = sup + x ) N x i j x fx). x Both of these are Frèchet spaces; see [3, Prop 8.]. Definition 3.. A tempered distribution on K is a continuous linear functional on S K ); that is, a continuous linear map S K ) C. Examples 3.. The point mass δ 0 defined by δ 0, f = f0) is a distribution. More generally, if dx is a Radon measure on K and is finite, integration against dx is a distribution. This is also true for infinite if there is some compact K 0 and real a, c > 0 such that, on compact sets containing K 0, dx is dominated by c x a dx.
TATE S THESIS 7 Denote the space of tempered distributions by S K ). For λ S K ) and f S K ) we will write λf) as λ, f. We can define an action of K on S K ) with aλ, f = λ, f xa ). Gien a quasicharacter ω of K, we can define another action of K on S K ) by multiplying by ωa). 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 S ω). The uncomfortable reader may be coninced this terminology is sound as follows. If λ is an eigendistribution of eery λ aλ, as defined by the first action, triially the eigenalues ωa) 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 proe 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.. Lemma 3.4 s proof. Lemma 3.4 is proen as coincidentally) Theorem 3.4 of []. 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-behaed 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 ersion of this claim. If is finite, let f be any Schwartz-Bruhat function with f0) 0. There is a short exact sequence 0 Cc K e ) S K ) 0 C 0, where Cc K ) is the space of functions in S K ) supported away from 0. We can split this, writing S K ) = Cc K ) Cf. A distribution is thus defined by its restriction to Cc K ) and its alue on f. Define ω-eigendistributions on K in the obious way, namely as linear λ : Cc K ) C for which λ, f xa ) = ωa)λ, f for all a K. The following facts are clear. ω-eigendistributions on K are spanned by ωx) d x. Any ω-eigendistribution in S ω) restricts to an ω-eigendistribution on K. δ 0 spans all distributions supported at 0, and is an ω-eigendistribution if and only if ω is triial. We can now erify Lemma 3.4 by assuming λ is an ω-eigendistribution and asking what this forces about λ, f. This will essentially reduce to a formal game. Case : is finite and ω is ramified. Let λ be an ω-eigendistribution. Then some c C must exist such that λ coincides with cωx) d x on Cc K ). We will check that this determines λ on all remaining functions. Let a be any element of U that ω is nontriial on. That λ is an ω-eigendistribution forces λ, f o = ωa) λ, f o
8 BAPTISTE DEJEAN to hold, so λ, f o = 0. It follows that λ, f o xπ n ) = 0 for all n Z; that is, λ is 0 on the characteristic function of any m n. Howeer, as any f S K ) is constant on a neighborhood m n of 0, f differs by one of these characteristic functions from a function supported away from 0; this implies λ, f = λ, f f0)f o ) xπ n = c fx)ωx) d x. K mn This shows that S ω) is at most one-dimensional, for λ is determined by c. Further, we can now check S ω) is populated by defining a nonzero ω-eigendistribution z 0 ω), f := K mn fx)ωx) d x. This is independent of our choice of n, so long as it s large enough for f to be constant on m n. We can reinterpret z 0 ω) in a more natural way as the principal alue integral z 0 ω), f = PV fx)ωx) d x = lim fx)ωx) d x. K n K mn Case : is finite and ω is unramified and nontriial. We proceed similarly. Let λ be an ω-eigendistribution. Again, some c C exists such that λ coincides with cωx) d x on Cc K ). As f is Schwartz-Bruhat, fx) f ) xπ is supported away from 0, implying λ, fx) f xπ As λ is an ω-eigendistribution, we also hae This gies us λ, fx) f xπ ) = c fx) f xπ ) = λ, f λ, f xπ )) ωx) d x. ) = ω π )) λ, f. fx) λ, f = c ω π )) )) f xπ ωx) d x, showing that S ω) is at most one-dimensional. By defining z 0 ω), f = fx) )) f xπ ωx) d x, we obtain a nonzero ω-eigendistribution. Case 3: ω is triial. Let λ be an ω-eigendistribution, i.e. let λ be K -inariant, and let c be as usual. We hae and λ, f o = λ, f o 0 = λ, f o x) f o ) xπ, xπ ) = c U d x
TATE S THESIS 9 follows. This forces c to be 0; that is, λ is triial on functions supported away from 0. Therefore λ is a scalar multiple of δ 0. Conersely, any scalar multiple of δ 0 is an ω-eigendistribution. Notice that, if we define z 0 ) as in Case, z 0 ) = δ 0 U d x. Case 4: is infinite. We summarize, but do not proe, the results in this case. If we define π s/ Γ ) s L ω) = π s+ Γ ) s+ ) n s+ π) Γ s + n if K = R and ωx) = x s if K = R and ωx) = sgnx) x s if K = C and ωx) =, x/ x /) n x s then we can holomorphically choose a generator z 0 ω) for S ω) whose restriction to K is L ω) ωx) d x. We refer the reader to [, Prop 3..8], [, Pages -], and [0] for more details. 3.3. Recap. For all ω, we hae just defined a distribution z 0 ω), which gies a nonzero generator for the one-dimensional ector space S ω). Another key point is that we hae parameterized z 0 ω) holomorphically in ω. This allows us to conclude that certain constants of proportionality are holomorphic. Unless ) ω is the triial character in the finite case, x n in the real case, or x/ x / n x n in the complex case, the restriction of z 0 ω) to K is nonzero. This means there is another generator, call it zω), whose restriction to K ωx) d x. By our calculations in proing Lemma 3.4, we obtain is 3.) zω) = z 0 ω) L ω), where ωπ ) if is finite and ω is unramified if is finite and is ramified 3.6) L ω) = π s/ Γ ) s if K = R and ωx) = x s π s+ Γ. ) s+ ) if K = R and ωx) = sgnx) x s n s+ π) Γ s + n if K = C and ωx) = x/ x /) n x s Remark 3.7. Notice zω) is integration against ωx) d x if ω has positie exponent; that is, if this integral conerges. Therefore zω) can be iewed as a meromorphic extension of ωx) d x. L ω) is the factor necessary to eliminate the zeroes and poles of zω), yielding a nonzero holomorphic family of distributions z 0 ω). That z 0 ω) be scaled exactly as we hae defined it is also critical; see Remark 4.. Remark 3.8. If χ is a Dirichlet character, we hae already described Construction.) how to pull back χ to obtain a character of A Q. If ω is this character s factor at a prime p, note L p ω s p ) = χp) p s is the factor at p of the Euler product Ls, χ). In Section, this will be the connection to Dirichlet and Hecke) L-functions.
0 BAPTISTE DEJEAN 3.4. Local Fourier analysis and the local functional equation. Much like in the real case, if K is a local field and ψ : K S is a nontriial character, y ψy ) gies an isomorphism of K with its character group; that is, K is self-dual. Identifying K with its dual in this way, the Fourier transform of a function f L K ) L K ) is gien by ˆfξ) = K fx)ψxξ) dx. The Fourier transform is not only an automorphism of L K ), but also of S K ). The choice of dx such that ˆf = f x) holds is called self-dual with respect to ψ. See [8, Sec 3.3] and [8, Prop 7-] for more details. For the rest of this subsection, fix a character ψ : K S, and define the Fourier transform using the corresponding self-dual measure dx. Definitions 3.9. If λ is a distribution, its Fourier transform ˆλ is defined by ˆλ, f = λ, ˆf. If ω is a character of K, its shifted dual is ˆω = ω. If λ is an ω-eigendistribution, ˆλ is a ˆω-eigendistribution, for if a K, ˆλ, f xa ) = λ, f xa ) = λ, a ˆfxa) = a λ, ˆfxa) = a ω a ) λ, ˆf = a ω a) ˆλ, f. Combining this with Lemma 3.4 and the fact that z 0 is holomorphic in ω, we obtain the following. Theorem 3.0 Local functional equation). Some holomorphic nonzero factor ε ω, ψ) exists such that ε ω, ψ) z 0 ω) = ẑ0 ˆω), and if ω is nontriial some meromorphic γ ω, ψ) exists such that γ ω, ψ) zω) = ẑ ˆω). These are related by the equation γ ω, ψ) = ε ω, ψ) Lˆω) L ω). Proposition 3.. For y K, ε ω, ψ y)) = y / ωy)ε ω, ψ). Proof. Work through definitions. Therefore it suffices to compute ε ω, ψ) for a single ψ. To compute ε ω, ψ), we need only choose a test function at which to ealuate z 0 ω) and ẑ0 ˆω). First, we will make some necessary calculations about dy and f o. If is finite, define the conductor ν of ψ by the condition that m ν is the kernel of ψ. By definition, f o x) = ψxy) dy. O K By Lemma.4, this works out to be f o x) = f o xπ ν ) O K dy. Similarly, f o x) = ; f o π ν dy) OK for Fourier inersion to hold, dy must assign OK the measure π ν/, so f o x) = π ν/ f o xπ ν ). Now we actually compute ε ω, ψ). If is finite and ω is unramified, we see that Tate proed this differently, using the formula zω), f zˆω), ĝ = zˆω), ˆf zω), g to establish this functional equation on the strip 0 < Res) < and using it to extend z. See [9, Thm.4.].
TATE S THESIS 3.) z 0 ω), f o = K f o x) f o As z 0 ˆω) is an ˆω-eigendistribution, ẑ0 ˆω), f o = z 0 ˆω), π ν/ f o )) xπ ωx) d x = d x + U 0 d x = K U ) xπ ν = ˆω π ν ) π ν/ z 0 ˆω), f o, U d x. and z 0 ω), f o = z 0 ˆω), f o directly from the definition of z 0 ; therefore ε ω, ψ) = ω π ν ) π ν/. The other cases are similar; we briefly describe them. If is finite and ω is ramified, let c be the conductor of ω. We use the test function g ω x) = { ψx) 0 else if x m ν c whose Fourier transform is gien by { π ν/ c if ξ mod m c ĝ ω ξ) = ). 0 else For the case K = R, we will only gie ε ω, ψ) for the character ψx) = e πi x. If ω ) =, use the test function e π x, and if ω ) =, use the test function x e π x. For the case K = C, we will only gie ε ω, ψ) for the character ψx) = e 4πi Rex) and let n Z be the integer such that ωx) = x n on S. If n 0, use the test function x n e π x, and if n 0 use x n e π x. For more details on the computation of ε, see [, Prop 3.8] or [9, Sec.]. Note that Tate s factor ρ is the inerse of our factor γ.) We now list the results: ω π ν ) π ν/ if is finite and ω is unramified π ν c U ω x)ψx) dx if is finite and ω is ramified with conductor c if K = R, ω ) =, 3.3) ε ω, ψ) = and ψx) = e πi x. i if K = R, ω ) =, and ψx) = e πi x i) n if K = C, ωx) = x n on S, and ψx) = e 4πi Rex) The remaining cases follow from Proposition 3.. We can rewrite the case where is finite and ω is unramified more coneniently as the Gauss sum 3.4) ε ω, ψ) = ω π c ν ) π ν/ x O K /m c ) ω x)ψ, π ν c x ).
BAPTISTE DEJEAN 4. Global theory 4.. From local analysis to global analysis. We hae already seen how local characters multiply to gie 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.) and Theorem 3.0. For more details and proofs, we refer the reader to [9, 3] and [, Sec 4]. Define a topological ector space S A K ) of Schwartz-Bruhat functions on A K as the restricted tensor product of all of our S K )s; that is, as the space spanned by all formal products f, where eery f is in S K ), cofinitely many f s are f o, and linearity in eery f is imposed. Topologize this as the direct limit of the tensor products of finitely many S K )s. Notice eery element i f i, of S A K ) defines a real function x ) i f i, x ) on A K. This assignment is easily seen to be injectie, so we will identify S A K ) with a space of functions of A K. As before, define a tempered distribution as a continuous linear functional on S A K ). A tempered distribution is of the form f λ, f, where eery λ is in S K ) and cofinitely many λ s are triial on f o ; see [, Lem 4.]. Denote this distribution λ. To multiply Haar measures, let G be a restricted direct product, and choose Haar measures dx on eery G of K such that cofinitely many O K s are assigned measure. Define a measure dx = dx on G as follows. Say we hae Borel X K, cofinitely many of which are O K. Gie X the measure X dx. 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.. By 3.), the normalizations required to multiply local Haar measures on K and to multiply our local z 0 s coincide: we need cofinitely many U s to be assigned measure. This is not a coincidence. For the surface of unramified ω, the holomorphic family z 0 ω) of distributions is determined up to a nonzero holomorphic factor by the requirement that it be a basis element of S ω). This holomorphic factor is fixed by the requirement that z 0 ω), f o = for all unramified ω. That is, we deliberately scaled z 0 ω) to make these conditions coincide. From now on, we will fix measures d x assigning cofinitely many U s measure. Notation 4.. Let dx be a Haar measure on A K, and let d x be a Haar measure on A K constructed as a product of the local measures we just fixed. Integrating Schwartz-Bruhat functions on A K is easy: if f is a generator of S A K ), 4.3) A K f dx = K f dx. We hae a global absolute alue that seres the same purpose as our local ones. Define : A K R by x ) = x ; that is, let be the quasicharacter whose local factors are. As a corollary of our construction of a global Haar measure from local ones, we see that dax) = a dx for all idèles a, therefore d x and dx x agree up to a constant.
TATE S THESIS 3 Notations 4.4. We collect all our global notation in one place so we don t lose the reader. A K is the restricted direct product K. A K is the restricted direct product K. dx is a Haar measure on A K. d x is a Haar measure on A K. is the global absolute alue we just defined. We will also need the following. U AK is group of idèles of absolute alue. 4.. Remarks about characters of A K. Proposition 4.. splits after restricting to its image. Proof. If K is a number field, K has an infinite aluation, from which one can construct a splitting of : send x 0, ) to ι x) if K = R or ι x) if K = C. If K has positie 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 A K = U AK im, and if x A K, let x U A K be the first component of x in this splitting. For a quasicharacter ω of A K, we might hope to write ω = ω x) x s for some character ω of U AK and let the exponent of ω be Res). Unfortunately, U AK is not compact, so we cannot always do this. Howeer, U AK /K is compact. One sees this by constructing a fundamental domain for K in U AK whose closure is compact by Tychonoff s theorem; this closure now surjects to U AK /K under the quotient map, forcing U AK /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 triial on K, the restriction of ω to U AK is in fact a character, so we can write ω = ω x) x s and define the exponent of ω to be Res). This is independent of our decomposition A K = U AK im. We will restrict our attention to these characters. The actual reason for doing so is to make Theorem 4. hold). Conention 4.6. A character of A K is assumed to be triial on K. As an important note, is still a quasicharacter; that is, x = for x K. We can show this by using the norm to reduce this to the cases K = Q and K = F p t), which we can directly check, or can show this by constructing a fundamental domain D for K in A K and showing that D and xd hae the same measure. We can now construct a picture of global quasicharacters much like the one for local quasicharacters. Gien a global character ω, the space of all quasicharacters of the form ωx) x s for s C forms a Riemann surface under this parameterization; this is a cylinder if K has positie characteristic and a plane if K is a number field. This Riemann surface structure is independent of our choice of representatie character ω; the characters correspond to the imaginary axis Res) = 0. The translates of this imaginary axis a circle if K has positie characteristic) correspond to the different exponents of quasicharacters. 4.3. From local eigendistributions to global eigendistributions. For a character ω of A K, define two actions of A K on the space S A K) of distributions by defining aλ as ωa)λ or by aλ, f = λ, f xa ), and define an ω-eigendistribution
4 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 z 0 ω) = z 0 ω ). We hae just defined a global ersion of z 0 ; we will now do the same to z. Definition 4.8. zω), f = proided this integral conerges for all f. A K fx)ωx) d x, By Remark 3.7, 4.3), and appropriate conergence theorems, this conerges for ω of exponent greater than, and zω), f = z ω ), f. Therefore, for such ω, multiplying the local instances of 3.) yields 4.9) zω) = z 0 ω) Λω), where Λω) = L ω ). Note that z and Λ are meromorphic in ω where they are defined, and further that z 0 is holomorphic in ω and nonzero. 4.4. Global Fourier analysis. Let ψ ) be nontriial characters of the completions K of K, cofinitely many of which hae kernel O K. By Proposition.9, ψ is a character of A K, and y ψy ) gies an isomorphism of A K with its character group; that is, A K is self-dual. If we further require ψ to be triial on K, this restricts to an isomorphism of K with the character group of A K /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, and that this ψ works is [9, Thm 4..4] and [9, Lem 4..]. Let ψ be a character of A K, triial on K. If f S A K ), define its Fourier transform by ˆfξ) = A K fx)ψxξ) dx. By 4.3), the Fourier transform of f is ˆf, defined by the local Fourier transforms gien by the local ψ s. Therefore the Fourier transform is an automorphism of S A K ), and we can normalize dx so ˆfx) = f x) holds. In fact, this self-dual dx is just the product dx of the local self-dual measures with respect to the local ψ s. We define the Fourier transform of a distribution λ, again, by ˆλ, f = λ, ˆf. We can now multiply the local instances of Theorem 3.0 and obtain a global analogue. Theorem 4.0. There is a nonzero holomorphic factor εω) such that εω) z 0 ω) = ẑ 0 ˆω); εω) is gien by ε ω, ψ ). This is in fact a finite product.) Notice εω) doesn t depend on ψ by Proposition 3..
TATE S THESIS 4.. Global functional equation. We can try to extend zω) by 4.9), but this requires a meromorphic extension of Λ. Instead, we reerse the situation, extending zω) meromorphically and defining Λω) as the factor making 4.9) true. Once this is done, 4.9) and Theorem 4.0 combine to yield the equation 4.) zω) = Λω) Λ ˆω) εω) ẑ ˆω). We may be tempted to call this the global functional equation. Instead, we will gie that name to the following equation, which is the critical computation of Tate s thesis. Theorem 4. Global functional equation). z has a meromorphic extension to all quasicharacters satisfying zω) = ẑ ˆω). Corollary 4.3. Λ has a meromorphic extension satisfying Λ ω) = εω) Λˆω). Proof. Combine Theorems 4. and 4.. Before proing Theorem 4., we need a prerequisite. Lemma 4.4 Poisson summation). For f S A K ), x K fx) = ξ K ˆfξ). Proof sketch. Say s K fx + y) conerges absolutely and uniformly on compact sets for any y A K, and the same holds for ˆf. Then this formula follows from repeating the classical proof; see [9, Lem 4..4] or [8, Thm 7-7]. Further, any f S A K ) satisfies these conditions by [8, Lem 7-6]. Remark 4.. The triiality of ψ on K forces the self-dual measure dx to assign the measure to a fundamental domain for K. Poisson summation, in this form, is equialent to this fact; Tate [9] uses it to proe Poisson summation. Ramakrishnan and Valenza [8] aoid using this fact, instead taking the scaling of dx into account ia Fourier inersion. Theorem 4.6 Riemann-Roch). For f S A K ) and a A K, a x K f xa ) = ξ K ˆfξa). Proof. Apply Lemma 4.4 to f xa ). If K has positie characteristic, the case f = f o recoers the classical Riemann-Roch theorem for cures oer finite fields; see [8, Thm 7.]. This can therefore can be iewed as a restatement and number-theoretic analogue of the classical Riemann-Roch theorem.
6 BAPTISTE DEJEAN Proof of Theorem 4.. We treat only the case that K is a number field. The positie-characteristic case is similar; see [8, Thm 7-6]. Choose a splitting A K = U A K im, identifying im = 0, ) with a subgroup of A K. Let dx be Lebesgue measure, and let dx r = dx/x. There is now a unique measure dx u on U AK such that d x on A K is the product dx u dx r. If ω has exponent greater than, we see zω), f = = = A K fx)ωx) d x = fxt)ωxt) dx u dt r 0 U AK fxt)ωxt) dx u dt r + fxt)ωxt) dx u dt r U AK 0 U AK fxt)ωxt) dx u dt r + f xt ) ω xt ) dx u dt r. U AK U AK We treat the second term. Let E be a fundamental domain for K in U K ; we remind the reader that [9, Sec 4.3] has a construction. Notice f xt ) ω xt ) dx u = xyt ) ω xt ) dx u. U AK E y K f We can almost apply Riemann-Roch 4.6; we first need to add the final term E f0)ω xt ) dx u. Once we hae, Riemann-Roch gies f xt ) ω xt ) dx u + f0)ω xt ) dx u U AK E = x t ˆf ξx t ) ω xt ) dx = ˆf ξx t ) ˆω x t ) dx u. E ξ K E ξ K Reersing steps yields that this is ˆfxt)ˆωxt) dxu + U AK We hae, therefore, zω), f = E ˆf0)ˆωxt) dx u. fxt)ωxt) dx u dt r + ˆfxt)ˆωxt) dxu U AK U AK f0) ω xt ) dx u dt r + ˆf0) ˆωxt) dx u dt r E If ω is unramified, that is if ω is nontriial on U AK, the last two terms are 0. Otherwise, ωx) = x s for some s C, so these terms are E f0) t s dt r E dx u + ˆf0) t s dt r dx u E f0) = + ˆf0) ) dx u. s s E
TATE S THESIS 7 In the notation of [9], we obtain 4.7) zω), f = U AK fxt)ωxt) dx u dt r + U AK ˆfxt)ˆωxt) dxu {{ f0) + ˆf0) ) s s E dx u }} where the expression {{gs)}} is 0 if ω is unramified and gs) if ω = s. 4.7) now conerges for any ω other than and, not just ω of exponent greater than. We can therefore define zω) by 4.7), and the functional equation zω) = ẑ ˆω) is immediate. As a bonus, z has simple poles exactly at and. With the parameterization s of the Riemann surface containing these, we can immediately read off the residues as δ 0 E dx u at and ˆδ 0 E dx u at.. Computations Now we will compute specific examples. We will identify prime integers with the corresponding finite aluations of Q. If K = Q, we can choose additie characters as follows. Let ψ x) = e πi x. For a finite prime p, Q p /Z p is algebraically) isomorphic to Z [/p] /Q, which algebraically) embeds into R/Z. Composing these gies a homomorphism λ : Q p /Z p R/Z. Let ψ p x) = e πi λx) ; this is an additie character of conductor 0. Eery ψ p is unramified, hence ψ = p ψ p gies a character of A Q triial on Q. For an arbitrary number field K and a aluation of K lying oer a aluation p of Q, we can define ψ by composing ψ p with the trace map K Q p. If p is finite, this has kernel diff K /Q p ). Cofinitely many of these differents are, so ψ = ψ gies a character of A K, which is also triial on K. For finite, let ν be the conductor of ψ ; that is, m ν = diff O K /Z p ). As we checked in Subsection 3.4, for finite, the measure on K which is self-dual with respect to ψ assigns O K the measure π ν/. For more details, see [9, Sec.] and [9, Lem 4..]. We will now fix this choice of ψ for computation of ε... Warm-up: Riemann zeta function. Let K be Q. For s C, let ω s be s, so ω s = s. The local factors of ω s are then s. Λ ω s ) is the Riemann zeta function ζs) times the extra factor L ω s ) = π s/ Γ s ). Our extension of Λ thus gies an extension of ζ. For our functional equation, eery ε p is, so we obtain that is, Λ ω s ) = Λ ω s ) ; s.) π s/ Γ ζs) = π ) s This is the classical functional equation for ζ. Γ s ) ζ s).,
8 BAPTISTE DEJEAN.. Dirichlet L-functions. Let χ be a primitie Dirichlet character mod m. As we hae already described Construction.), there is an idèlic character ω, triial on Q, such that L p ω s ) = χp) p for finite primes p. Therefore s Λ ω s ) = Ls, χ) L ω s ) for s with Res) >. This equation gies a meromorphic extension of Ls, χ), and by Corollary 4.3 it satisfies Ls, χ)l ω s ) = εω s )L s, χ )L ω s ). We will now compute our local ε factors. For finite p relatiely prime to m, ω p is unramified and ψ p has conductor ν = 0, so by 3.3) ε p ω p s p, ψ p ) =. As ω is a character of R, we only hae two choices for ω, determined by { the image of. ω ) = χ ) = χ ), giing ε ω s, ψ ) = if χ ) = i if χ ) =. We now need only to deal with the remaining ε factors. The conductor c p of ω p is the multiplicity with which p diides m. Computing, we obtain p m ε p ω p p, ψ p ) = p m p cp Z p = m m s p m Z p = m s χ x) x Z/m) ω p ω x) x s ψ p x) dx p ι p x p ) ψ p x p ) dx p m p m e πi xp/m = m s p m x Z/m) χx) e πi x/m, where x p denotes the image of x under the composition Z/m Z/p cp Z/m of maps gien by the Chinese remainder theorem. We collect our final functional equation: it is.) Ls, χ)l ω s ) = ε L s, χ )L ω s) m s where ε = L ω s ) = and L ω s) = { if χ ) = i if χ ) =, { π s/ Γ ) s if χ ) = π s+ Γ ) s+ if χ ) =, {π s Γ ) s if χ ) = π s Γ ) s if χ ) =. x Z/m) χx) e πi x/m,
TATE S THESIS 9.3. Hecke characters. Let K be a number field. Fix an ideal f of O K, and let f = p pcp be its factorization into prime ideals. For x O K, let x denote the residue of x residue mod f. Definition.3. A Hecke character mod f is a character χ of the group of fractional ideals relatiely prime to f satisfying the following condition: a character ω of arch. K exists such that, for all x mod f), χx)) = ω x). Notice χ uniquely specifies ω. Definition.4. A Hecke character mod f is called primitie if it is not induced by restriction) by a Hecke character mod f for any f properly diiding f. If χ is a Hecke character mod f, χ f x) = χx)) ω x) gies a well-defined character of O K /f). Hence we can define Hecke characters as characters which factor as χ f x) ω x), for some χ f and ω, on principal ideals coprime to f. Gien a Hecke character χ mod f, we can define an L-function.) Ls, χ) = I χi) I s = p χp) p s, where the sum is taken oer ideals of O K relatiely prime to f, the product is taken oer primes of O K not diiding f, and I = O K /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 gie a ery explicit and computational, yet not ery clean, construction of ω. ) If p is a prime not diiding f, the requirement L p ω p s p = χp) p s forces ω p to be unramified with ω p π ) = χ p). For infinite, let ω be the factor of ω at. For eery remaining aluation, let ω be the factor of ω ram at, where ω ram : p f K p S is defined as follows. First, define ω ram on p f U p as the pullback of χ f under the quotient p f U p O K /f). The factorization χx)) = χ f x) ω x) now guarantees that, for whateer extension ω ram we choose, ωx) = ω x ) is triial on eery α K coprime to f. We now need to extend ω ram to the rest of f K S to make ω triial on all of K. This is obtained by the weak approximation theorem.. Continuity follows by translating to the case x f U, where we hae already explicitly constructed a continuous ω ram.) This gies 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 eery prime p, p cp f where c p is the conductor ) of ω ). Then we can define a Hecke character χ by letting χp) = ω p πp and extending by linearity. The associated ω is then arch. ω. These are inerse bijections between primitie characters and characters of A K. Hence, the classical and idèlic pictures are equialent. χ f is with some obious identifications) the product of the ramified ω s, restricted to their U s. ω is ω. For the unramified ω s, ω π ) encode χ itself. The remaining ω π )s, though not seen in the classical picture, are there to make ω triial on K. As a corollary of the equialence between the classical and idèlic pictures, for K a number field, the content of Corollary 4.3 is exactly the functional equation
0 BAPTISTE DEJEAN for Ls, χ) for primitie Hecke characters χ. In particular, if we define Λs, χ) = Λ ω s ), where χ and ω are corresponding Hecke and idèlic characters, we obtain.6) Λs, χ) = Ls, χ) arch. L ω s ) for Res) >. 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 = corresponds to χ ) = and ω = sgn corresopnds to χ ) =. The distinction between these two kinds of Dirichlet characters comes in this iew) from their haing different infinite parts..4. Example: Hecke characters of Q ) mod ). To illustrate the situation just described, we will work out the exercise on [, Page 30]: compute the functional equations of Ls, χ) for Hecke characters of Q ) mod f = ). First, we will determine which ω s are allowed. There are two infinite places of Q ), namely a + b = a + b and a + b = a b, and both of these are real. ω is a character of R R ; therefore we may write it uniquely as ω x, x ) = x a x a sgn x ) s sgn x ) s for a, a R i and s, s Z/. For ω to correspond to a Hecke character χ, ω x) = χx)) for x mod ) ) forces ω to be triial on units of O Q ) which are congruent to mod ). As and + generate the units of O Q ), the group of units congruent to mod ) + is principal and generated by ). ω is triial + on ) if and only if + a a) ) s+s = ; that is, some n Z + exists such that a a = πin/ log ) and s + s n mod ). Say we hae such an ω. As O Q ) is a PID and its units surject to those of O Q ) / ) = F, we can retriee the χ with this ω by defining χx/y)) = ω x) ω y) for x, y mod ) ). This gies a well-defined character on all ideals coprime to ). Computing the corresponding χ f, we see χ f 3) = χ f + = ω + ) χ + )) = ω + ), and 3 generates O Q ) / ) = F. Let ω be the idèlic character corresponding to χ. At eery finite place other than, ) ) = and ω is unramified, so ω =. As ω =, ). This, that ω has conductor, and ω ) ω = ω ) completely determine ω. ω + 3) = χ f 3 ) = The description of ω for other finite is straightforward. Pick a generator x of p which is congruent to mod ). ω is unramified, and ωx) = forces ω π ) = ω x) = ω x) to hold.
TATE S THESIS ) We can now finally compute all our local ε factors. diff O Q ) /Z = ), so ε ω s, ψ ) = for eery finite other than. From 3.4), we obtain ) ε ω s, ψ = / s ω ) χ f x)e 4πi x/. x F Our infinite ε-factors are { ε i ω i s ) if a i = 0 i, ψ i = i if a i =. Our functional equation is therefore where Λs, χ) = Λ s, χ ) ε / s ω ) χ f x)e 4πi x/, x F Λs, χ) = Ls, χ) L ω s ) L ω s ), Λ s, χ ) = L s, χ ) L ω s ) L ω s ), if a = a = 0 ε = i if a a, if a = a = and L and L are as in.... Final remarks. We refer the reader to [8] for classical applications; among these are the Tchebotare density theorem. If K is a field of positie characteristic containing F q, then ζ K s) = Λ s ) is in fact a rational function in q s! See Exercise in Chapter 7 of [8] for a proof. The properties of this rational function are generalized by Weil s celebrated conjectures, now resoled.
BAPTISTE DEJEAN References [] Bump, Daniel A. Automorphic Forms and Representations. Cambridge: Cambridge Uniersity Press, 998. [] Cassels, J. W. S. Global Fields. In Algebraic Number Theory, edited by J. W. S. Cassels and A. Fröhlich, 4-84. London: Academic Press Inc., 967. [3] Folland, Gerald B. Real Analysis: Modern Techniques and Their Applications. New York: John Wiley & Sons, Inc., 999. [4] Hecke, E. Eine neue Art on Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen. Mathematische Zeitschrift, Issue -, Volume 6 March 90): -. [] Kudla, Stephen S. Tate s Thesis. In An Introduction to the Langlands Program, edited by Joseph Bernstein and Stephen Gelbart, 09-3. New York: Springer Science+Business Media, 004. [6] Lang, Serge. Algebraic Number Theory. New York: Springer Science+Business Media, 986. [7] Neukirch, Jürgen. Zeta Functions and L-series. In Algebraic Number Theory, 49-49. Translated by Norbert Schappacher. New York: Springer Science+Business Media, 999. [8] Ramakrishnan, Dinakar, and Valenza, Robert J. Fourier Analysis on Number Fields. London: Academic Press Inc., 967. [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, 30-347. London: Academic Press Inc., 967. [0] Weil, André. Fonction zêta et distributions. Séminare Bourbaki, Volume 9 964-966), Talk no. 3, 3-3.