2.1 Automorphic forms for GL(1, A Q )

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1 CHAPTER II AUTOMORPHIC REPRESENTATIONS AND L-FUNCTIONS FOR GL1, A Q 2.1 Automorhic forms for GL1, A Q Let A Q denote the adele ring over Q as in definition The key oint is that GL1, A Q is just A Q, the multilicative subgrou of ideles of Q. In conformity with modern notation we shall use the notation g = {g, g 2, g 3,... } to denote an element of the grou GL1, A Q. Here g v Q v for all v and g Z for all but finitely many finite rimes. The multilicative grou Q is diagonally embedded in A Q and acts on A Q by left multilication. Proosition tells us that a fundamental domain for this action is given by Q \A Q = 0, Z, where the roduct is over all finite rimes. An idelic function f : A Q C is said to be factorizable if it is determined by local functions f v : Q v C v, where f 1 on Z for all but finitely many finite rimes, and where fg = f v g v, g = {g, g 2, g 3,... } A Q. v Note that this definition is slightly different than the notion of factorizability of adelic functions which was given in Definition Unitary Hecke character of A Q A Hecke character of A Q is defined to be a continuous homomorhism ω : Q \A Q C. A Hecke character is said to be unitary if all its values have absolute value 1. A unitary Hecke character of A Q is characterized by the following four roerties: i ωgg = ωgωg, g, g A Q ; ii ωγg = ωg, γ Q, g A Q ; iii ω is continuous at {1, 1, 1,... }. iv ω = 1. 1 Tyeset by AMS-TEX

2 2 II. AUTOMORPHIC REPRESENTATIONS AND L-FUNCTIONS FOR GL1, A Q Definition Moderate growth We say that a function φ : A Q C is of moderate growth if, for each g = {g, g 2, g 3,... } A Q, there exist ositive constants C and M such that φ {tg, g 2, g 3,... } < C 1 + t M for all t R. Definition Automorhic form Fix a unitary Hecke character ω as in An automorhic form for GL1, A Q with character ω is a function which satisfies the following conditions: φ : GL1, A Q C 1 φγg = φg, g A Q, γ Q ; 2 φzg = ωzφg, g A Q, z A Q ; 3 φ is of moderate growth as in definition Let S ω denote the set of all automorhic forms for GL1, A Q with character ω as in definition If c 1, c 2 C, are arbitrary comlex constants, and φ 1, φ 2 S ω, then it is easy to see that c 1 φ 1 + c 2 φ 2 is again automorhic with character ω. The sace S ω is, therefore, a vector sace over C. Setting g = {1, 1, 1,... } it immediately follows from definition that φz = cωz, with c = φ{1, 1, 1,... }. Thus S ω is a one-dimensional sace. The reader may ask why we simly do not define an automorhic form as a Hecke character as in 2.1.2? The reason is that we want to give a uniform definition of automorhic form for GLn, A Q that holds for all n = 1, 2, 3,... In the case of n = 1, definition becomes suerfluous since z, g both lie in the same sace. This is not the case for n > 1 as we shall see later. Definition may seem rather imosing at first sight but it turns out that the automorhic forms for GL1, A Q are just classical Dirichlet characters in disguise. For a fixed integer q > 1, a Dirichlet character χ mod q is a homomorhism χ : Z/qZ C. That is, χab = χaχb, a, b Z/qZ. Because a ϕq = 1 for all a Z/qZ, such a function must take values in the ϕqth roots of unity. In articular, Here ϕq is Euler s ϕ function. χa = 1, a Z/qZ.

3 II. AUTOMORPHIC REPRESENTATIONS AND L-FUNCTIONS FOR GL1, A Q 3 It is standard ractice in analytic number theory see [Davenort, 2000] to lift a Dirichlet character χ to Z by defining a new function χ 1 : Z C which satisfies χ 1 a = 0, a Z with a, q 1; χ 1 a + mq = χa, a, m Z with a, q = 1, χ 1 ab = χ 1 aχ 1 b, a, b Z. We shall follow the standard ractice of denoting the character χ 1 by the symbol χ. Remarkably, it is also ossible to lift χ to the idele grou A Q. Since A Q is a urely multilicative grou, the value of the lifted character can never be 0. The result is an automorhic form as in definition We now exlicitly describe and rove the existence of this lifting which was found by Tate and aeared in his thesis [Tate, 1950]. Definition Idelic lift of a Dirichlet character Let χ be a Dirichlet character of conductor f as in 2.1.5, where f is a fixed rime ower. We define the idelic lift of χ to be the unitary Hecke character χ idelic : Q \A Q C defined as χ idelic g = χ g χ 2 g 2 χ 3 g 3, g = {g, g 2, g 3,... } A Q, where and where 1, χ 1 = 1, χ g = 1, χ 1 = 1, g > 0, 1, χ 1 = 1, g < 0, { χv m, if g v v m Z v and v, χ v g v = χj 1, if g v k j + f Z with j, k Z, j, = 1 and v =. To see that definition actually defines a unitary Hecke character satisfying we make the following observations. First of all, for every rime v, it is clear from the definition that χ v g v g v = χ v g v χ v g v for all g v, g v Q v. Consequently, χ idelic must satisfy i. Secondly, if l is any finite rime, then χ l l = χl. Also χ l = χl 1, χ v = 1 for any finite rime v and χ v l = 1 for any finite rime v l and v. It follows that χ idelic l = 1 for all rimes l. Also χ idelic { 1, 1, 1,..., } = χidelic {1, 1, 1,..., } = 1. When combined with i, this establishes ii. Thirdly, we can see directly that the kernel of χ idelic is an oen neighborhood of {1, 1, 1,... }. It is also obvious that χ idelic = 1 since χ has this roerty on Z/ n Z. The above

4 4 II. AUTOMORPHIC REPRESENTATIONS AND L-FUNCTIONS FOR GL1, A Q observations establish that χ idelic is indeed a Hecke character satisfying the four conditions of More generally, every Dirichlet character χ of conductor q = r i=1 fi i, where 1, 2,... r are distinct rimes and f 1, f 2,... f r 1 can be factored as χ = r χ i, i=1 where χ i is a Dirichlet character of conductor fi i. It follows that χ may be lifted to a Hecke character χ idelic on A Q where χ idelic = r i=1 χ i idelic. Theorem Every automorhic form φ on GL1, A Q, as in definition 2.1.4, can be uniquely exressed in the form φg = c χ idelic g g it A, g A Q, where c C, t R, are fixed constants, and χ idelic is an idelic lift of a fixed Dirichlet character χ as in definition and Here, if g = {g, g 2, g 3,... }, then g A = g v v is the idelic absolute value. v Proof: It follows from definition that we may take φg = c ωg, g A Q with c = φ {1, 1, 1,... }, and where ω is a unitary Hecke character satisfying For each rime v, consider the embedding Then, if we define i v g v = {1,..., 1, g v }{{} v th osition, 1..., 1}, g v Q v. ω v g v := ωi v g v, g v Q v then ω v is a character of Q v for every rime v. Furthermore, ωg = ω v g v v where ω g 1, g Z for all but finitely many finite rimes. The Hecke characters ω can then be determined if we can classify the local characters ω v for all rimes v. First of all, every unitary continuous multilicative character ω : R C is of the form ω g = g it, or ω g = g it signg,

5 II. AUTOMORPHIC REPRESENTATIONS AND L-FUNCTIONS FOR GL1, A Q 5 { +1, if g > 0, for some fixed constant t R, where signg = 1, if g < 0. To continue the roof and classify the characters ω v for v <, we introduce some definitions. Definition Unramified local character Fix a finite rime. A local character ω, occurring in the decomosition , is said to be unramified if ω u = 1 for all u Z. At, the local character g it for some fixed t R is said to be unramified. Fix a finite rime. The unramified local characters of Q are easy to describe. Every g Q is an element of m Z for some integer m. Let g = m u with u Z. Then if ω is unramified, it follows that ω g = ω m u = ω m = ω m. So once we know the value of ω at the one oint, we know its value everywhere. Definition Ramified local character and its conductor Fix a finite rime. We say a local character ω, occurring in the decomosition , is ramified if ω u 1 for some u Z. The conductor of ω is defined to be k where k is the smallest ositive integer such that 1 + k Z is contained in the kernel of ω. We say the local character ω is ramified if ω u = ω u for all u R. Fix a finite rime. The ramified local characters of Q follows. Let g = m u with u Z. Then can be described as ω g = ω m u = ω m ω u. Since ω u 1 for some u Z it follows, by continuity, that the kernel of ω contains an oen subgrou of the form 1 + k Z. We shall assume k is minimal and term k the conductor of ω. Claim: Z / 1 + k Z = Z/ k Z. To see this note that the multilicative cosets in Z / 1 + k Z are all of the form j 1 + k Z = j + k Z, 1 j < k, j, = 1. Furthermore j 1 + k Z j 1 + k Z = j j 1 + k Z from which it follows that we may assume j, j, j j Z/ k Z. The above claim imlies that the ramified local character ω must satisfy: ω g = ω m χ k j 1, for some fixed Dirichlet character χ k of Z/ k Z, for all j Z/ k Z, and whenever g m j + k Z.

6 6 II. AUTOMORPHIC REPRESENTATIONS AND L-FUNCTIONS FOR GL1, A Q The relations , , and secify all the ossible local characters that may occur in the factorization of ω. Now the local characters ω v must be chosen so that ωγg = ω v γg v = ωg v for all g A Q, and for all γ Q. The relation ωγg = ωγωg = ωg imlies that ωγ = 1 for all γ Q. Let S = { 1, 2,..., r } denote the finite set of rimes where ω i is ramified with conductor ki i for i = 1, 2,..., r. Define χ = r i=1 χ k i i where each χ k i i is determined by for i = 1, 2,..., r. Let l > 0 be a rime where l S. Then combining with , , , it follows that This forces ωl = 1 = l it ω l l ω l l = l it r i=1 On the other hand, if l S, we have So in this case, ωl = 1 = l it ω l l ω l l = l it r r i=1 χ k i i l 1. χ k i i l = l it χl. i=1 i l r i=1 i l χ k i i l 1. χ k i i l 1. Finally we consider the case where γ = 1 in It follows that ω 1 = 1 = ω 1 r χ k i 1 1 = ω 1χ 1. i=1 Equations , , together with definition imly theorem

7 II. AUTOMORPHIC REPRESENTATIONS AND L-FUNCTIONS FOR GL1, A Q The L-function of an automorhic form We have shown in theorem that every automorhic form φ on GL1, A Q is associated to a uniquely defined Dirichlet character χ. To each such automorhic form φ we shall attach an L-function: Ls, φ = 1 χ 1 s, where the above roduct converges absolutely for Rs > 1. The L-function is just the classical Dirichlet L-function associated to the Dirichlet character χ. What we have done here is to take some classical objects: Dirichlet character, Dirichlet L-function, and dress them in very fancy clothes so that they become almost unrecognizable extremely elegant objects with a new ersonality. Nevertheless, we are undaunted because we know them for what they truly are. However, there is much to be gained by such an aroach and the later rewards will be very significant. Our next goal is to obtain the analytic continuation and functional equation of Ls, φ by the adelic method of [Tate, 1950], [Iwasawa, 1952, 1992]. In general, the L-function can be constructed by integrating over the idele grou the automorhic form φ against a suitable test function. Let us begin by discussing the simlest case when the automorhic form is the trivial function, i.e., the constant one. In this case, the L-function is just the Riemann zeta function ζs = s. n=1 n s = In order to construct the Riemann zeta function as an idelic integral, we must introduce a set of test functions and aroriate measures to do idelic integration. Here are the recise definitions we need. Definition Adelic Schwartz sace Let S denote the set of all adelic Schwartz functions as defined in definition Recall that these are just linear combinations of functions f : A Q C where f = f v with f v the characteristic v function of Z v for all but finitely many v and f is Schwartz on R and f v is a locally constant comactly suorted function at all finite rimes v. Definition Idelic integral As in definition 1.7.5, we define the idelic integral for factorizable idelic functions f = v f v such that f is the characteristic function 1 Z for almost all rimes by fxd x = f v x v d x v, A Q v S Q v where S is a finite set containing and all the rimes such that f 1 Z. Also, { dx d x, if v =, x v = 1 dx 1 1 x, if v = is a finite rime.

8 8 II. AUTOMORPHIC REPRESENTATIONS AND L-FUNCTIONS FOR GL1, A Q Thus, d x is normalized so that Z d x = 1. In the usual manner, we extend the definition of adelic integration to define E fxd x for functions f : A Q C, and subsets E of A Q such that f 1 E is a linear combination of factorizable functions of the tye considered above. As in the additive theory, we have f v x v d x = A Q v v Q v f v x v d x v. We summarize this information as defining an idelic differential: d x = d x v. v Definition Idelic absolute value For x = {x, x 2,... } A Q, we recall the definition x A = x v v. v Definition Secial choice of test function For x = {x, x 2,... } A Q, we define the test function hx = e πx2 1 Zv x v S v< where 1 Zv is the characteristic function of Z v at all the finite rimes v <. The function h has the nice roerty that h = ĥ, i.e., it is its own Fourier transform. Let s C with Rs > 1. Following examle 1.5.8, the Riemann zeta function, ζs, then aears naturally in the following comutation: A Q hx x s A d x = e πt 2 R = π s 2 Γ s 2 = π s 2 Γ s 2 t s dt t 1 s 1 ζs. Z {0} x s d x The meromorhic continuation and functional equation of ζs can be obtained by use of the adelic Poisson summation formula Since h = ĥ, and more secifically, ĥ0 = h0 = 1, we may rewrite the adelic Poisson summation formula in the form: hαx = x α Q A x A α Q h α x.

9 II. AUTOMORPHIC REPRESENTATIONS AND L-FUNCTIONS FOR GL1, A Q 9 Recall roosition which says that A Q = α Q α Q \A Q. To obtain the analytic continuation and functional equation of ζs we roceed as follows: hx x s d x = hx x s d x A A A Q = = α Q Q \A Q x A 1 α Q \A Q α Q hαx αx s A d x α Q hαx x s A d x + x A 1 α Q hαx x s A d x. In the above we have used the roduct formula theorem which says that α A = 1 for α Q. We aly the Poisson summation formula to the first term in the last line of to obtain x A 1 α Q hαx x s A d x = = = x A 1 x A 1 x A 1 x s 1 x s A A 1 x A + 1 x A d x + x s 1 x s d x + A A α Q h h x Q \A α Q Q x A 1 x A 1 By roosition strong aroximation for ideles we have It follows that for Rs > 0, x A 1 Q \A Q = 0, Z. x s A d x = 1 0 y s dy y = 1 s. α 1 x s A x d x α x s 1 d x A x α Q hαx x 1 s A d x.

10 10 II. AUTOMORPHIC REPRESENTATIONS AND L-FUNCTIONS FOR GL1, A Q Combining the above comutations with 2.2.6, 2.2.8, yields π s 2 Γ s 2 ζs = = 1 s 1 1 s + α Q hαx x s A d x x A 1 hαx x s + x 1 s d x. A A α Q Note that the integral on the right side of converges for all s C and defines an entire function. Consider x on the right side of Since x is restricted to be in the fundamental domain for Q \A Q, it follows that x Z for all finite rimes. It further follows from the definition of h that αx Z for all finite rimes. This imlies that α Z. After some elementary maniulations it is easy to show that equation is really the same as the classical identity π s 2 Γ s 2 ζs = 0 e πn2 y 2 y s n Z n 0 = 1 s 1 1 s + 1 dy y n Z n 0 One immediately obtains the following theorem. e πn2 y 2 y s + y 1 s dy y. Theorem The Riemann zeta function, ζs = n s, has a meromorhic continuation to all s C with a simle ole at s = 1. It satisfies the functional equation π s s 2 Γ ζs = π 1 s 1 s 2 Γ ζ1 s. 2 2 Tate, in his thesis [Tate, 1950] realized that an identity of tye could be obtained for any test function f in the adelic Schwartz sace S defined in If we rewrite the adelic Poisson summation formula in the form α Q fαx = ˆf0 x A + 1 x A α Q ˆf n=1 α x f0, and we relicate the stes in the roof of , using a more general function f instead of h, it immediately follows that α Q fαx x s A d x = ˆf0 s 1 f0 s + x A 1 α Q fαx x s A 1 s + ˆfαx x d x. A

11 II. AUTOMORPHIC REPRESENTATIONS AND L-FUNCTIONS FOR GL1, A Q 11 Note that the right side of does not change if we make the simultaneous transformations: s 1 s, f ˆf. Now, let us assume that f S is factorizable, as in definition Then f is the characteristic function of Z for all but finitely many rimes. Let S = { 1, 2,..., l } denote the finite set of rimes i where f i is not equal to the characteristic function of Z i i = 1, 2,..., l. It follows that A Q fx x s A d x = R f y y s = f s = f s S S Q Q dy y Q f x x s d x f x x s d x 1 s 1 S f x x s d x 1 s 1 ζs, where f denotes the Mellin transform of f given by fs = 0 fyy s dy y. We immediately conclude that the right hand side of is invariant under the simultaneous transformations s 1 s, f ˆf. INTERLUDE Three remarks of Ivan Fesenko: Remark 1: Equations and are further generalized in Tate s thesis [Tate, 1950]. The method is a very owerful and amazingly simle way to deduce the key roerties of zeta and L-functions. Exactly the same reasoning as above gives the functional equation and meromorhic continuation of the comleted zeta function of an algebraic number field K. The analogue of the right hand side of is 1 c s 1 1 s + x K \A K x A 1 α K hαx x s A + x 1 s A d x where c is the volume of K \A K, and A K = {x A K x A = 1}.

12 12 II. AUTOMORPHIC REPRESENTATIONS AND L-FUNCTIONS FOR GL1, A Q As first observed in [Iwasawa, 1952], the above formula imlies the finiteness of c which in turn imlies the comactness of K \A K and the finiteness of the class grou of K. It also easily imlies the Dirichlet theorem of units in K. So, the most fundamental theorems of classical algebraic number theory follow as easy and fast corallaries of the adelic comutation of the zeta function. Remark 2: Ideles were first introduced by Chevalley in the 1930 s to simlify and make more concetual the exosition of class field theory. The adelic method of Tate and Iwasawa works with functions on adelic objects but does not use any result of class field theory. An extension of the adelic method from GL1 to GLn was first found by Godement and Jacquet see [Godement-Jacquet, 1972] and is a major theme of this book. The extension gives a noncommutative theory over global fields, which is related to noncommutative class field theory and the Langlands rogram. The latter is about 40 years old and is still actively involved in the discovery of new fundamental concets. Remark 3: Instead of going from GL1 to GLn, it is also ossible to extend the work of Tate and Iwasawa in an entirely different asect. From the oint of view of modern algebraic geometry, the field Q may be thought of as the field of rational functions on the one dimensional arithmetic scheme SecZ. A two dimensional adelic analysis seeks to develo and use harmonic analysis on adelic saces associated to an arithmetic surface i.e., a two dimensional object. For examle, to an ellitic curve E over a global field K, given by equation Y 2 = X 3 + ax + b, one can associate an arithmetic surface E. Two dimensional abelian class field theory [Parshin, Kato, Saito, Fesenko] and others describes abelian extensions of the field of rational functions on E via certain adelic objects and their quotients. The adelic objects are restricted roducts of certain two-dimensional local fields, e.g., Q t, Rt, etc., which are not locally comact grous. See [Fesenko, 2003, 2008] for a develoment of a twodimensional adelic theory and a generalization of in this context. The two-dimensional adelic theory studies a zeta integral which is closely related to the arithmetic zeta function of E which itself is essentially ζ K s times ζ K s 1 divided by the Hasse-Weil L-function of E. Note that, as in the classical, one-dimensional case, the study of zeta functions uses adelic objects which naturally show u in class field theory, but does not deend on results from class field theory. We shall now show how the adelic method of Tate and Iwasawa can be used to obtain the meromorhic continuation and functional equation of all GL1, A Q L-functions. Let φ be a non-zero automorhic form for GL1, A Q as in definition By theorem 2.1.9, the automorhic form φ is determined by a constant c 0, a real number t, and the idelic lift χ idelic of a Dirichlet character χ. Let f S be an adelic Schwartz function as in definition For s C with Rs > 1, the Dirichlet L-function associated to φ will aear naturally in the comutation of the adelic integral A Q fx χ idelic x x s d x = c 1 A A Q fx φx x s it A d x.

13 II. AUTOMORPHIC REPRESENTATIONS AND L-FUNCTIONS FOR GL1, A Q 13 Let S = { 1, 2,..., l } be the finite set of rimes where χ is ramified see definition Then χ is unramified for S. We shall now also make the assumtion that f is equal to the characteristic function of Z for S. The comutation of easily generalizes to the situation of We have fx χ idelic x x s A A Q = d x R f y χ y y s = f χ s = f χ s S S Q Q = f χ s S Q dy y Q f x χ x x s d x f x χ x x s d x 1 χ s 1 S f x χ x x s d x 1 χ s 1 Ls, χ f x χ x x s d x Ls, χ where f χ s = f y χ y y s dy R y denotes the Mellin transform of f χ. Note, also, that if χ is ramified then χ = 0. This fact is used to obtain the last line in Furthermore, equation also easily generalizes to this situation and we may obtain fx χ idelic x x s d x A A Q = fαxχ idelic αx x s d x A α Q = ˆf0 x s 1 A f0 x s A χ idelic x d x x A 1 + x A 1 α Q fαxχ idelic x x s A + ˆfαxχ idelic x x 1 s A d x.

14 14 II. AUTOMORPHIC REPRESENTATIONS AND L-FUNCTIONS FOR GL1, A Q In the last line above we have used the fact that χ idelic 1/x = χ idelic x, which holds because χ idelic is unitary, i.e., has absolute value = 1. Note that the last integral in converges absolutely for all s C and defines an entire function. It is clear that the right most integral on the right side of equation is invariant under the simultaneous transformations: s 1 s f ˆf, χ idelic χ idelic. One may also show that ˆf0χidelic x x s 1 A f0χ idelic x x s A d x = 0 x A 1 if χ idelic has ramification as in definition at some finite rime. The fact that the left hand side of vanishes is essentially due to the fact that a non-trivial Dirichlet character χ mod q satisfies q j=1 j,q=1 χj = 0. It follows from that if χ idelic has ramification, then the L-function Ls, χ has a holomorhic continuation to all s C and satisfies a functional equation: s 1 s. Note that the holomorhy follows since there is a articular choice of f in for which ˆf is everywhere nonvanishing. If χ idelic is everywhere unramified, then we are in the situation of the Riemann zeta function and the formulas take the form of Let χ be a rimitive Dirichlet character mod q. It is known see [Davenort, 2000] that the Dirichlet L-function Ls, χ has holomorhic continuation to the entire comlex lane and satisfies the exlicit functional equation ξ s, χ := 1 π q 2 s+a Γ s + a 2 Ls, χ = τχ i a q ξ 1 s, χ where is the Gauss sum and τχ = q a=1 a,q=1 χae ia q { 0, if χ 1 = 1, a = 1, if χ 1 = 1.

15 II. AUTOMORPHIC REPRESENTATIONS AND L-FUNCTIONS FOR GL1, A Q 15 Definition Global conductor and root number The integer q in is called the conductor of Ls, χ while the constant τχ i a q is called the root number. Tate defined a different root number which deends on s and is not of s 1 2 π absolute value one. Tate s root number is defined to be q τχ i a q. Tate realized that the identities and which are invariant under the transformations must encode the classical functional equation of the Dirichlet L-function. The fact that and hold for any adelic Schwartz function f S suggests that there must be further symmetries hidden in these identities. This led him to discover the so called local functional equations and the fact that the analytic conductor and root number can be determined by local conditions. In the next two sections we shall introduce the local L-functions and their functional equations and exlain how the global functional equation follows from them. 2.3 The local L-functions and their functional equations In this section let v determine a local field Q v. Definition Local L-function Fix s C with Rs > 0. Let f v : Q v C be a locally constant comactly suorted function as in definition if v < and a Schwartz function if v =. Let ω v : Q v C be a local unitary character, i.e., a continuous homomorhism of absolute value 1. We define Ls, f v, ω v = f v x ω v x x s v d x, Q v to be the local L-function associated to ω v and f v. Here d x denotes the differential associated to the multilicative Haar measure as in definition if v < and d x = dx/ x if v =. Remark 2.3.5: The local integral is absolutely convergent if Rs > 0. To see this we write Ls, f v, ω v = f v x ω v x x s v d x + f v x ω v x x s v d x. x Q v x v>1 x Q v x v 1 The first integral on the right above is clearly convergent. Since f v is bounded in the region x v 1, the second integral is bounded by x σ v d x, x Q v x v 1 an integral which converges absolutely for Rs = σ > 0 for any fixed v. We now state the main theorem in the local theory.

16 16 II. AUTOMORPHIC REPRESENTATIONS AND L-FUNCTIONS FOR GL1, A Q Theorem Local functional equation Let s C with 0 < Rs < 1. The local L-function Ls, f v, ω v, as defined in 2.3.3, satisfies the functional equation Ls, f v, ω v = ρ ωv s L1 s, ˆf v, ω v where ˆf v denotes the v-adic Fourier transform of f v see 1.6.1, 1.6.8, ω v = ω 1 v is the comlex conjugate of ω v, and ρ ωv s is a meromorhic function which is indeendent of the choice of f v. Proof: To show that ρ ωv is indeendent of f v it is enough to show that Ls, f v, ω v L1 s, ˆf v, ω v = Ls, g v, ω v L1 s, ĝ v, ω v for any function g v : Q v C with the same roerties as f v. We will rove that by showing that Ls, f v, ω v L1 s, ĝ v, ω v = L1 s, ˆf v, ω v Ls, g v, ω v Ls, f v, ω v L1 s, ĝ v, ω v is symmetric in f v and g v. By remark 2.3.5, if 0 < Res < 1, we can write as an absolutely convergent double integral f v xĝ v yω v xy 1 x s v y 1 s v d x d y. Q v Q v The measure d y is invariant under the transformation y xy. It follows that can be rewritten in the form f v xĝ v xyω y x v y 1 s v d x d y. But Q v Q v ĝ v xy = g v ze v xyzdz. Q v Consequently f v xĝ v xyω y x v y 1 s v d x d y Q v Q v = Q v Q v Q v f v xg v ze v xyzω y y 1 s v dx d y dz which is clearly symmetric in f v and g v. { v v 1, if v, 1, if v =, Proosition Let s C with 0 < Rs < 1. Let ρ ωv s be defined as in theorem Then ρ ωv s ρ ωv 1 s = 1.

17 II. AUTOMORPHIC REPRESENTATIONS AND L-FUNCTIONS FOR GL1, A Q 17 Proof: By definition Ls, f v, ω v = ρ ωv sl1 s, ˆf v, ω v, L1 s, ˆf v, ω v = ρ ωv 1 sls, f, ω v. Consequently which imlies that Ls, f, ω v = ρ ωv sρ ωv 1 sls, f v, ω v, ρ ωv s ρ ωv 1 s = 1. Theorem Exlicit comutation of ρ ωv s Let s C, 0 < Rs < 1. The function ρ ωv s can be exlicitly given as follows. Case 1: v = ρ ω s = π s 2 Γ s 2 π 1 s 2 Γ 1 s 2, if ω x = 1 for x R, 1 i π s+1 2 Γ s+1 2 π 1 s+1 2 Γ 1 s+1 2, if ω x = signx for x R. Case 2: v = < ρ ω s = / 1 ω 1 ω 1 s, if ω s is unramified, rs 1 ω r r j=1 j,=1 ω je ij r, ω is ramified and has conductor r. Remarks: The local L-function Ls, f v, ω v can easily be = 0 if the Schwartz function f v is chosen in a stuid way. In the roof below we have made smart choices f v = fv in each situation. The factor 1/i that aears in ρ ω s when ω x = signx, occurs in the global root number, of for the case when χ 1 = 1. When ω is ramified and has conductor r then ρ ω s contains the Gauss sum r j=1 j,=1 ω je i j r, and when these Gauss sums are multilied together over all ramified rimes we obtain the Gauss sum occurring in the root number

18 18 II. AUTOMORPHIC REPRESENTATIONS AND L-FUNCTIONS FOR GL1, A Q Proof of theorem : We first consider the case v =, f x = e πx2, and ω 1. Then the local L-function is Ls, f, ω = e πx2 x s Since f = ˆf and ω = ω, it follows that dx = π s s 2 Γ x 2 ρ ω s = Ls, f, ω L1 s, f, ω = π s 2 Γ s 2 2 Γ. 1 s π 1 s Next, consider the case v =, f x = xe πx2, and ω x = signx = x x. Consequently Ls, f, ω = e πx2 x s+1 A simle comutation shows that f x = L1 s, f, ω = i It follows that e πx2 x 1 s+1 ρ ω s = Ls, f, ω L1 s, f, ω = 1 i 2 dx = π s+1 x 2 Γ s f ye ixy dy = if x. Hence dx = iπ 1 s+1 2 Γ x π s+1 π 1 s+1 1 s Γ s Γ 1 s+1 We now consider the non-archimedean situation where v = is a finite rime and ω is unramified. Choose Then Ls, f, ω = 1 Q { 1, if x f Z, x = 0, otherwise. f xω x x s dx x = 1 = 1 + ω s + ω 2 2s + ω 3 3s + = 1 ω 1 s. 2 Z {0} 2. ω x x s dx x

19 II. AUTOMORPHIC REPRESENTATIONS AND L-FUNCTIONS FOR GL1, A Q 19 Since f = f, we also have Consequently L1 s, f, ω = 1 ω 1 s 1 ω ρ ω s = 1 s. 1 ω s 1. Finally, we consider v = and ω is ramified with conductor r as in definition In this case, we choose { e f i{x}, if x r Z, x = 0, otherwise. where {x} denotes the fractional art of x Q as in definition With these choices we obtain Ls, f, ω = e i{x} ω x x s dx 1 x = 1 f Z {0} r = r 1 r l=1 j=1 l j+ r Z j,=1 r ls ω l l=1 = r+1 1 rs ω r r j=1 In the above comutation, we used the fact that r j=1 j,=1 e i{x} ω x x s dx x r j=1 j,=1 e ij l ω j = 0 e ij r ω j. e ij l ω j if l < r. In order to comute L1 s, f, ω we need to comute the Fourier transform f. We have f x = f ye i{xy} dy = Q e i{y1 x} dy r Z { r, if x 1 + r Z, = 0, otherwise.

20 20 II. AUTOMORPHIC REPRESENTATIONS AND L-FUNCTIONS FOR GL1, A Q It follows that L1 s, f, ω = r ω x x 1 s 1 1+ r Z = r dx 1 x = r Z dx x Theorem can be used to show that the local L-functions, as defined in 2.3.3, have meromorhic continuation to the whole comlex lane. Theorem Meromorhic continuation of the local L-functions Fix s C with Rs > 0. Let Ls, f v, ω v denote the local L-function as in definition Then Ls, f v, ω v has meromorhic continuation to all s C. Proof: The exlicit comutation of ρ ωv s given in theorem shows that in all cases, ρ ωv s has meromorhic continuation to all s C. When this is combined with the functional equation of theorem we immediately obtain the meromorhic continuation of the local L-function to the whole comlex lane. 2.4 Classical L-functions and root numbers The founders of the theory of L-functions over Q, Riemann, Dirichlet, Hecke, etc., defined L-functions so that their Euler roducts were as simle as ossible and so that their functional equations were as symmetric as ossible. The classical L-functions over Q are just the Dirichlet L-functions as in or the Riemann zeta function. If we are given a Schwartz function f : A Q C as in definition and an adelic automorhic form ω : GL1, A Q C as in definition 2.1.4, then we have intensively studied Tate s global L-function which was defined by Ls, f, ω := fx ωx x s d x A GL1,A Q for Rs > 1. It is natural to ask which choices of f and ω lead to the classical L-functions? Let s C with Rs > 1. If we assume that f = v f v is factorizable and ω = v ω v then Ls, f, ω = f v x v ω v x v x v s v d x v = L v s, f v, ω v, v Q v v which reduces Tate s global L-function to a roduct of local L-functions. Recall that a local character ω v : Q v C may be ramified or unramified as in definition

21 II. AUTOMORPHIC REPRESENTATIONS AND L-FUNCTIONS FOR GL1, A Q Accordingly, we say the rime v is ramified or unramified, resectively. In order to construct the classical L-functions it is necessary to choose f v for every v so that the local integral is as simle as ossible and the global functional equation of Ls, f, ω is as symmetric as ossible. It is easy to do this if v is unramified. In this case, we choose f v for all v so that ˆf v = f v, i.e., f v is its own Fourier transform. This amounts to the choices: { e πx 2, if v = is unramified, f v x v = 1 Z x, if v = < is unramified. Define the local L-function, denoted L v s, ω v, by setting L v s, ω v = L v s, f v, ω v for the above choice of f v. It follows from and the roof of theorem that the local L-functions take the form L v s, ω v = π s 2 Γ s 2, if v = is unramified, 1 ω s 1, if v = < is unramified. It is not as clear, however, how to make the choices when v is ramified. Again, using theorem as our guide, we will choose f v so that L v s, ω v := L v s, f v, ω v and { π s+1 2 Γ s L v s, ω v = 2, if v = is ramified, 1, if v = < is ramified. We may then define the classical comleted L-function L s, ω = L v s, ω v L s, ω = v where L v s, ω v is defined in 2.4.2, Then π s 2 Γ s 2 unramified π s+1 2 Γ s+1 2 unramified 1 ω s 1, if is unramified, 1 ω s 1, if is ramified. Definition Local root number Fix v. Let L v s, ω v be given by 2.4.2, 2.4.3, and for a Schwartz function f v : Q v C, let L v s, f v, ω v be a non identically vanishing local L-function as in We define the local root number to be the comlex valued function ɛ v s, ω v which satisfies L v 1 s, ˆf v, ω v L v 1 s, ω v = ɛ v s, ω v L vs, f v, ω v. L v s, ω v Note that by theorem 2.3.6, the local root number ɛ v s, ω v is indeendent of the choice of f v. Furthermore, by theorem , the local root number ɛ v s, ω v takes the value 1 at unramified rimes. It follows that the infinite roduct ɛs, ω := ɛ v s, ω v v

22 22 II. AUTOMORPHIC REPRESENTATIONS AND L-FUNCTIONS FOR GL1, A Q is really just a finite roduct which can be exlicitly comuted using theorem Proosition Global functional equation For Rs > 1, let L s, ω be defined as in Then L s, ω has a meromorhic continuation to all s C with at most simle oles at s = 0, 1, and satisfies the functional equation L s, ω = ɛs, ωl 1 s, ω where ɛs, ω is given by If ω is not the trivial character then L s, ω is an entire function of s. Proof: For each rime v, define a local L-function Ls, fv, ω v as in , , , For Rs > 1 let Ls, f, ω = Ls, fv, ω v. v Then by , we know that Ls, f, ω has a meromorhic continuation to all s C, with at most simle oles at s = 0, 1, and satisfies the global functional equation Ls, f, ω = L1 s, f, ω. Now Ls, f, ω is almost the same as L s, ω. When these functions are defined for Rs > 1 as roducts, then they differ only at the rimes where ω is ramified. In fact, for Rs > 1, we may write Ls, f, ω L s, ω = ramified Ls, f, ω = Ls, ω ramified Ls, f, ω. It follows from that if ω is ramified then Ls, f, ω is just a constant times a ower of s. Thus the right side of is holomorhic everywhere and never zero. This shows that L s, ω has the same roerties as Ls, f, ω. In articular it has meromorhic continuation to all comlex s with at most simle oles at s = 0, 1. The simle oles can only occur when ω is trivial and we are in the situation of the Riemann zeta function. Further, by the definition of the local root number 2.4.6, 2.4.7, we obtain from that Ls, f, ω L s, ω = ɛs, ω 1 ramified L1 s, f, ω. L1 s, ω The function on the right side of has meromorhic continuation to Rs < 0, and in that region is equal to ɛs, ω 1 L1 s, f, ω L 1 s, ω,

23 II. AUTOMORPHIC REPRESENTATIONS AND L-FUNCTIONS FOR GL1, A Q 23 from which we deduce that L1 s, f, ω L 1 s, ω = ɛs, ω Ls, f, ω L s, ω. When is combined with 2.4.9, roosition follows. Remarks on roots numbers: The global root number ɛs, ω is recisely Tate s root number as defined in Automorhic reresentations for GL1, A Q We begin with the abstract definition of a reresentation of a grou on a vector sace. Definition Reresentation of a grou on a vector sace Let G be a grou and let V be a vector sace. A reresentation of G on V is a homomorhism π : G EndV = {set of all linear mas: V V }. Here πg. v denotes the action of πg on v and πg g = πg. πg for all g, g G. We call V the sace of π and refer to the ordered air π, V as a reresentation. Remarks: Very often we shall consider reresentations π, V where V is a sace of functions f : G C and the action is given by right translation, i.e., πg. fg = fgg, g, g G. In this situation we see that πg g. fg = πg. πg. fg. Note that a reresentation defined by a left action on a sace of functions would satisfy the reverse identity πg g = πg. πg for g, g G. If the grou G and the vector sace V are equied with toologies, then we shall also require the ma G V V, given by g, v πg. v, to be continuous. Definition Irreducible reresentation A reresentation π, V as in is said to be irreducible if V 0, and V has no closed π-invariant subsace other than 0 and V. Definition Intertwining mas and isomorhic reresentations Let π 1 : G EndV 1, π 2 : G EndV 2, be two reresentations as in definition An intertwining ma, or intertwining oerator, is a linear ma L : V 1 V 2 such that L. π 1 g. v = π 2 g. L. v for all g G, v V 1. Here L. v denotes the action of L on v V 1. If there is an intertwining oerator L : V 1 V 2 which an isomorhism of vector saces, then the two reresentations are said to be isomorhic.

24 24 II. AUTOMORPHIC REPRESENTATIONS AND L-FUNCTIONS FOR GL1, A Q Roughly seaking, an automorhic reresentation of GLn for n = 1, 2,... is an irreducible reresentation of GLn, A Q on a toological vector sace V which consists of comlex valued functions on GLn, A Q satisfying certain growth, smoothness, and invariance roerties. We shall now make the definition recise for the case of GL1, A Q. Definition Automorhic reresentation for GL1, A Q Fix a unitary Hecke character ω of A Q as in Define V ω to be the one-dimensional vector sace over C of all automorhic forms with character ω as defined in We may define a reresentation by requiring that for all φ V ω and all g, x GL1, A Q. π : GL1, A Q End V ω πg. φx := φx g = ωgφx Remarks: It is clear that the Hecke character ω is factorizable, as a function, in the sense described at the beginning of this chater. But in fact, the reresentation π, V ω also factors as a reresentation. Once everything is viewed in the right manner, this fact is very close to being a tautology. We use this as an oortunity to introduce the notion of factorizable which is aroriate to reresentations. We first introduce the aroriate roduct, which is the restricted tensor roduct. In a sense, this roduct is to tensor roducts what the restricted direct roduct used to form the adeles is to Cartesian roducts. We first recall the ordinary tensor roduct of two vector saces see [Atiyah- Macdonald, 1969]. Suose V and W are vector saces over a field k, with ossibly infinite bases B and C. Then the tensor roduct of V and W is a vector sace, denoted V W, generated by a basis consisting of the symbols v w, where v B and w C. It may also be realized as the vector sace sanned by the set of all symbols v w with v V and w W, such that the only relations are those coming from relations in V, relations in W, or relations of the form α v w = α v w = v α w, for all α k, v V, w W, v 1 + v 2 w = v 1 w + v 2 w, for all v 1, v 2 V, w W, v w 1 + w 2 = v w 1 + v w 2, for all v V, w 1, w 2 W. An element of the form v w is called a ure tensor. Because the ure tensors san, it is often enough to rove theorems only for ure tensors. This construction generalizes in a straightforward way to threefold and n-fold tensor roducts, and even to infinite tensor roducts. The tensor roducts of interest to us will be infinite and indexed by the set of rimes. For each rime v let V v be a vector sace. For these, we write v V v for the tensor roduct and v ξ v for a ure tensor, where ξ v V v for each v. To define a restricted tensor roduct, one must fix a nonzero vector ξv in V v for all but finitely many v. Then the restricted tensor roduct of {V v } relative to {ξv} is the subsace of v V v sanned by the set of ure tensors which are of the form This vector sace is denoted vv v. ξ = v ξ v ξ v = ξ v for almost all v.

25 II. AUTOMORPHIC REPRESENTATIONS AND L-FUNCTIONS FOR GL1, A Q 25 Definition Factorizability for reresentations of A Q Let V be a vector sace. A reresentation π, V of A Q is factorizable if there exists: 1 For each v, a reresentation π v of Q v, on a vector sace V v. 2 For almost all v, a distinguished vector ξ v V v. 3 An isomorhism of vector saces l : vv v V, such that l v πv g v. ξ v = πg. l v ξ v, for all g = {g, g 2, g 3,... } A Q and all vξ v such that ξ v = ξ v for almost all v. Remarks: The linear ma l in definition gives an isomorhism of vv v and V. Note that is recisely what is required see definition to make the reresentation v π v, vv v isomorhic to the reresentation π, V. Proosition All automorhic reresentations for GL1, A Q are factorizable The reresentation π, V ω as defined in is factorizable. Proof: For each v, we take an abstract one-dimensional comlex vector sace, sanned by a vector, which we denote ξv. Thus V v is simly {cξv c C} for each v. We define an action of Q v on V v by π v g. cξ v = c ω v g ξ v, g Q v. Now, we need to define a linear ma l from vv v to V ω. By linearity, it is enough to define it on the element v ξv, which sans vv v. Recall that V ω is a one-dimensional vector sace sanned by the function ω. The definition of l is simly l v ξv = ω. One then easily verifies that l v πv g v. ξ v = πg. l v ξ v. Both sides are equal to ωgω, i.e, the function in V ω which takes the value ωgωx at the oint x. 2.6 Hecke oerators for GL1, A Q The theory of Hecke oerators for automorhic forms on GL2, R is well known see [Goldfeld, 2006]. Very roughly, Hecke oerators act on automorhic forms and transform them to other automorhic forms. If an automorhic form is an eigenfunction of all the Hecke oerators then it is called a Hecke eigenform. Hecke roved the remarkable theorem that the L-function associated to a Hecke eigenform has an Euler roduct. We shall now show that a similar result can be established for automorhic forms for GL1, A Q.

26 26 II. AUTOMORPHIC REPRESENTATIONS AND L-FUNCTIONS FOR GL1, A Q Definition Hecke oerator for GL1, A Q Let φ be an automorhic form for GL1, A Q with unitary Hecke character ω as in definition For every idele n = {n, n 2,... } A Q we define the Hecke oerator T n which acts on φ as follows for all ideles g GL1, A Q. By definition , we see that T n φg = φng T n φg = ωnφg so that every automorhic form is a Hecke eigenform. Actually, definition is not very interesting because the sace of automorhic forms on GL1, A Q with fixed character ω is just a one-dimensional sace. Hecke s theorem is trivial in this situation because we have already shown that the only ossible L-functions we can obtain for GL1, A Q are Dirichlet L-functions which do have Euler roducts. We state Hecke s theorem for comleteness. Theorem Let φ be an automorhic form for GL1, A Q with unitary Hecke character ω as in definition If φ is an eigenfunction of all the Hecke oerators T n as defined in then the L-function associated to φ has an Euler roduct. 2.7 The Rankin-Selberg method The urose of section 2.7 is to show that the classical Rankin-Selberg method see [Goldfeld, 2006] has an analogue on GL1, A Q. This very brief section is for cognoscenti who already have familiarity with the classical Rankin-Selberg method. Let s C with Rs > 0. Let f : A Q C be an adelic Schwartz function as in definition The Tate series associated to f is defined to be T x, s, f := α Q fαx αx s A = α Q fαx x s A for all x A Q. Because we are averaging over the multilicative grou Q it is easy to see that T γx, s, f = T x, s, f for all γ Q and x A Q. If φ 1, φ 2 are automorhic forms for GL1, A Q with characters ω 1, ω 2, resectively, as in definition 2.1.4, then it is clear that φ 1 φ 2 is again an automorhic form with character ω 1 ω 2. The classical Rankin-Selberg unfolding comutation takes the following form on GL1, A Q : φ 1 xφ 2 xt x, s, f d x = φ 1 xφ 2 xfx x s A d x Q \A Q A Q where the right side of is the comleted L-function roduct of local L- functions associated to the automorhic form φ 1 φ 2.

27 II. AUTOMORPHIC REPRESENTATIONS AND L-FUNCTIONS FOR GL1, A Q The -adic Mellin transform Definition adic Mellin transform Let f : Q C be a locally constant comactly suorted function. Assume that fy = h y for some other function h : { l l Z} C. For s C, we define the -adic Mellin transform fs := Q fu u s d u. Proosition adic Mellin inversion Let f : Q C be a locally constant comactly suorted function. Assume that fy = h y for some other function h : { l l Z} C. Let f be the Mellin transform as in definition Then fy = log log 0 fit y it dt. Proof: It follows from definition that for y = l, we have log log 0 fit y it dt = log log 0 = m Z h m [ = h l. Q m Z fu u it d u log log 0 ] ilt dt im lt dt d u The above Mellin transform and its inverse can be generalized to comactly suorted locally constant functions f : Q C where fy is not necessarily a function of y. It is convenient to make the following definition. Definition Conductor of a locally constant comactly suorted function Let f : Q C be a locally constant comactly suorted function. Then f must be identically equal to zero on m Z for all but a finite number of integers m. Let M denote the finite set of such integers m. For each m M, there exists an integer N m such that the function f m y must be constant on the cosets of 1 + N Z in Z. Choose N to be the largest of the values of all such N m with m M. Then N is defined to be the conductor of f. Definition Normalized unitary character of Q A continuous function ψ : Q C which satisfies ψyy = ψyψy, y, y Q P ; ψy C = 1, y Q, ψ = 1,

28 28 II. AUTOMORPHIC REPRESENTATIONS AND L-FUNCTIONS FOR GL1, A Q is called a normalized unitary multilicative character of Q. Let N be the smallest integer k 0 such that 1 + k Z is contained in the kernel of ψ. Then ψ is said to have conductor N. We also call ψ a character mod N. The number of ψ mod N is ϕ N, where ϕ is Euler s function. This follows from , along with the corresonding fact about classical Dirichlet characters. With these reliminaries in lace, we may now resent the more general Mellin transform. Definition General -adic Mellin transform Let f : Z C be a locally constant comactly suorted function. For s C, and ψ : Q C, a normalized unitary character as in 2.8.4, we define the Mellin transform fs, ψ := fu ψu u s d u. Q Proosition General -adic Mellin inversion Let f : Z C be a locally constant comactly suorted function of conductor N as in Let f be the Mellin transform as in definition Then we have fy = 1 ϕ N ψ mod N log log 0 fit, ψ ψy 1 y it dt. Proof: It follows from definition that if y = l, then 1 ϕ N ψ mod N = 1 ϕ N = 1 φ N = 1 ϕ N = fy. log log 0 ψ mod N fit, ψ ψy 1 y it dt log ψ mod N Q ψ mod N l Z log 0 Q fu ψu u it d u ψy 1 y it dt fu ψu ψy 1 log fu ψu ψy 1 d u log 0 u/y it dt d u The last ste in the above argument follows from the orthogonality relation 1 { 1, if u y N ϕ N ψuψy 1 Z, = 0, otherwise, ψ mod N and the fact that N l.

29 II. AUTOMORPHIC REPRESENTATIONS AND L-FUNCTIONS FOR GL1, A Q 29 Exercises for Chater Let t R be nonzero, and consider the automorhic form φ : GL1, A Q C defined by φg = g it A. Prove that φ is continuous, but show that its kernel is not an oen subgrou. Deduce that φ cannot be the idelic lift of a Dirichlet character. 2.2 From Theorem 2.19, we see that any automorhic form for GL1, A Q can be written as φg = c χ idelic g g it A for some Dirichlet character χ and some constants c and t. Find an exlicit formula for t in terms of nice values of φ. 2.3 Let G be a toological grou e.g., R, Q, or A Q and let ψ : G C be a homomorhism not necessarily continuous. a Show that ψ is continuous if and only if it is continuous at the identity of G. b If the image of ψ is a finite set, show that ψ is continuous if and only if kerψ is an oen subgrou. For examle, G = A Q and ψ = χ idelic for some Dirichlet character χ. 2.4 Suose ω : Q v C is a character such that ω 1 = 1, and let f be an even Schwartz function i.e., f x = fx for all x Q v. Prove that the local L-function satisfies Ls, f, ω = Let v be a rime and ω v : Q v C a ramified local character. Recall from that we defined the local factor of the classical comleted L-function to be { π s+1 2 Γ s+1 L v s, ω v = 2, if v =, 1 if v = <. Find a Schwarz function f v : Q v C such that L v s, f v, ω v = L v s, ω v. Hint: Do the cases v = and v = < searately. 2.6 For 0 < Rs < 1 and ω v any local character, show that ɛ v s, ω v = L vs, ω v L v 1 s, ω v ρ ω v s 1. Deduce that ɛ v s, ω v 1 if either v = or v = < and ω is unramified. If v = < and ω is ramified with conductor f, then ɛ v s, ω v = fs 1 ω f f a=1 a,q=1 ωae ia/f 2.7 For any character ω : Q v C and any comlex number s with 0 < Rs < 1, show that ρ ω s = ω 1ρ ω s. 1.

30 30 II. AUTOMORPHIC REPRESENTATIONS AND L-FUNCTIONS FOR GL1, A Q Conclude that ρ ω 1/2 = 1. When v = < and ω is ramified with conductor f, deduce the following classical fact about Gauss sums: f ωae ia/f = f. a=1 a,=1 2.8 Let χ mod f be a nontrivial Dirichlet character for some rime and some integer f 1. a Show that f j=1 j,=1 χj = 0. Hint: Make the change of variables j aj for some a Z with χa 0. b Show that Z χ idelic x d x = c Deduce equation f j=1 j,=1 χj = Let G be a grou, V a comlex vector sace, and π : G AutV an irreducible reresentation. Prove that if G is Abelian, then dimv 1. Hint: For a fixed g 0 G, consider how G acts on the eigensaces of πg In this exercise we will show that any continuous unitary character ω : R C is of the form ωg = g it, or ωg = g it signg, for some fixed constant t R as in In order to accomlish this, we will assume the following fact from the theory of covering saces in algebraic toology: for any continuous ma f : R + C with values in the unit circle, there exists a continuous ma λ : R + R such that λ1 = 0 and fg = e iλg. Here R + is the multilicative grou of ositive real numbers. The existence of such a ma λ is reasonably obvious, but the nontrivial art is that we may take λ to be continuous. For the remainder of the roblem, let ω be any continuous unitary character as above, and let f = ω R + be the restriction of ω to the subgrou of ositive reals. a Define a ma Λ : R + R + R by the formula Λg, h = λgh λg λh. Show that e iλg,h = 1 for all g, h R +. Conclude that λ is a grou homomorhism. Hint: You will need to use the continuity of λ for the final conclusion. b Show that there exists t R such that λe r = rt for all real numbers r. Deduce that fg = g it for all g R +.