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178 STEPHEN GELBART, The unifying role of group symmetry in geometry so penetratingly ex. pounded by Felix Klein in his 1872 Erlanger Program has led to a century of. progress A worthy successor to the Erlanger Program seems to be Langlands. program to use infinite dimensional representations of Lie groups to illuminate. number theory, That the possible number fields of degree n are restricted in nature by the. irreducible infinite dimensional representations of GL was the visionary. conjecture of R P Langlands His far reaching conjectures present tantaUzing. problems whose solution will lead us to a better understanding of representa. tion theory number theory and algebraic geometry Impressive progress has. already been made but very much more Hes ahead, The purpose of this paper is to explain what the Langlands program is about. what new perspectives on number theory it affords and what kinds of. results it can be expected to prove, To begin with Langlands program is a synthesis of several important. themes in classical number theory It is also and more significantly a. program for future research This program emerged around 1967 in the form of. a series of conjectures and it has subsequently influenced recent research in. number theory in much the same way the conjectures of A Weil shaped the. course of algebraic geometry since 1948, At the heart of Langlands program is the general notion of an automor.
phic representation m and its L function L s IT These notions both defined. via group theory and the theory of harmonic analysis on so called adele. groups will of course be explained in this paper The conjectures of Langlands. just alluded to amount roughly to the assertion that the other zeta functions. arising in number theory are but special realizations of these L s IT. Herein lies the agony as well as the ecstasy of Langlands program To. merely state the conjectures correctly requires much of the machinery of class. field theory the structure theory of algebraic groups the representation theory. of real and adic groups and at least the language of algebraic geometry In. other words though the promised rewards are great the initiation process is. forbidding, Two excellent recent introductions to Langlands theory are Bo and Art. However the first essentially assumes all the prerequisites just mentioned. while the second concentrates on links with Langlands earlier theory of. Eisenstein series, The idea of writing the present survey came to me from Professor Paul. Halmos and I am grateful to him for his encouragement Although the. finished product is not what he had in mind my hope is that it will still make. accessible to a wider audience the beauty and appeal of this subject in. particular I shall be pleased if this paper serves as a suitable introduction to. the surveys of Borel and Arthur, One final remark This paper is not addressed to the experts Readers who. wish to find additional information on such topics as the trace formula. 0 series L indistinguishability zeta functions of varieties etc are referred to. the annotated bibliography appearing after Part IV I am indebted to Martin. Karel and Paul Sally for their help in seeing this paper through to its. publication,THE LANGLANDS PROGRAM 179,I INTRODUCTION. In this article I shall describe Langlands theory in terms of the classical. works which anticipated as well as motivated it Examples are the local global. methods used in solving polynomial equations in integers especially Hasse s. principle for quadratic forms the use of classical automorphic forms and. zeta functions to study integers in algebraic number fields and the use of. groups and their representations to bridge the gap between analytic and. algebraic problems Thus more than one half of this survey will be devoted to. material which is quite well known though perhaps never before presented. purely as a vehicle for introducing Langlands program. To give some idea of the depth and breadth of Langlands program let me. leisurely describe one particular conjecture of Langlands the rest of this paper. will be devoted to adding flesh and pretty clothes to this skeletal sketch as. well as defining all the terms alluded to in this Introduction. In algebraic number theory a fundamental problem is to describe how an. ordinary prime factors into primes in the ring of integers of an arbitrary. finite extension E of Q Recall that the ring of integers OE consists of those x. in E which satisfy a monic polynomial with coefficients in Z Though 0E need. not have unique factorization in the classical sense every ideal of OE must. factor uniquely into prime ideals the primes of 0E Thus in particular. with each 9t a prime ideal of 0E and the collection PJ completely de. termined by, Now suppose in addition that E is Galois over Q with Galois group.
G Gal Q This means that E is the splitting field of some monic. polynomial in Q x and G is the group of field automorphisms of E fixing Q. pointwise According to a well known theorem each element of G moves. around the primes 9t dividing and G acts transitively on this set Thus the. splitting type of p in 0E is completely determined by the size of the. subgroup of G which fixes any i e by the size of the isotropy groups Gt. which are conjugate in G, For simplicity we shall now assume that the primes in are distinct i e. the prime p is unramified in E In this case the afore mentioned isotropy. groups are cyclic To obtain information about the factorization of such. attention is focused on the so called Frobenius element Fr of G the canonical. generator of the subgroup of G which maps any 9 into itself We shall discuss. all these matters in more detail in II C 2 To be sure F is an automorphism. of E over Q determined only up to conjugacy in G Nevertheless the resulting. conjugacy class Frp completely determines the factorization type of For. example when Fr is the class of the identity alone then and only then p. splits completely in E i e p factors into the maximum number of primes in 0E. namely r E Q G, In general one seeks to describe Fvp and hence the factorization of p in. E intrinsically in terms of and the arithmetic of Q To see what this means. consider the example,E Q i a i a j 8 e Q,180 STEPHEN GELBART. with OE Z n mi n m G Z In this case G Gal Q, complex conjugation and some elementary algebra shows that. i f 1 is a quadratic residue mod p,conjugation otherwise.
For convenience let us identify Ga E Q with the subgroup 1 of C via. the obvious isomorphism a G 1 Then we have, with l p the Legendre symbol equal to 1 or 1 according to whether 1 is. or is not a quadratic residue mod p To express this condition in terms of a. congruence condition on p instead of on 1 we appeal to a part of the. quadratic reciprocity law for Q which states that for odd precisely those p. unramified in Q,l l 2 i e o Frp l 2, This is the type of intrinsic description of Fr we sought from it and the. l I 2 l sl 4, we conclude that the factorization of p in Z i depends only on its residue. modulo 4 In particular all primes in a given arithmetic progression mod 4. have the same factorization type in Z i Moreover since all the prime ideals. of Z are principal and of the form n or n im we obtain the following. THEOREM FERM T 1640 EULER 1754 Suppose p is an odd prime Then p. can be written as the sum of two squares n2 m2 if and only if p 1 4. PROOF p n2 m2 n im n im if and only if splits completely. A major goal of class field theory is to give a similar description of Fr for. arbitrary Galois extensions E However this goal is far from achieved and in. general is probably impossible, In general we cannot expect there to be a modulus N such that Fr. if and only if p lies in some arithmetic progression mod N However if E is. abelian i e G Gal Q is abelian then a great deal can be said Indeed. suppose E is such an extension and a G Cx is a homomorphism Then it is. known that there exists an integer Na 0 and a Dirichlet character. Xo Z NZ X C such that o Frp Xo p, for all primes p unramified in E This is E Artin s famous and fundamental.
reciprocity law of abelian class field theory 2 It implies just as in the special. case E Q z that the splitting properties of p in E depend only on its. The more familiar form of this law directly identifies Gal Q with the idele class group of Q. modulo the norms from we stress the dual form of this assertion only because its. formulation seems more amenable to generalization i e nonabelian E. THE LANGLANDS PROGRAM 181, residue modulo some fixed modulus N depending on E To see how this result. directly generalizes the classical result of Ferm t and Euler we note that when. E Q 0 and a G 1 is as before,with X Z 4Z C defined as follows. X0 n lYn l 2, For more general abelian extensions Artin s theorem not only implies the. general quadratic reciprocity law in place of the supplementary rule l p. 1 2 but also the so called higher reciprocity laws of abelian class field. theory For a discussion of such matters see for example Goldstein Tate or. The question remains for non abelian Galois extensions how can the family. Fr be described in terms of the ground field Q, Recognizing the utility of studying groups in terms of their matrix represen. tations Artin focused attention on homomorphisms of the form a Gal Q. GLW C i e on n dimensional representations of the Galois group G In this. way he was able to transfer the problem of analyzing certain conjugacy. classes in G to an analogous problem inside GLW Q where such classes as. o Frp are completely determined by their characteristic polynomials. det o Frp p s By also introducing the Artin L functions. L s o R det ln o Ftp p y, whose exact definition will be given in II C 2 Artin was further able to.
reduce this problem to one involving the analytic objects L s a. Problem Can the L functions L s a be defined in terms of the arithmetic. of Q alone, It was in the context of this problem that Artin proved his fundamental. reciprocity law Indeed for abelian E over Q and one dimensional a Artin. proved that his L s a is identical to a Dirichlet L series. L s x u x p p rl,for an appropriate choice of character x Z NZ X C. For arbitrary E and a Artin was able to derive important analytic properties. of L s9 a However what he was unable to do was discover the appropriate. dimensional analogues of Dirichlet s characters and L functions Although. some such 2 dimensional automorphic L functions were being studied nearby. and concurrently by Hecke it remained for Langlands 40 years later to see. the connection and map out some general conjectures. Roughly speaking here is what Langlands did He isolated the notion of an. automorphic representation of the group GL over the adeles of Q as the. appropriate generalization of a Dirichlet character Furthermore he associated. L functions with these automorphic representations generalizing Dirichlet s. 182 STEPHEN GELBART, L functions in the case n 1 Finally he conjectured that each dimensional. Artin L function L s9 a is exactly the L function L s ira for an appropriate. automorphic representation ira of GLn This is discussed with an arbitrary. number field F in place of Q in Part IV of the present paper cf Conjecture. The conjectured correspondence a ira is to be regarded as a far reaching. generalization of Artin s reciprocity map o x0 I 1 c a s e n 2 when ira. corresponds to a classical automorphic form f z in the sense of Hecke see. LB the map o ira affords an interpretation of the classes Fr in terms of. certain conjugacy classes in GL2 C determined by the Fourier coefficients of. the form f z In general the proper formulation of this conjecture and other. conjectures of Langlands requires a synthesis and further development of all. the themes alluded to heretofore local global principles automorphic forms. group representations etc, In Part II of this paper I motivate the use of adic numbers and adeles and. survey Hecke s theory of automorphic forms the functions of Artin and. Hecke and the use of group representations in number theory Perforce this. brings us to the theory of infinite dimensional representations of real and adic. In Part III these classical themes and ingredients are mixed together to. produce the all important notion of an automorphic representation of GLn. over Q Finally in Part IV I survey the high points of Langlands general. program with an emphasis on its historical perspective and a brief description. of techniques and known results,II CLASSICAL THEMES.
A The local global principle One of the major preoccupations of number. theory in general has been finding integer solutions of polynomial equations of. 1 P xl9x2 xH 0, For convenience let us assume that P is actually a homogeneous polynomial. and let us agree that only nonzero solutions are of interest The difficulty in. solving 1 is illustrated by Fermat s famous unproved assertion that the. particular equation, has no nontrivial solutions in integers for n 2 Indeed much of the develop. ment of the theory of algebraic numbers is linked to attempts by people. contemporary with Kummer to solve this problem, On the other hand a question which is more easily decided is the existence. of integral solutions modulo m Clearly a necessary condition that integer. solutions of 1 exist is that the congruence,2 P xl9 9xn 0 modm. be solvable for every value of the modulus m This observation leads naturally. to the local methods we shall now explain,THE LANGLANDS PROGRAM 183.
Suppose m NM with N and M relatively prime By the Chinese Re. mainder Theorem 2 has a solution if and only if the similar congruences for. N and M do In other words to solve 2 it is sufficient to solve congruences. modulo pk for any prime p and all positive integers k. Whenever we focus on a fixed prime we say we are working locally So. suppose we fix a prime and ask whether the congruence. 3 x j SE O mod, has a solution for all natural numbers k It was Hensel who reformulated this. question in a formal yet significant way in 1897 For each prime p he. introduced a new field of numbers the adic numbers and he showed. that the solvability of 3 for all k is equivalent to the solvability of 1 in the. 7 adic numbers Thus the solvability of the congruence 2 for all n is. equivalent to the solvability of 1 in the 7 adic numbers for all. Let us return now to the original problem of solving 1 in ordinary integers. In addition to being able to solve 2 modulo all integers m it is also clearly. necessary to be able to find real solutions for 1 The question of when these. obviously necessary conditions are also sufficient is much more difficult since. the assertion that an equation is solvable if and only if it is solvable modulo. any integer and has real solutions is in general false or at least not known. For example the Ferm t equation has been known to be solvable 7 adically. for all since around 1909, On the other hand there are important instances where this local global. principle is known to work,THEOREM HASSE MINKOWSKI Suppose. Q xl9 xH atjXtXj, is a quadratic form with atj in Z and det a y 0 Then Q xx xn 0 has a. nontrivial integer solution if and only if it has a real solution and a p adic solution. for each p, In order to give a more symmetric form to this example of the local global.
principle let me recall how the 7 adic numbers can be constructed analogously. to the real numbers Fixing a prime we can express any fraction x in the. form pan m with n and m relatively prime to each other and to Then an. absolute value is defined on Q by, and the field of 7 adic numbers is just the completion of Q with respect to this. metric jp Note that the integer a called the 7 adic order of JC can be. negative and the integers that are close to zero 7 adically are precisely the. ones that are highly divisible by p Though perhaps jarring at first this 7 adic. notion of size is entirely natural given our earlier motivations the congruence. n 0 with k large translates into the statement that n is close to zero. 7 adically,184 STEPHEN GELBART, Because R is the completion of Q with respect to the usual absolute value. it is customary to write 1 1 for x Q for R and then call R the. completion of Q at the infinite prime oo The result is a family of locally. compact complete topological fields Q which contain Q one for each oo. Each Qp is called a local field and Q itself is called a global field With. this terminology the Hasse Minkowski theorem takes the following symmetric. form a quadratic form over Q has a global solution if and only if it has a local. solution for each prime p, For the purposes of this article the significance of the local global principle is. this global problems should be analyzed purely locally and with equal attention. paid to each of the local places Q, Note For a leisurely discussion of adic numbers and instances of the. local global principle the reader is urged to browse through the Introduction. to BoShaf and Cassels Also highly recommended is the expository article. B Hecke theory and the centraliry of automorphic forms In the 19th century. the arithmetic significance of automorphic forms was clearly recognized and. examples of such forms were used to great effect in number theory. Around 1830 Jacobi worked with the classical theta function 0 z in order. to obtain exact formulas for the representation numbers of n as a sum of r. squares Then 30 years later Riemann exploited this same function in order to. derive the analytic continuation and functional equation of his famous zeta. function f s, Before explaining these matters in more detail let us briefly recall the.
classical notion of an automorphic form, 1 Basic notions Let H denote the upper half plane in C and regard the. SL II j a b c d real ad be 1, as the group of fractional linear transformations of H An automorphic form. of weight A is a function f z which is holomorphic in H and almost. invariant for the transformations y ac in some discrete subgroup T of. for all y acbd in V, The most famous example of an automorphic form is the classical theta. 0 z 2 e 2 1 2 2e7ri 2,THE LANGLANDS PROGRAM 185, This is an automorphic form of weight for the group. c j S L 2 Z 6 c 0 2 a d l 2,0 z iz l 26 z, More generally let Qr xl9 9xr denote the quadratic form 2 1 xf9 and set.
the sum extending over all integral vectors nl9 9nr Then dr z is again. an automorphic form this time of weight r 2 This example has special. number theoretic significance because the coefficients in the Fourier expansion. of this periodic function are the representation numbers of the quadratic form. Qr Indeed if r n9 Qr denotes the number of distinct ways of expressing n as. the sum of r squares then,er z 0 z r ir e,Here are some more examples of automorphic forms. i Let A z denote the function defined in H by,A z e2wiz U U e2 inzf4 2 r n e2winz. It is an automorphic form of weight 12 for the full modular group T SL2 Z. and its Fourier coefficients r n carefully investigated by Ramanujan in 1916. are closely related to the classical partition functionp n. ii For k 1 the function,c d 0 0 cz 4 d, is called the normalized Eisenstein series of weight 2k It is again an. automorphic form with respect to the full modular group SL2 Z this time. with Fourier expansion,0 l i y 2 2 fc e 2, with Bk the so called nth Bernoulli number and or n 2 dr. From these few examples it is already clearly indicated that automorphic. forms comprise an integral part of number theory Indeed invariance of the. form with respect to translations of the type z z h implies the existence of. a Fourier expansion 2ane2 rrikz h9 with the an of number theoretic significance. In general the automorphy property 1 implies f z is determined by its. values on a fundamental domain D for the action of T in H More precisely. 186 STEPHEN GELBART, D is a subset of H such that every orbit of T with respect to the action.
z az cz d has exactly one representative in D For example for. T SL2 Z the fundamental domain D looks like this, Note that any other fundamental domain must be obtained by applying to this. D some J in T In particular the domain D l pictured above is precisely the. image of D by the inversion element J t n e point at infinity for D. being mapped to the cusp at 0 in the boundary of the fundamental domain. To be able to apply convenient methods of analysis to the study of. automorphic forms it is customary to impose additional technical restrictions. on the regularity of at cusps along the boundary of a fundamental domain. especially at infinity This implies in particular that f z always has a. Fourier expansion of the form,2 f z 2 aS, For example for A z or E2k z we can take h 1 but for 0 z which is an. automorphic form only on T 2 which does not contain the translation. z z 1 the period is no longer 1 and we must take h 2. Let us denote by Mk T the vector space of automorphic forms of weight k. for T which are regular at the cusps of T and by 5 1 the subspace off z. in Mk T which actually vanish at the cusps Functions in this latter space are. called cusp forms for such functions like the modular discriminant A z. the constant term a0 in the expansion 2 is zero, We have already remarked that automorphic forms in general have number. theoretic interest because their Fourier coefficients involve solution numbers of. THE LANGLANDS PROGRAM 187, number theoretic problems For example by relating 04 z to certain Eisen. stein series on T 2 we obtain Jacobi s remarkable formula. Thus the need for analyzing this space Mk T is clearly indicated. As we shall soon see the subsequent theory developed by Hecke was so. successful that it suggested new ways to look at automorphic forms in number. theory as well as immediately providing the tools to solve existing classical. 2 Hecke s theory Hecke s key idea was to characterize the properties of an. automorphic form in terms of a corresponding Dirichlet series The most. famous Dirichlet series around is of course Riemann s zeta function. 1 i n i p rl p cc, So let us first sketch Riemann s original analysis of s which Hecke so.
brilliantly generalized,Recall the gamma function identity. valid for Re s 0 In modern parlance we say that T s is the Mellin. transform of e at s With this identity we derive the relation. w T s S 2s f Hit 1, with 6 the classical theta function already encountered In other words 2s is. essentially the Mellin transform of 6 it From this fact it is a simple matter to. derive the desired analytic properties of f j in terms of the automorphic. properties of 0 z and conversely Here are the key steps. 2 s o 2 J0,using the change of variable t t,jusing the automorphy property 01 tl 2 it. 188 STEPHEN GELBART, Note that invariance with respect to the substitution s s is already. obvious To reverse the process and derive the functional equation i e. automorphy condition of 0 z from that of f s we require Mellin inversion. By generahzing this proof Hecke was able to explain the symmetry of a. large number of Dirichlet series and also pave the way towards finding. automorphic forms seemingly everywhere in number theory. Given a sequence of complex numbers a0 ax an with an 0 nc for. some c 0 and given h 0 k 0 C 1 consider the series. and the function defined in H by, THEOREM 1 HECKE The following two conditions are equivalent.
A O a0 s C k s is entire bounded in every vertical strip and. satisfies the functional equation k s C s,B f z C z i kf z. In other words the holomorphic function z is automorphic of weight k. for the group of transformations generated by z z h and z l z if. and only if its associated Dirichlet series Zan ns is nice We shall often use. the term nice to describe a Dirichlet series satisfying certain analytic. properties similar to f s, The second part of Hecke s theory answers the question when does j s. 2an ns have an Euler product expansion of the form j s Up O0Lp s with. Lp s a power series in 5 A formal computation shows that s factors as. pprime m X P, whenever the coefficients an are multiplicative i e anm anam if n and m are. relatively prime, Characterizing such multiplicativity is crucial Indeed since the coefficients. an always have number theoretic significance it is of great interest to know. when knowledge of these an9s can be reduced to knowing ap for prime. Note that when the an9s are completely multiplicative i e anm anam for all. n and m the Euler product expansion above reduces to the familiar expression. THE LANGLANDS PROGRAM 189, Such is of course the case for the Riemann zeta function.
2 7 n i J p s r 1, where the Euler product expansion discovered appropriately by Euler is. tantamount to the fundamental theorem of arithmetic Another example is. provided by the coefficients an x n w t n X a character of the integers. modulo some N i e x s completely multiplicative of period N and. x 0 0 such a character is called a Dirichlet character mod N and the. corresponding series,2 2 2 i n i x 1,is a Dirichlet L series. In both these examples the Euler factors Lp s are of degree 1 in p s In. general such an expansion as 2 is too much to ask for usually we ask for. only ordinary multiplicativity and then the factors Lp s turn out to be of the. second not first degree in 7 5, Hecke s contribution was to characterize the multiplicativity of the an or. t s intrinsically and even locally in terms of f z by introducing a. certain ring of Hecke operators T p defined in a space of automorphic. forms of fixed weight,THEOREM 2 HECKE Assume for convenience that. belongs to 5A SL2 Z and ax 1 Then the an s are multiplicative and f s has. an Euler product expansion if and only if f is an eigenfunction f or all the Hecke. operators T p with T p f apf In this case,Lp s l app pk l y.
EXAMPLE Since 5 12 SL2 Z is one dimensional and T p preserves this. space the condition T p A pA is automatic Thus one obtains the multi. plicativity of the coefficients r n conjectured by Ramanujan and first proved. by L J Mordell, REMARKS 1 Hecke s Theorem 1 really says that an automorphic eigenform. of weight k on SL2 Z is indistinguishable from an Euler product of degree 2. with prescribed analytic behavior and functional equation involving the. substitution s k s This observation sheds a new light on the theory of. automorphic forms since there are many number theoretical situations where. data an leads to a nice L function hence by Hecke to an automorphic form. Following up on this idea A Weil in 1967 completed Hecke s theory by. 190 STEPHEN GELBART, similarly characterizing automorphic forms not just on SL2 Z but also on the. so called congruence subgroups such as,c j e S L 2 Z c o. These subgroups have in general many generators whereas Hecke s theorem. deals with automorphic forms only for the groups generated by z z X and. In this way Weil was led to an extremely interesting conjecture By carefully. analyzing the zeta function attached to an elliptic curve E over Q with. Lp s 1 app s p 2s and 1 p ap the number of points on the. reduced curve modulo Weil was able to conjecture that such a zeta func. tion is the Dirichlet series attached to an automorphic form in some S2 T0 N. cf Wel In other words the study of these curves might perversely be. regarded as a special chapter in the theory of automorphic forms. 2 Perhaps it is now clear to the reader that an automorphic form z like. an elliptic curve or a quadratic form should be regarded as a global object. over Q and that the ap or the Euler factors Lp s comprise local data for in. much the same way that adic solutions comprise local data for rational i e. global solutions of Diophantine equations This turns out to be the case. but must remain a fuzzy notion until the language of automorphic representa. tions is introduced in Part III, Note Two excellent sources on Hecke theory which we have followed. closely are Ogg and Rob 3, C Artin and other L functions Around 1840 Dirichlet succeeded in.
proving the existence of infinitely many primes in an arithmetic progression by. replacing Euler s analysis of the series 21 5 by his own analysis of the. Dirichlet L functions L s x 2x w 5 Soon afterwards Riemann. focused on such Dirichlet series as functions of a complex variable thereby. inspiring a spate of applications of Dirichlet series to number theory in. general and the theory of prime numbers in particular Finally in 1870. Dedekind introduced a new kind of zeta function to study the integers in an. arbitrary number field E i e any finite extension of Q This kind of zeta func. tion now called a Dedekind zeta function and denoted E s 9 made it possible. to relate the primes of Q to those of E and to analyze the distribution of. primes within E alone, Despite this widespread use of L series in the nineteenth century and the. concomitant need to generalize these functions further a full understanding of. the arithmetic significance of L functions awaited twentieth century develop. 1 Abelian L functions In 1916 Hecke was able to establish the analytic. continuation and functional equation of Dirichlet s L functions and to gener. alize them to the setting of an arbitrary number field To describe Hecke s. achievement properly we must first recall how to generalize the family of. adic fields Q considered in II A, Fix a finite extension E of Q and let OE denote the ring of integers of E By. a finite place or prime v of E we understand a prime ideal 9 in OE and we. THE LANGLANDS PROGRAM 191, often confuse the notations v and 9 by a fractional ideal of 0E we. understand an O submodule of E with the property that x C 0E for some. x G Ex It is a basic fact that the prime ideals are invertible in the sense that. 9 9 l 0 for some fractional ideal P 1 and that every fractional ideal. factors uniquely into powers of prime ideals, Now if x is in Ex we define ordg jc to be the positive or negative power. of 9 appearing in the factorization of the principal ideal JC and we set. x v x N9 OTd x, with N9 the cardinality of the field 0E 9 By analogy with the case of Q we.
also define a real place v of E to be a norm x a x with a R a. real embedding Complex infinite places are defined analogously The. result is a family of completions Ev of E9 one for each prime or place v of E. Following the lead of the local global principle for Q we treat all these. finite or infinite places equally, Recall that a classical Dirichlet character is just a homomorphism of. Z NZ X into Cx extended to Z by composition with the natural homomor. phism Z Z NZ and with the convention that x 0 if N 1 The. appropriate generalization of such a character to the number field E is called a. Hecke character or grossencharacter x This is a family of homomorphisms. X Ex C x one for every place v of is such that for any x in Ex regarded as. embedded in each Ex l S vxv x 1 I e Hecke characters are trivial on. Ex Implicit here is the fact that all but finitely many of the Xt are unramified. i e Xv xv f r xv i 1 E such t n a t 1 1 1 The fact that Dirichlet. characters give rise to such Hecke characters is spelled out in We 3 p 313 we. shall return to these matters in II D l, Now we can define Hecke s abelian L series attached to x xv by. LE X s 2 IIO X 9 N9 1, Here 51 is an ordinary ideal of 0E and x T is defined multiplicatively on. the ideals relatively prime to those such that xv s ramified the conductor of. X In particular x 0 whenever xv i s ramified otherwise x Xv v. with bv in Ev such that 1 1 N9 l a uniformizing variable for Ev. If x is the unit character i e xv i or a t n e n X reduces to. Dedekind s zeta function E s On the other hand when E Q and x is of. finite order L x s reduces to a familiar Dirichlet L series L s x In. general using ingenious and complicated arguments Hecke was able to derive. the analytic continuation and functional equation for all these L series in 1917. This settled a natural arithmetic question was how does a series like E s. factor into L series involving only the field Q Partly in an attempt to solve this. problem E Artin was led around 1925 to define yet another new L series. 2 Nonabelian L jfunctions Suppose AT is a number field and E is a finite. Galois extension of K with Galois group G G i E K By a representation. of G we understand a homomorphism a of G into GL F the group of. invertible linear transformations of a complex vector space of dimension n.

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