An Introduction to the Langlands Program PDF
Preview An Introduction to the Langlands Program
D. Bump J. W. Cogdell E. de Shalit D. Gaitsgory E. Kowalski S.S. Kudla An Introduction to the Langlands Program Joseph Bernstein Stephen Gelbart Editors Springer Science+ Business Media, LLC Joseph Bemstein Stephen Gelbart Tel Aviv University The Weizmann Institute of Science Department of Mathematics Department of Mathematics RamatAviv Rehovot76100 Tel Aviv 69978 Israel Israel Ubrary of Congress Cataloging·in·Publlcation Data An introduction to the Langlands program 1 Joseph Bernstein, Stephen Gelbart, editors ; [with contributions by] D. Bump ... [et al.). p. cm. Includes bibliographical references. ISBN 0-8176-3211-5 (alk. paper)-ISBN 3-7643-3211-5 (alk. paper) 1. Automorphic forms. 2. L-functions. 1. Bernstein, Joseph, 1945-II. Gelbart, Stephen s .. 1946- QA353.A9159 2003 515'-dc21 2003043653 CIP ISBN 978-0-8176-3211-3 ISBN 978-0-8176-8226-2 (eBook) DOI 10.1007/978-0-8176-8226-2 AMS Subject Classifications: IIMxx, 11Fxx, 14Hxx, 22Exx Printed on acid-frec paper. ~® Birkhăuser a()?} © 2004 Springer Science+Business Media New York Originally published by Birkhauser Boston in 2004 Ali rights reserved. This work may not be translated or copied in whole or in part without the written permission ofthe publisher (Springer Science+Business Media, LLC), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information stor· age and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not tobe taken as an expression of opinion as to whether or not they are subject to propeny rights. 98765432 SPIN 10995211 www.birhauser-science.com Contents Preface ............................................................ vii 1. Elementary Theory of L-Functions I E. Kowalski ......................................................... 1 2. Elementary Theory of L-Functions II E. Kowalski ....................................................... 21 3. Classical Automorphic Forms E. Kowalski ....................................................... 39 4. Artin L Functions Ehud de Shalit ..................................................... 73 5. L-Functions of Elliptic Curves and Modular Forms Ehud de Shalit ..................................................... 89 6. Tate's Thesis Stephen S. Kudla .................................................. 109 7. From Modular Forms to Automorphic Representations StephenS. Kudla .................................................. 133 8. Spectral Theory and the Trace Formula Daniel Bump ..................................................... 153 9. Analytic Theory of L-Functions for GLn J. W. Cogdell ...................................................... 197 10. Langlands Conjectures for GLn J. W. Cogdell ...................................................... 229 11. Dual Groups and Langlands Functoriality J. W. Cogdell ...................................................... 251 12. Informal Introduction to Geometric Langlands D. Gaitsgory ..................................................... 269 Preface During the last half-century the theory of automorphic forms has become a major focal point of development in number theory and algebraic geometry, with appli cations in many diverse areas, including combinatorics and mathematical physics. The 12 chapters presented in this book are based on lectures that were given in the School of Mathematics of the Institute for Advanced Studies at the Hebrew University of Jerusalem, March 12-16, 2001. The goal of these lectures was to introduce young researchers to the theory of automorphic forms, to explain its connection with the theory of £-functions, as well as to indicate connections to other fields of mathematics. The central premise of the School was to formulate the Langlands program, which gives a very broad picture connecting automorphic forms and £-functions. To describe it, our lecturers used different technical methods to establish special cases of the program. The Langlands program roughly states that, among other things, any L-function defined number-theoretically is the same as the one which can be defined as the automorphic L-function of some GL(n). In this loose way, every L-function is (conjecturally) viewed as one and the same object. To introduce the theory of automorphic forms, we have not tried to give a complete formal introduction to the subject; this would require much more time. Instead, our lecturers concentrated on a variety of topics, and gave an "informal" personal view of number theory from the classical zeta function up to the Langlands program. We hope that the present book will be able to serve the same informal role. The contribution of each of the lecturers and their articles may now be briefly described as follows. The first eight chapters are devoted to the case of GL( 1) and GL(2): E. Kowalski classically focuses on the basic zeta-function of Riemann and its generalizations to Dirichlet and Heeke L-functions, class field theory, and a selection of topics devoted to classical automorphic functions; E. de Shalit carefully surveys the conjectures of Artin and Shimura-Taniyama-Weil. After discussing Heeke's L-functions, S. Kudla examines classical modular (automorphic) L-functions as GL(2) ones, thereby bringing into play the theory of representations. One way to study those representations which are "automorphic" viii Preface is via Selberg's theory of the "trace formula"; this is introduced in D. Bump's chapter. The last four chapters, by J. Cogdell and D. Gaitsgory, are more abstract. After starting with discussion of cuspidal automorphic representations of GL(2, (A)), Cogdell quickly gets to Langlands' theory for GL(n, (A)); then he explains why one needs the Langlands' dual group in order to formulate the general conjectures for a reductive group G different from GL(n). Gaitsgory gives an informal introduction to the geometric Langlands program. This is a new and very active area of research which grew out of the theory of automorphic forms and is closely related to it. Roughly speaking, in this theory we everywhere replace functions-like automorphic forms-by sheaves on algebraic varieties; this allows us to use powerful methods of algebraic geometry in order to construct "automorphic sheaves." The Editors are grateful to all six authors for their considerable skill in pulling these diverse pieces together. We also wish to thank C. J. Mozzochi for the pho tograph of Langlands that appears on the cover and Shlomit Davidzon for other graphic elements. Finally, we thank the Institute for Advanced Studies of the He brew University of Jerusalem-in particular, Dahlia Aviely, Smedar Danziger, Pnina Feldman, Shani Freiman, and Alex Levitzki, for making this Workshop possible. Joseph Bernstein Tel Aviv University Ramat Aviv, Israel Stephen Gelbart Nicki and J. Ira Harris Professorial Chair The Weizmann Institute of Science Rehovot, Israel December 2002 An Introduction to the Langlands Program 1 Elementary Theory of L-Functions I E. Kowalski 1 Introduction In this first chapter we will define and describe, in a roughly chronological order from the time of Euler to that of Heeke, some interesting classes of holomorphic functions with strange links to many aspects of number theory. Later chapters will explain how at least some of the mysterious aspects are understood today. But it should be emphasized that there are still many points that are not fully explained, even in a very sketchy, philosophical way. I will particularly try to mention some of the more peculiar features of the theory of L-functions (and of automorphic forms) which arise from the point of view of analytic number theory. I will also give indications at the places where future chapters after mine will bring new perspectives. The next chapter will develop the points presented here, and in particular will sketch proofs of some of the most important ones, especially when such a proof yields new insights into the theory. The mathematicians whose name are most important for us now are Euler, Gauss, Dirichlet, Riemann, Dedekind, Kronecker, Heeke, Artin, and Hasse. 2 The Riemann zeta function The first L-function has been given Riemann's name. This fact is convenient for us: it seems to call for some explanation since no one denies that other mathematicians, most notably Euler, considered this function before, and these explanations are the best entrance to our subject. The function in question is defined by the series 1 s(s) = "-. ~ns n~I For integers s ~ 1, this was studied even before Euler, and even for s ~ 2, it is well known that Euler first found an exact formula (see (5.1)). However the starting point for the theory of L-functions is Euler's discovery that the existence and uniqueness of factorization of an integer as a product of prime powers im- 2 E. Kowalski plies that n--11 ' ~(s) = (2.1) p 1- pS a product over all prime numbers. From the divergence of the harmonic series, Euler deduced from this a new proof of Euclid's theorem that there are infinitely many primes, and with some care be obtained the more precise formulation that L-1 =log log X+ 0(1) p~X p as X --+- +oo. 1 Thus (2.1) clearly contained new information about the distribution of prime numbers. Riemann's insight [Rie] was to remark that the function ~(s) thus defined, if s + is taken to be a complex variable (s = a it in his notation), is holomorphic in its region of convergence. This justifies looking for its (maximal) analytic contin uation, but that too had been done before (see [We]). It is the combination of the Euler product expansion (2.1) and the analytic continuation given by the functional equation described below, which is the cause for all our rejoicing, as it reveals the strange "duality" between the complex zeros of ~(s) and prime numbers. To be more specific, Riemann stated that ~ (s) has a meromorphic continuation to the whole complex plane, the only singularity being a simple pole with residue 1 at s = 1, and that moreover this analytic continuation satisfied the following property, aptly named the functional equation: the function A(s) = rr-sf2r(s/2)~(s) (2.2) is meromorphic except for simples poles at s = 0 and s = 1 and satisfies for all s E C the relation = A(l - s) A(s). (2.3) From the simple poles of ~(s) at s = 1 and of f(s/2) at s = 0 one deduces in particular that ~(0) = -rr-112r(1/2)/2 = -1/2. Moreover, the other poles at s = -2n, n ~ 1 integer, of r(s/2) show that (2.3) implies that~( -2n) = 0, for n ~ 1: those zeros are called the trivial zeros of ~(s). This, and in fact all of (2.3) for integers s ~ 1, was already known to Euler in the language of divergent series! 1To avoid any controversy, here are the definitions of Landau's 0(· ··)and Vinogradov's « sym bols: f = O(g) as x ~ xo means that there exists some (unspecified) neighborhood U of xo. and a constant C ~ 0 such that lf(x)l ~ Cg(x) for x E U; this is equivalent to f « g for x E U where now U is specified beforehand, and in this latter case one can speak of the "implicit constant" C in«. However we sometimes also speak of estimates involving 0(· ··)being "uniform" in some parameters: this means that the U and C above can be chosen to be independent of those parameters.
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