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Automorphic Forms, Representation Theory and Arithmetic TATA INSTITUTE of FUNDAMENTAL RESEARCH STUDIES in MATHEMATICS

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Automorphic Forms, Representation Theory and Arithmetic TATA INSTITUTE of FUNDAMENTAL RESEARCH STUDIES in MATHEMATICS Automorphic Forms, Representation Theory and Arithmetic TATA INSTITUTE OF FUNDAMENTAL RESEARCH STUDIES IN MATHEMATICS General Editor : K. G. Ramanathan 1. M. Herve´ : Several Complex Variables 2. M. F. Atiyah and others : Differential Analysis 3. B. Malgrange : Ideals of Differentiable Functions 4. S. S. Abhyankar and others : Algebraic Geometry 5. D. Mumford : Abelian Varieties 6. L. Schwartz : Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures 7. W. L. Baily, Jr., and others : Discrete Subgroups of Lie Groups and Applications to Moduli 8. C. P. RAMANUJAM : A Tribute 9. C. L. Siegel : Advanced Analytic Number Theory 10. S. Gelbart and others : Automorphic Forms, Representation Theory and Arithmetic Automorphic Forms, Representation Theory and Arithmetic Papers presented at the Bombay Colloquium 1979, by GELBART HARDER IWASAWA JACQUET KATZ PIATETSKI–SHAPIRO RAGHAVAN SHINTANI STARK ZAGIER Published for the TATA INSTITUTE OF FUNDAMENTAL RESEARCH, BOMBAY SPRINGER–VERLAG Berlin Heidelberg New York (1981) © TATA INSTITUTE OF FUNDAMENTAL RESEARCH, 1981 ISBN 3 - 540 - 10697 - 9. Springer Verlag, Berlin - Heidelberg - New York ISBN 0 - 387 - 10697 - 9. Springer Verlag, New York - Heidelberg - Berlin No part of this book may be reproduced in any form by print, microfilm or any other means without written permission from the Tata Institute of Fundamental Research, Bombay 400 005 Printed by Spads Phototype Setting Ind., (P.) Ltd. 101 A, Poonam Chambers, Dr. Annie Besand Road, Worli, Bombay 400 018, and Published by H. Goetze Springer-Verlag, Heidelberg, West Germany © Tata Institute of Fundamental Research, 1969 PRINTER IN INDIA INTERNATIONAL COLLOQUIUM ON AUTOMORPHIC FORMS REPRESENTATION THEORY AND ARITHMETIC BOMBAY, 8–15 January 1979 REPORT An International Colloquium on Automorphic forms, Representation theory and Arithmetic was held at the Tata Institute of Fundamental Research, Bombay, from 8 to 15 January 1979. The purpose of the Colloquium was to discuss recent achievements in the theory of auto- morphic forms of one and several variables, representation theory with special reference to the interplay between these and number theory, e.g. arithmetic automorphic forms, Hecke theory, Representation of GL2 and GLn in general, class fields, L-functions, p-adic automorphic forms and p-adic L-functions. The Colloquium was jointly sponsored by the International Mathe- matical Union and the Tata Institute of Fundamental Research, and was financially supported by them and the Sir Dorabji Tata Trust. An Organizing Committee consisting of Professors P. Deligne, M. Kneser, M.S. Narasimhan, S. Raghavan, M.S. Raghunathan and C.S. Se- shadri was in charge of the scientific programme. Professors P. Deligne and M. Kneser acted as representatives of the International Mathemati- cal Union on the Organising Committee. The following mathematicians gave invited addresses at the Collo- quium: W. Casselman, P. Deligne, S. Gelbart, G. Harder, K. Iwasawa, H. Jacquet, N.M. Katz, I. Piatetski-Shapiro, S. Raghavan, T. Shintani, H.M. Stark and D. Zagier. Professor R. Howe was unable to attend the Colloquium but has sent a paper for publication in the Proceedings. 6 Report Professors A. Borel and M. Kneser who accepted our invitation, were unable to attend the Colloquium. The invited lectures were of fifty minutes’ duration. These were followed by discussions. In addition to the programme of invited ad- dresses, there were expository and survey lectures by some invited speak- ers giving more details of their work. Besides the mathematicians at the Tata Institute, there were also mathematicians from other universities in India who were invitees to the Colloquium. The social programme during the Colloquium included a Tea Party on 8 January; a programme of Western music on 9 January; a pro- gramme of Instrumental music on 10 January; a dinner at the Institute to meet the members of the School of Mathematics on 11 January; a per- formance of classical Indian Dances (Bharata Natyam) on 12 January; a visit to Elephanta on 13 January; a programme of Vocal music on 13 January and a dinner at the Institute on 14 January. Contents 1 ON SHIMURA’S CORRESPONDENCE FOR MODULAR FORMS OF HALF-INTEGRAL WEIGHT∗ 1 1 The Metaplectic Group . 4 2 Admissible Representations . 6 3 Whittaker Models . 8 4 The Theta-Representations rχ ............... 10 5 A Functional Equation of Shimura Type . 12 6 L and ǫ-Factors ...................... 14 7 A Local Shimura Correspondence . 17 8 The Metaplectic Group . 19 9 Automorphic Representations of Half-Integral Weight . 20 10 Fourier Expansions . 21 11 Theta-Representations . 25 12 A Shimura-Type Zeta Integral . 26 13 An Euler Product Expansion . 29 14 A Generalized Shimura Correspondence . 34 15 TheTheorem ....................... 34 16 Applications and Concluding Remarks . 39 2 PERIOD INTEGRALS OF COHOMOLOGY CLASSES WHICH ARE REPRESENTED BY EISENSTEIN SERIES 46 2 TheEisensteinSeries ................... 76 4 Arithmetic Applications . 118 7 8 CONTENTS 3 WAVE FRONT SETS OF REPRESENTATIONS OF LIE GROUPS 131 1 Generalities . 131 2 Examples .........................144 4 ON P-ADIC REPRESENTATIONS ASSOCIATED WITH Zp-EXTENSIONS 157 5 DIRICHLET SERIES FOR THE GROUP GL(N). 171 1 Introduction........................171 2 Maassforms........................172 3 Fourier expansions . 173 4 TheMellinTransform. .175 5 Theconvolution......................176 6 Functional Equations . 179 6 CRYSTALLINE COHOMOLOGY, DIEUDONNE´ MODULES, AND JACOBI SUMS 182 7 ESTIMATES OF COEFFICIENTS OF MODULAR FORMS AND GENERALIZED MODULAR RELATIONS 272 8 A REMARK ON ZETA FUNCTIONS OF ALGEBRAIC NUMBER FIELDS1 281 9 DERIVATIVES OF L-SERIES AT S = 0 288 1 Introduction........................288 2 Complex quadratic ground fields . 288 3 L-series considered over Q ................296 10 EISENSTEIN SERIES AND THE RIEMANN ZETA-FUNCTION 302 11 EISENSTEIN SERIES AND THE SELBERG TRACE FORMULA I 332 0 Introduction........................332 CONTENTS 9 1 Statement of the main theorem . 336 2 Eisenstein series and the spectral decomposition... 342 3 Computation of I(s) for (s) > 1. ............353 ℜ 4 Analytic continuation of I(s),...............364 ON SHIMURA’S CORRESPONDENCE FOR MODULAR FORMS OF HALF-INTEGRAL WEIGHT∗ By S. Gelbart and I. Piatetski-Shapiro Introduction 1 G. Shimura has shown how to attach to each holomorphic cusp form of half-integral weight a modular form of even integral weight. More pre- cisely, suppose f (z) is a cusp form of weight k/2, level N, and character χ. Suppose also that f is an eigenfunction of all the Hecke operators 2 N 2 2 T (p ), say T(p ) f = ωp f . If k 5, then the L-function k,χ ≥ ∞ s s 2k 2 2s 1 A(n)n− = 1 ωp p− + χ(p)p − − − − Xn=1 pY< ∞ is the Mellin transform of a modular cusp form of weight k 1, level − N/2, and character χ2. for further details, see [Shim] or [Niwa]. Our purpose in this paper is to establish a Shimura correspondence for any (not necessarily holomorphic) cusp form of half-integral weight defined over a global field F(not necessarily Q). Our approach is similar to Shimura’s in that we use L-functions. Out point of view is new in that we use the theory of group representations. 1Talk presented by S.G. 1 2 S. Gelbart and I. Piatetski-Shapiro Roughly speaking, suppose π = πv is an automorphic cuspidal v representation of the metaplectic groupN which doesn’t factor through GL2. Then we introduce an L-factor L(s, πv) for each v and we prove that the L-function L(s, π) = L(s, πv) Yv belongs to an automorphic representation of GL2(AF) in the sense of [Jacquet-Langlands]. Since we characterize those π which correspond to cuspidal (as opposed to just automorphic) representations of GL2(AF) we refine as well as generalize Shimura’s results. Let us now describe our correspondence in more detail. Suppose π is an automorphic cuspidal representation of the metaplectic group. Since π is determined by its local components πv, we want to describe its “Shimura image” S (π) in purely local terms. Thus we construct a local correspondence S : πv πv → by “squaring” the representation πv; if πv is an induced representation, x this means squaring the characters of Fv which parametrize πv. In gen- eral, this process of “squaring” tends to smooth out representations, as we shall now explain. 3 Suppose we consider the theta-representations of the metaplectic group. These representations generalize the classical modular forms of half-integral weight given by the theta-series ∞ 2πin2 θχ(z) = χ(n)e z nX= −∞ where χ is an (even) Dirichlet character of Z. Since these representa- tions arise by pasting together a grossencharacter χ of F with the “even or odd” part of the canonical metaplectic representation constructed in [Weil], we denote these representations by rχ and call them Weil repre- sentations. Locally, rχ is supercuspidal when χv( 1) = 1. Almost ev- v − − erywhere, however, χv( 1) = 1, rχ is the class 1 quotient of a reducible − v On Shimura’s Correspondence for Modular Forms... 3 principal series representation at s = 1/2, and the global representation rχ = rχv Ov is “distinguished” from several different points of view. Most signifi- cantly, these rχ exhaust the automorphic forms of half-integral weight which are determined by just one Fourier coefficient; this is the principal result of [Ge PS2]. Now if πv is an even Weil representation rχ (i.e. χv( 1) = 1), its v − Shimura image will be the one-dimensional representation χv of GL2(Fv), whereas if πv is an “odd” Weil representation, S (πv) will be the special representation Sp(χv); cf. §7. The Shimura correspondence thus takes cuspidal rχ to automorphic representations of GL(2) which almost ev- erywhere are one-dimensional and hence not cuspidal. The main result of this paper, however, guarantees that these representations are the only cuspidal π which map to non-cuspidal automorphic forms of GL(2). This explains the restriction k 5 in [Shim] and ultimately resolves ≥ “Open question (C)” of that paper; cf.
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