http://dx.doi.org/10.1090/surv/133 Traces of Hecke Operators Traces of Hecke Operators Andrew Knightly Charles Li American Mathematical Society EDITORIAL COMMITTEE Jerry L. Bona Peter S. Landweber Michael G. Eastwood Michael P. Loss J. T. Stafford, Chair 2000 Mathematics Subject Classification. Primary 11-XX. For additional information and updates on this book, visit www.ams.org/bookpages/surv-133 Library of Congress Cataloging-in-Publication Data Knightly, Andrew, 1972- Traces of Hecke operators / Andrew Knightly, Charles Li. p. cm. — (Mathematical surveys and monographs ; v. 133) Includes bibliographical references and index. ISBN-13: 978-0-8218-3739-9 (alk. paper) ISBN-10: 0-8218-3739-7 (alk. paper) 1. Hecke operators. 2. Trace formulas. 3. Number theory. I. Li, Charles, 1973- II. Title. QA243.K54 2006 515/.7246-dc22 2006047814 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. © 2006 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 11 10 09 08 07 06 Dedicated to: Dwight, Leah, Mom, Dad -A.K. Mom, Dad -C.L. Contents Traces of Hecke Operators 1. Introduction 1 2. The Arthur-Selberg trace formula for GL(2) 3 3. Cusp forms and Hecke operators 7 3.1. Congruence subgroups of SL2(Z) 7 3.2. Weak modular forms 10 3.3. Cusps and Fourier expansions of modular forms 12 3.4. Hecke rings 20 3.5. The level N Hecke ring 24 3.6. The elements T(n) 29 3.7. Hecke operators 34 3.8. The Petersson inner product 37 3.9. Adjoints of Hecke operators 42 3.10. Traces of the Hecke operators 45 Odds and Ends 4. Topological groups 49 5. Adeles and ideles 52 5.1. p-adic Numbers 52 5.2. Adeles and ideles 54 6. Structure theorems and strong approximation for GL2(A) 59 6.1. Topology of GL2(A) 59 6.2. The Iwasawa decomposition 61 6.3. Strong approximation for GL2(A) 63 6.4. The Cart an decomposition 66 6.5. The Bruhat decomposition 69 7. Haar measure 69 7.1. Basic properties of Haar measure 69 7.2. Invariant measure on a quotient space 74 7.3. Haar measure on a restricted direct product 82 7.4. Haar measure on the adeles and ideles 85 7.5. Haar measure on B 87 7.6. Haar measure on GL(2) 90 7.7. Haar measure on SL2(R) 92 7.8. Haar measure on GL(2) 94 7.9. Discrete subgroups and fundamental domains 95 7.10. Haar measure on Q\A and Q*\A* 102 7.11. Quotient measure on GL2(Q)\GL2(A) 103 7.12. Quotient measure on 5(Q)\G(A) 105 viii CONTENTS 8. The Poisson summation formula 108 9. Tate zeta functions 119 9.1. Definition and meromorphic continuation 119 9.2. Functional equation and behavior at s = 1 124 10. Intertwining operators and matrix coefficients 128 10.1. Linear algebra 129 10.2. Representation theory 134 10.3. Orthogonality of matrix coefficients 140 11. The discrete series of GL2(R) 151 11.1. if-type decompositions 151 11.2. Representations of 0(2) 155 11.3. K-type decomposition of an induced representation 156 11.4. The (s, If)-modules (VX)K 162 11.5. Classification of irreducible (g, K)-modules for GL2(R) 165 11.6. A detailed look at the Lie algebra action 181 11.7. Characterization of the weight k discrete series of GL2(R) 186 Groundwork 12. Cusp forms as elements of LQ(UJ) 195 12.1. From Dirichlet characters to Hecke characters 195 12.2. From cusp forms to functions on G(A) 197 12.3. Comparison of classical and adelic Fourier coefficients 198 12.4. Characterizing the image of Sk(iV,u/) in LQ(U) 201 13. Construction of the test function / 206 13.1. The non-archimedean component of / 206 13.2. Spectral properties of R(f) 213 14. Explicit computations for /k and /<*> 221 The Trace Formula 15. Introduction to the trace formula for R(f) 227 16. Terms that contribute to K(x, y) 230 17. Truncation of the kernel 231 18. Bounds for £7 l/fcrSfe)! 240 19. Integrability of kj(g) 248 20. The hyperbolic terms as weighted orbital integrals 259 21. Simplifying the unipotent term 270 22. The trace formula 276 Computation of the Trace 23. The identity term 279 24. The hyperbolic terms 280 24.1. A lemma about orbital integrals 280 24.2. The archimedean orbital integral and weighted orbital integral 281 24.3. Simplification of the hyperbolic term 283 24.4. Calculation of the local orbital integrals 283 24.5. The global hyperbolic result 286 25. The unipotent term 288 25.1. Explicit evaluation of the zeta integral at oo 288 25.2. Computation of the non-archimedean local zeta functions 291 CONTENTS ix 25.3. The global unipotent result 293 26. The elliptic terms 294 26.1. Properties of the elliptic orbital integrals 295 26.2. The archimedean elliptic orbital integral 300 26.3. Orders and lattices in an imaginary quadratic field 302 26.4. Local-global theory for lattices 307 26.5. From G(Afin) to lattices in E 310 26.6. The non-archimedean orbital integral for N — 1 314 26.7. The case of level N 318 Applications 27. Dimension formulas 333 27.1. The elliptic terms 333 27.2. The unipotent term 337 27.3. The dimension of 5k(7V, a/) and some examples 338 28. Computing Hecke eigenvalues 340 28.1. Obtaining eigenvalues from knowledge of the traces 340 28.2. Integrality of Hecke eigenvalues 343 28.3. The r-function 344 28.4. An example with nontrivial character 347 29. The distribution of Hecke eigenvalues 351 29.1. Bounds for the Eichler-Selberg trace formula 352 29.2. Chebyshev polynomials 355 29.3. Distribution of eigenvalues 356 29.4. Further applications and generalizations 359 30. A recursion relation for traces of Hecke operators 360 Bibliography 363 Tables of notation 368 Statement of the final result 370 Index 373 Bibliography [Arl] Arthur, J., A trace formula for reductive groups I. Duke Math J., 45, (1978), 911-952. [Ar2] , The invariant trace formula, I & II, J. Amer. Math. Soc. 1 (1988), no. 2,3. [Ar3] , The L2-Lefschetz numbers of Hecke operators, Invent. Math. 97 (1989), 257-290. [Ar4] , An introduction to the trace formula, in "Harmonic analysis, the trace formula, and Shimura varieties", Clay Math. Proa, 4, Amer. Math. Soc, Providence, RI, (2005), 1-263. [Ax] Axler, S., Linear algebra done right, Second edition, Springer-Verlag, New York, 1997. [Ba] Baldoni, M. W., General representation theory of real reductive Lie groups, in [Rep]. [Bar] Bargmann, V., Irreducible unitary representations of the Lorentz group, Annals of Math., 48, (1947), 568-640. [Bo] Bourbaki, N., Integration I, II, translation by S. Berberian, Springer- Verlag, Heidelberg, 2004. [Bu] Bump, D., Automorphic Forms and Representations, Cambridge Stud­ ies in Advanced Mathematics 55, Cambridge University Press 1998. [CD] Clozel, L., and Delorme, P., Pseudo-coefficients et cohomologie des groupes de Lie reductifs reels, C. R. 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