Graduate Texts in Mathematics 228 Editorial Board S. Axler F.W. Gehring K.A. Ribet More information about this series at http://www.springer.com/series/136 Graduate Texts in Mathematics 1 TAKEUTI/ZARING. Introduction to 34 SPITZER. Principles of Random Walk. Axiomatic Set Theory. 2nd ed. 2nd ed. 2 OXTOBY. Measure and Category. 2nd ed. 35 ALEXANDER/WERMER. Several Complex 3 SCHAEFER. Topological Vector Spaces. Variables and Banach Algebras. 3rd ed. 2nd ed. 36 KELLEY/NAMIOKA et al. Linear 4 HILTON/STAMMBACH. A Course in Topological Spaces. Homological Algebra. 2nd ed. 37 MONK. Mathematical Logic. 5 MAC LANE. Categories for the Working 38 GRAUERT/FRITZSCHE. Several Complex Mathematician. 2nd ed. Variables. 6 HUGHES/PIPER. Projective Planes. 39 ARVESON. An Invitation to C*-Algebras. 7 J.-P. SERRE. A Course in Arithmetic. 40 KEMENY/SNELL/KNAPP. Denumerable 8 TAKEUTI/ZARING. Axiomatic Set Theory. Markov Chains. 2nd ed. 9 HUMPHREYS. Introduction to Lie Algebras 41 APOSTOL. 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Lectures in Abstract Algebra I. Classical Mechanics. 2nd ed. Basic Concepts. 61 WHITEHEAD. Elements of Homotopy 31 JACOBSON. Lectures in Abstract Algebra II. Theory. Linear Algebra. 62 KARGAPOLOV/MERLZJAKOV. Fundamentals 32 JACOBSON. Lectures in Abstract Algebra of the Theory of Groups. III. Theory of Fields and Galois Theory. 63 BOLLOBAS. Graph Theory. 33 HIRSCH. Differential Topology. (continued after references) Fred Diamond Jerry Shurman A First Course in Modular Forms Fred Diamond Jerry Shurman Department of Mathematics Department of Mathematics King’s College London Reed College Strand, London WC2R 2LS Portland, OR 97202 United Kingdom USA [email protected] [email protected] Editorial Board S. Axler F.W. Gehring K.A. Ribet Mathematics Department Mathematics Department Mathematics Department San Francisco State East Hall University of California, University University of Michigan Berkeley San Francisco, CA 94132 Ann Arbor, MI 48109 Berkeley, CA 94720-3840 USA USA USA [email protected] [email protected] [email protected] Mathematics Subject Classification (2000): 25001, 11019 Library of Congress Cataloging-in-Publication Data Diamond, Fred. A first course in modular forms / Fred Diamond and Jerry Shurman. p. cm. — (Graduate texts in mathematics ; 228) Includes bibliographical references and index. ISBN 0-387-23229-X 1. Forms, Modular. I. Shurman, Jerry Michael. II. Title. III. Series. QA243.D47 2005 512.7′3—dc22 2004058971 ISBN 0-387-23229-X Printed on acid-free paper. ISBN 978-0 - 387 - 23229 -4 (hardcover) ISBN 978-0 - 387 - 27226 - 9 (eBook) ISBN 978-1-4419-2005-8 (softcover) DOI 10.1007/978-0-387-27226-9 © Springer Science+Business Media New York 2005, Corrected at 4th printing 2016 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media New York 2005, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage 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 to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. 987654 Springer is part of Springer Science+Business Media (www.springer.com) For our parents Contents Preface ........................................................ xi 1 Modular Forms, Elliptic Curves, and Modular Curves ..... 1 1.1 Firstdefinitionsandexamples............................. 1 1.2 Congruencesubgroups ................................... 11 1.3 Complextori ........................................... 24 1.4 Complex tori as elliptic curves . 31 1.5 Modularcurvesandmodulispaces......................... 37 2 Modular Curves as Riemann Surfaces ..................... 45 2.1 Topology............................................... 45 2.2 Charts................................................. 47 2.3 Elliptic points . 52 2.4 Cusps.................................................. 57 2.5 ModularcurvesandModularity........................... 63 3DimensionFormulas....................................... 65 3.1 Thegenus.............................................. 65 3.2 Automorphicforms...................................... 71 3.3 Meromorphicdifferentials................................. 77 3.4 DivisorsandtheRiemann–RochTheorem.................. 83 3.5 Dimension formulas for even k ............................ 85 3.6 Dimension formulas for odd k ............................. 89 3.7 More on elliptic points . 92 3.8 Moreoncusps .......................................... 98 3.9 Moredimensionformulas.................................106 4 Eisenstein Series ...........................................109 4.1 Eisenstein series for SL2(Z)...............................109 4.2 Eisenstein series for Γ (N)whenk ≥ 3......................111 4.3 Dirichlet characters, Gauss sums, and eigenspaces . 116 vii viii Contents 4.4 Gamma, zeta, and L-functions............................120 4.5 Eisenstein series for the eigenspaces when k ≥ 3 .............126 4.6 Eisensteinseriesofweight2...............................130 4.7 Bernoulli numbers and the Hurwitz zeta function . 133 4.8 Eisensteinseriesofweight1...............................138 4.9 The Fourier transform and the Mellin transform . 143 4.10NonholomorphicEisensteinseries..........................148 4.11Modularformsviathetafunctions.........................155 5 Hecke Operators ...........................................163 5.1 Thedoublecosetoperator................................163 5.2 The d and Tp operators.................................168 5.3 The n and Tn operators ................................179 5.4 The Petersson inner product . 182 5.5 AdjointsoftheHeckeOperators...........................184 5.6 OldformsandNewforms..................................188 5.7 TheMainLemma.......................................190 5.8 Eigenforms.............................................196 5.9 The connection with L-functions ..........................201 5.10Functionalequations.....................................208 5.11 Eisenstein series again . 210 6 Jacobians and Abelian Varieties ...........................215 6.1 Integration, homology, the Jacobian, and Modularity . 216 6.2 MapsbetweenJacobians .................................221 6.3 ModularJacobiansandHeckeoperators....................230 6.4 Algebraicnumbersandalgebraicintegers...................234 6.5 Algebraiceigenvalues ....................................237 6.6 Eigenforms, Abelian varieties, and Modularity . 244 7 Modular Curves as Algebraic Curves ......................253 7.1 Elliptic curves as algebraic curves . 254 7.2 Algebraiccurvesandtheirfunctionfields...................261 7.3 Divisorsoncurves.......................................272 7.4 The Weil pairing algebraically . 279 7.5 Function fields over C ....................................283 7.6 Function fields over Q ....................................291 7.7 Modular curves as algebraic curves and Modularity . 294 7.8 Isogeniesalgebraically....................................299 7.9 Heckeoperatorsalgebraically .............................304 8 The Eichler–Shimura Relation and L-functions ............313 8.1 Elliptic curves in arbitrary characteristic . 314 8.2 Algebraic curves in arbitrary characteristic .