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Elliptic Curves, Modular Forms, and Their L-Functions STUDENT MATHEMATICAL LIBRARY IAS/PARK CITY MATHEMATICAL SUBSERIES Volume 58 Elliptic Curves, Modular Forms, and Their L-functions Álvaro Lozano-Robledo American Mathematical Society Institute for Advanced Study stml-58-lozano-cov.indd 1 1/5/11 1:17 PM http://dx.doi.org/10.1090/stml/058 Elliptic Curves, Modular Forms, and Their L-functions STUDENT MATHEMATICAL LIBRARY IAS/PARK CITY MATHEMATICAL SUBSERIES Volume 58 Elliptic Curves, Modular Forms, and Their L-functions Álvaro Lozano-Robledo American Mathematical Society, Providence, Rhode Island Institute for Advanced Study, Princeton, New Jersey Editorial Board of the Student Mathematical Library Gerald B. Folland Brad G. Osgood (Chair) Robin Forman John Stillwell Series Editor for the Park City Mathematics Institute John Polking Cover art courtesy of Karl Rubin, using MegaPOV, which is based on POV-Ray, both of which are open source, freely available software. 2000 Mathematics Subject Classification. Primary 14H52, 11G05; Secondary 11F03, 11G40. For additional information and updates on this book, visit www.ams.org/bookpages/stml-58 Library of Congress Cataloging-in-Publication Data Lozano-Robledo, Alvaro,´ 1978– Elliptic curves, modular forms, and their L-functions / Alvaro´ Lozano-Robledo. p. cm. — (Student mathematical library ; v. 58. IAS/Park City mathemat- ical subseries) Includes bibliographical references and index. ISBN 978-0-8218-5242-2 (alk. paper) 1. Curves, Elliptic. 2. Forms, Modular. 3. L-functions. 4. Number theory. I. Title. QA567.2.E44L69 2010 516.352—dc22 2010038952 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]. c 2011 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/ 10987654321 161514131211 A mis padres, que me enseñaron todo lo importante, a mi abuela, por su sonrisa que no aparece en fotografías, a Marisa, por lograr que siempre me supere, y a “Sally”, que nacerá pronto. Contents Preface xi Chapter 1. Introduction 1 §1.1. Elliptic curves 1 §1.2. Modular forms 7 §1.3. L-functions 11 §1.4. Exercises 15 Chapter 2. Elliptic curves 17 §2.1. Why elliptic curves? 17 §2.2. Definition 20 §2.3. Integral points 23 §2.4. The group structure on E(Q) 24 §2.5. The torsion subgroup 32 §2.6. Elliptic curves over finite fields 35 §2.7. The rank and the free part of E(Q) 43 §2.8. Linear independence of rational points 46 §2.9. Descent and the weak Mordell-Weil theorem 49 §2.10. Homogeneous spaces 59 §2.11. Selmer and Sha 66 vii viii Contents §2.12. Exercises 69 Chapter 3. Modular curves 77 §3.1. Elliptic curves over C 77 §3.2. Functions on lattices and elliptic functions 82 §3.3. Elliptic curves and the upper half-plane 84 §3.4. The modular curve X(1) 87 §3.5. Congruence subgroups 90 §3.6. Modular curves 91 §3.7. Exercises 94 Chapter 4. Modular forms 99 §4.1. Modular forms for the modular group 99 §4.2. Modular forms for congruence subgroups 105 §4.3. The Petersson inner product 110 §4.4. Hecke operators acting on cusp forms 111 §4.5. Exercises 118 Chapter 5. L-functions 123 §5.1. The L-function of an elliptic curve 123 §5.2. The Birch and Swinnerton-Dyer conjecture 127 §5.3. The L-function of a modular (cusp) form 135 §5.4. The Taniyama-Shimura-Weil conjecture 137 §5.5. Fermat’s last theorem 140 §5.6. Looking back and looking forward 142 §5.7. Exercises 143 Appendix A. PARI/GP and Sage 147 §A.1. Elliptic curves 147 §A.2. Modular forms 154 §A.3. L-functions 156 §A.4. Other Sage commands 158 Appendix B. Complex analysis 159 Contents ix §B.1. Complex numbers 159 §B.2. Analytic functions 160 §B.3. Meromorphic functions 163 §B.4. The complex exponential function 165 §B.5. Theorems in complex analysis 166 §B.6. Quotients of the complex plane 168 §B.7. Exercises 169 Appendix C. Projective space 171 §C.1. The projective line 171 §C.2. The projective plane 173 §C.3. Over an arbitrary field 174 §C.4. Curves in the projective plane 175 §C.5. Singular and smooth curves 176 Appendix D. The p-adic numbers 179 §D.1. Hensel’s lemma 181 §D.2. Exercises 182 Appendix E. Parametrization of torsion structures 185 Bibliography 189 Index 193 Preface This book grew out of the lecture notes for a course on “Elliptic Curves, Modular Forms and L-functions” that the author taught at an undergraduate summer school as part of the 2009 Park City Mathe- matics Institute. These notes are an introductory survey of the theory of elliptic curves, modular forms and their L-functions, with an em- phasis on examples rather than proofs. The main goal is to provide the reader with a big picture of the surprising connections among these three types of mathematical objects, which are seemingly so distinct. In that vein, one of the themes of the book is to explain the statement of the modularity theorem (Theorem 5.4.6), previously known as the Taniyama-Shimura-Weil conjecture (Conjecture 5.4.5). In order to underscore the importance of the modularity theorem, we also discuss in some detail one of its most renowned consequences: Fermat’s last theorem (Example 1.1.5 and Section 5.5). It would be impossible to give the proofs of the main theorems on elliptic curves and modular forms in one single course, and the proofs would be outside the scope of the undergraduate curriculum. However, the definitions, the statements of the main theorems and their corollaries can be easily understood by students with some stan- dard undergraduate background (calculus, linear algebra, elementary number theory and a first course in abstract algebra). Proofs that are accessible to a student are left to the reader and proposed as exercises xi xii Preface at the end of each chapter. The reader should be warned, though, that there are multiple references to mathematical objects and results that we will not have enough space to discuss in full, and the student will have to take these items on faith (we will provide references to other texts, however, for those students who wish to deepen their understanding). Some other objects and theorems are mentioned in previous chapters but only explained fully in later chapters. To avoid any confusion, we always try to clarify in the text which objects or results the student should take on faith, which ones we expect the stu- dent to be familiar with, and which will be explained in later chapters (by providing references to later sections of the book). The book begins with some motivating problems, such as the congruent number problem, Fermat’s last theorem, and the represen- tations of integers as sums of squares. Chapter 2 is a survey of the algebraic theory of elliptic curves. In Section 2.9, we give a proof of the weak Mordell-Weil theorem for elliptic curves with rational 2- torsion and explain the method of 2-descent. The goal of Chapter 3 is to motivate the connection between elliptic curves and modular forms. To that end, we discuss complex lattices, tori, modular curves and how these objects relate to elliptic curves over the complex num- bers. Chapter 4 introduces the spaces of modular forms for SL(2, Z) and other congruence subgroups (e.g., Γ0(N)). In Chapter 5 we define the L-functions attached to elliptic curves and modular forms. We briefly discuss the Birch and Swinnerton-Dyer conjecture and other related conjectures. Finally, in Section 5.4, we justify the statement of the original conjecture of Taniyama-Shimura-Weil (which we usu- ally refer to as the modularity theorem, since it was proved in 1999); i.e., we explain the surprising connection between elliptic curves and certain modular forms, and justify which modular forms correspond to elliptic curves. In order to make this book as self-contained as possible, I have also included five appendices with concise introductions to topics that some students may not have encountered in their classes yet. Appen- dix A is a quick reference guide to two popular software packages: PARI and Sage. Throughout the book, we strongly recommend that the reader tries to find examples and do calculations using one of these Preface xiii two packages. Appendix B is a brief summary of complex analysis. Due to space limitations we only include definitions, a few exam- ples, and a list of the main theorems in complex analysis; for a full treatment see [Ahl79], for instance. In Appendix C we introduce the projective line and the projective plane. The p-adic integers and the p-adic numbers are treated in Appendix D (for a complete refer- ence, see [Gou97]).
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