Advanced topics in the arithmetic of elliptic curves, joseph h silverman

Graduate Texts in Mathematics S Axler 151 Editorial Board F.W Gehring K.A Ribet Springer Science+Business Media, LLC Graduate Texts in Mathematics 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 T AKEUTIIZARING Introduction to Axiomatic Set Theory 2nd ed OXTOBY Measure and Category 2nd ed SCHAEFER Topological Vector Spaces 2nded HILTON/STAMM BACH A Course in Homological Algebra 2nd ed MAC LANE Categories for the Working Mathematician 2nd ed HUGHEs/PIPER Projective Planes SERRE A Course in Arithmetic T AKEUTIIZARING Axiomatic Set Theory HUMPHREYS Introduction to Lie Algebras and Representation Theory COHEN A Course in Simple Homotopy Theory CONWAY Functions of One Complex Variable I 2nd ed BEALS Advanced Mathematical Analysis ANDERSON/fuLLER Rings and Categories of Modules 2nd ed GoLUBITSKy/GUILLEMIN Stable Mappings and Their Singularities BERBERIAN Lectures in Functional Analysis and Operator Theory WINTER The Structure of Fields ROSENBLATT Random Processes 2nd ed HALMOS Measure Theory HALMOS A Hilbert Space Problem Book 2nded HUSEMOLLER Fibre Bundles 3rd ed HUMPHREYS linear Algebraic Groups BARNEs/MACK An Algebraic Introduction to Mathematical Logic GREUB Linear Algebra 4th ed HOLMES Geometric Functional Analysis and Its Applications HEWITT/STROMBERG Real and Abstract Analysis MANES Algebraic Theories KELLEY General Topology ZARISKI/SAMUEL Commutative Algebra Vol.I ZARISKIISAMUEL Commutative Algebra Vol.lI JACOBSON Lectures in Abstract Algebra I Basic Concepts JACOBSON Lectures in Abstract Algebra II Linear Algebra JACOBSON Lectures in Abstract Algebra III Theory of Fields and Galois Theory 33 HIRSCH Differential Topology 34 SPITZER Principles of Random Walk 2nded 35 ALEXANOERIWERMER Several Complex Variables and Banach Algebras 3rd ed 36 KELLEy/NAMIOKA et al Linear Topological Spaces 37 MONK Mathematical Logic 38 GRAUERTIFRITZSCHE Several Complex Variables 39 ARVESON An Invitation to C*-Algebras 40 KEMENy/SNELUKNAPP Denumerable Markov Chains 2nd ed 41 APOSTOL Modular Functions and Dirichlet Series in Number Theory 2nd ed 42 SERRE Linear Representations of Finite Groups 43 GILLMAN/JERISON Rings of Continuous Functions 44 KENDIG Elementary Algebraic Geometry 45 LoEVE Probability Theory I 4th ed 46 LoEVE Probability Theory II 4th ed 47 MOISE Geometric Topology in Dimensions and 48 SACHS/WU General Relativity for Mathematicians 49 GRUENBERG/WEIR Linear Geometry 2nd ed 50 Eow AROS Fermat's Last Theorem 51 KLINGENBERG A Course in Differential Geometry 52 HARTSHORNE Algebraic Geometry 53 MANIN A Course in Mathematical Logic 54 GRAVERIWATKINS Combinatorics with Emphasis on the Theory of Graphs 55 BROWNIPEARCY Introduction to Operator Theory I: Elements of Functional Analysis 56 MASSEY Algebraic Topology: An Introduction 57 CROWELLlFox Introduction to Knot Theory 58 KOBLITZ p-adic Numbers p-adic Analysis and Zeta-Functions 2nd ed 59 LANG Cyclotomic Fields 60 ARNOLD Mathematical Methods in Classical Mechanics 2nd ed 61 WHITEHEAD Elements of Homotopy Theory (continued after index) Joseph H Silverman Advanced Topics in the Arithmetic of Elliptic Curves With 17 Illustrations t Springer Joseph H Silvennan Mathematics Department Brown University Providence, RI 02912 USA jhs@math.brown.edu Editorial Board S Axler F.W Gehring K.A Ribet Department of Mathematics San Francisco State University San Francisco, CA 94132 USA Mathematics Department East HalI University of Michigan Ann Arbor, MI 48109 USA Department of Mathematics University ofCalifomia at Berkeley Berkeley, CA 94720-3840 USA Mathematics Subject Classifications (1991): 14-01, llGxx, 14Gxx, 14H52 Library of Congress Cataloging-in-Publication Data Silverman, Joseph H., 1955Advanced topics in the arithmetic of elliptic curves I Joseph H Silverman p cm - (Graduate texts in mathematics; v 151) Includes bibliographical references and index ISBN 978-0-387-94328-2 ISBN 978-1-4612-0851-8 (eBook) DOI 10 1007/978-1-4612-0851-8 Curves, Elliptic Curves, Algebraic Arithmetic Title II Series QA567.S442 1994 94-21787 516.3'52-dc20 Printed on acid-free paper © 1994 Springer Science+Business Media New York Originally published by Sprioger-Verlag New York, loc io 1994 All rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher Springer Scieoce+Busioess Media, LLC, 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 of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely byanyone Production managed by Hal Henglein; manufacturing supervised by Vincent Scelta Photocomposed copy prepared from the author's TeX file (Corrected second printing, 1999) ISBN 978-0-387-94328-2 SPIN 10727882 For Susan Preface In the introduction to the first volume of The Arithmetic of Elliptic Curves (Springer-Verlag, 1986), I observed that "the theory of elliptic curves is rich, varied, and amazingly vast," and as a consequence, "many important topics had to be omitted." I included a brief introduction to ten additional topics as an appendix to the first volume, with the tacit understanding that eventually there might be a second volume containing the details You are now holding that second volume Unfortunately, it turned out that even those ten topics would not fit into a single book, so I was forced to make some choices The following material is covered in this book: I Elliptic and modular functions for the full modular group II Elliptic curves with complex multiplication III Elliptic surfaces and specialization theorems IV Neron models, Kodaira-Neron classification of special fibers, Tate's algorithm, and Ogg's conductor-discriminant formula V Tate's theory of q-curves over p-adic fields VI Neron's theory of canonical local height functions So what's still missing? First and foremost is the theory of modular curves of higher level and the associated modular parametrizations of elliptic curves There is little question that this is currently the hottest topic in the theory of elliptic curves, but any adequate treatment would seem to require (at least) an entire book of its own (For a nice introduction, see Knapp [lJ.) Other topics that I have left out in order to keep this book at a manageable size include the description of the image of the £-adic representation attached to an elliptic curve and local and global duality theory Thus, at best, this book covers approximately half of the material described in the appendix to the first volume I apologize to those who may feel disappointed, either at the incompleteness or at the choice of particular topics In addition to the complete areas which have been omitted, there are several topics which might have been naturally included if space had been available These include a description of Iwasawa theory in Chapter II, viii Preface the analytic theory of p-adic functions (rigid analysis) in Chapter V, and Arakelov intersection theory in Chapter VI It has now been almost a decade since the first volume was written During that decade the already vast mathematical literature on elliptic curves has continued to explode, with exciting new results appearing with astonishing rapidity Despite the many omissions detailed above, I am hopeful that this book will prove useful, both for those who want to learn about elliptic curves and for those who hope to advance the frontiers of our knowledge I offer all of you the best of luck in your explorations! Computer Packages There are several computer packages now available for performing computations on elliptic curves PARI and SIMATH have many built-in elliptic curve functions, there are packages available for commercial programs such as Mathematica and Maple, and the author has written a small stand-alone program which runs on Macintosh computers Listed below are addresses, current as of March 1994, where these packages may be acquired via anonymous ftp PARI (includes many elliptic curve functions) math.ucla.edu 128.97.4.254 megrez.ceremab.u-bordeaux.fr 147.210.16.17 (directory pub/pari) (unix, mac, msdos, amiga versions available) SIMATH (includes many elliptic curve functions) ftp.math.orst.edu ftp.math.uni-sb.de apecs (arithmetic of plane elliptic curves, Maple package) math.mcgill.ca 132.206.1.20 (directory pub / apecs) Elliptic Curve Calculator (Mathematica package) Elliptic Curve Calculator (stand-alone Macintosh program) gauss.math.brown.edu 128.148.194.40 (directory dist/EllipticCurve) A description of many of the algorithms used for doing computations on elliptic curves can be found in H Cohen [1, Ch 7J and Cremona [1] Acknowledgments I would like to thank Peter Landweber and David Rohrlich for their careful reading of much of the original draft of this book My thanks also go to the many people who offered corrections, suggestions, and encouragement, including Michael Artin, Ian Connell, Rob Gross, Marc Hindry, Paul Lockhart, Jonathan Lubin, Masato Kuwata, Elisabetta Manduchi, Michael Rosen, Glenn Stevens, Felipe Voloch, and Siman Wong As in the first volume, I have consulted a great many sources while writing this book Citations have been included for major theorems, but Preface ix many results which are now considered "standard" have been presented as such In any case, I claim no originality for any of the unlabeled theorems in this book, and apologize in advance to anyone who may feel slighted Sources which I found especially useful included the following: Chapter I Apostol [1], Lang [1,2,3]' Serre [3], Shimura [1] Chapter II Lang [1], Serre [6], Shimura [1] Chapter IV Artin [1], Bosch-Liitkebohmert-Raynaud [1], Tate [2] Chapter V Robert [1], Tate [9J Chapter VI Lang [3,4]' Tate [3] I would like to thank John Tate for providing me with a copy of his unpublished manuscript (Tate [9]) containing the theory of q-curves over complete fields This material, some of which is taken verbatim from Professor Tate's manuscript, forms the bulk of Chapter V, Section In addition, the description of Tate's algorithm in Chapter IV, Section 9, follows very closely Tate's original exposition in [2], and I appreciate his allowing me to include this material Portions of this book were written while I was visiting the University of Paris VII (1992), IRES (1992), Boston University (1993), and Harvard (1994) I would like to thank everyone at these institutions for their hospitality during my stay Finally, and most importantly, I would like to thank my wife Susan for her constant love and understanding, and Debby, Danny, and Jonathan for providing all of those wonderful distractions so necessary for a truly happy life Joseph H Silverman March 27, 1994 Acknowledgments for the Second Printing I would like to thank the following people who kindly provided corrections which have been incorporated in this second revised printing: Andrew Baker, Brian Conrad, Guy Diaz, Darrin Doud, Lisa Fastenberg, Benji Fisher, Boris Iskra, Steve Harding, Sharon Kineke, Joan-C Lario, Yihsiang Liow, Ken Ono, Michael Reid, Ottavio Rizzo, David Rohrlich, Samir Siksek, Tonghai Yang, Horst Zimmer Providence, Rhode Island February, 1999 Contents Preface Computer Packages Acknowledgments Introduction CHAPTER I Elliptic and Modular Functions §l The Modular Group §2 The Modular Curve X(l) §3 Modular Functions §4 Uniformization and Fields of Moduli §5 Elliptic Functions Revisited §6 q-Expansions of Elliptic Functions §7 q-Expansions of Modular Functions §8 Jacobi's Product Formula for ~(T) §9 Hecke Operators §10 Hecke Operators Acting on Modular Forms §11 L-Series Attached to Modular Forms Exercises CHAPTER II Complex Multiplication §l Complex Multiplication over C §2 Rationality Questions §3 Class Field Theory - A Brief Review §4 The Hilbert Class Field §5 The Maximal Abelian Extension §6 Integrality of j §7 Cyclotomic Class Field Theory §8 The Main Theorem of Complex Multiplication §9 The Associated Gr6ssencharacter §10 The L-Series Attached to a CM Elliptic Curve Exercises vii viii viii 14 23 34 38 47 55 62 67 74 80 85 95 96 104 115 121 128 140 151 157 165 171 178 514 Hecke operator (continued) relations satisfied by, 68, 79, 90 self-adjoint, 92 simultaneous eigenfunction for all, 78, 79, 80, 92, 93 T(p), 74, 77 Heegner, 141 Height additivity, 256, 261 and linear equivalence, 256 associated to a divisor, 256, 265 associated to a morphism, 258 associated to a positive divisor, 285 associated to an ample divisor, 257 associated to diagonal of curves, 285 associated to exceptional curve, 285 canonical, 217, 247, 265, 266, 269, 281, 286 canonical See Canonical height duplication formula, 213, 257, 280 explicit 0(1) estimate, 280 finitely many points of bounded, 257, 265 for algebraically equivalent divisors, 285 functoriality, 256 geometric transformation properties, 213 infinitely many elements of bounded, 213, 220 Neron-Tate, 217, 247, 265, 266, 269, 281, 286 See also Canonical height normalization, 256 on a function field, 212 on a variety, 256 on an elliptic surface, 212, 213, 265 on curves, 257, 264, 265 on projective space, 255 parallelogram law, 213 quasi-parallelogram law, 280 regulator, 273 specialization of, 265, 266, 269, 281, 286 sum of local, 212 Height Machine, 256 verification of, 262 standard properties, 267, 268 Hensel's lemma, 330, 331, 401, 443 multi-variable version, 401 Henselian ring, 330, 401 complete ring is, 330 strictly, 330, 402 surjectivity of reduction, 330, 333, 337, 401 Henselization fraction field, 332 of a DVR, 331 strict, 337 universal mapping property, 332, 401 Index Hermitian inner product, 92 Higher ramification group, 379 index function for, 405 inertia group, 380 is normal, 380 over 2-adic field, 404 over 3-adic field, 404 Hilbert class field, 95, 118, 130, 132, 142, 181 abelian extension of, 134 Artin map for, 118 Galois group of, 118 generated by j(E), 121, 122, 166 of !QJ(v'-15), 180 of!QJ( v'=i3), 180 of quadratic imaginary field, 121, 122 Hilbert irreducibility theorem, 272 Hilbert theorem 90, 193, 421 Hindry, M., 277, 278, 486, 487 Holomorphic differential form, 27, 207 on a curve, 198 Homogeneous space of an elliptic curve, 199 Homology group, 37, 43 of a curve, 198 Homomorphism of groups, 427 of group varieties, 292 Homothetic lattices, 6, 9, 14, 37, 101, 103, 161 Homothety operator, 68, 90 relations satisfied by, 68 Horizontal curve, 237 intersection with, 283 Horizontal divisor, 237 on an arithmetic surface, 340 Hurwitz, A 101, 294, 485 Hyperelliptic curve integer points on, 277 Jacobian variety, 199, 287, 486 of genus two, 287 over function field, 277 Picard group, 287 Ideal of an idele, 119, 152, 159 principal, 132 Ideal class group, 99, 180 See also Hilbert class field acts on e,c£1 RK), 99 acts on e,c£1 RK) transitively, 100, 113 algorithm to compute, 85 class contains degree one primes, 118, 123 is finite, 86 is Picard group, 192 of a function field, 192 of quadratic imaginary field, 85 Idele Index ideal of, 119, 152, 159 multiplication by, 159, 170 Idele group, 119 characterization of ray class fields, 120, 162 class field theory using, 118-120 contains K*, 119 contains K~, 119 norm map, 119 topology on, 119 Identity component of a group variety, 115, 292 of Neron model is Weierstrass equation, 362, 378 Identity element Neron model, 326 Identity section of a group scheme, 306 Image of a rational map, 204, 279 Incidence matrix, 240, 241, 243, 283, 350, 402, 403, 486 Incomplete gamma function, 93 Index function for higher ramification groups, 405 Inert prime, 184 Inertia group, 120, 149, 169, 331, 380, 445 absolute, 380 Inner product Hermitian, 92 Petersson, 92 positive definite, 92 Inseparable isogeny, 127 Integer points effective methods, 277, 288 on an elliptic surface, 275 Integers of a function field, 275 Integral scheme, 311 Integral elliptic, 178 not path independent, 198 Intersection self, 234, 238, 243, 283, 342, 349, 351 transversal, 232, 282 Intersection index Arakelov, 344 computation of, 283 example, 349 linear equivalence, 341 local, 233, 283 on 340, 341 on an arithmetic surface, 339, 341 sum of local indices, 234 symmetric, 341 Intersection pairing, 233 Arakelov,344 incidence matrix, 240, 241, 243, 283 on an arithmetic surface, 341, 342, 353 on fibered surface, 238 symmetric, 341 Invariant differential, 43, 97, 134, 183 Inverse on a group scheme, 306 Irreducibility theorem of Hilbert, 272 IP'k, 515 Irreducible topological space, 280 Isogeny, 67, 182 between CM curves, 178, 180 between Tate curves, 453 comparison of discriminants, 453 conductors are equal, 404 degree preserved under reduction, 124 dual, 67, 125, 229, 400 equations for, 183 field of definition of, 105 is a homomorphism, 396 lift of Frobenius map, 130, 132, 162 of Neron model, 400 purely inseparable, 127 reduction mod