Graduate Texts in Mathematics Editorial Board F W Gehring P R Halmos 112 Graduate Texts in Mathematics I S JO 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 T AKEUTI/ZARING Introduction to Axiomatic Set Theory 2nd ed OXTOBY Measure and Category 2nd ed SCHAEFFER Topological Vector Spaces HILTON/STAMMBACH A Course in Homological Algebra MACLANE Categories for the Working Mathematician HUGHES/PIPER Projective Planes SERRE A Course in Arithmetic T AKEUTI/ZARING Axiomatic Set Theory HUMPHREYS Introduction to Lie Algebras and Representation Theory COHEN A Course in Simple Homotopy Theory CONWAY Functions of One Complex Variable 2nd ed BEALS Advanced Mathematical Analysis ANDERSON/FuLLER Rings and Categories of Modules GOLUBITSKy/GUILl.EMIN 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 2nd cd., revised HUSEMOLLER Fibre Bundles 2nd ed HUMPHREYS Linear Algebraic Groups BARNES/MACK An Algebraic Introduction to Mathematical Logic GRElJB 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 ZARISKI/SAMlJEL Commutative Algebra Vol II 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 HIRSCH Differential Topology SPITZER Principles of Random Walk 2nd ed WERMER Banach Algebras and Several Complex Variables 2nd cd KELl.Ey/NAMIOKA et aJ Linear Topological Spaces MONK Mathematical Logic GRAUERT/FRITLSCHE Several Complex Variables ARVESON An Invitation to C*-Algebras KEMENy/SNELL/KNAPP Denumerable Markov Chains 2nd ed APOSTOL Modular Functions and Dirichlet Series in Number Theory SERRE Linear Representations of Finite Groups GILLMAN/JERISON Rings of Continuous Functions KENDIG Elementary Algebraic Geometry LOEVE Probability Theory I 4th ed LOEVE Probability Theory II 4th ed MOISE Geometric Topology in Dimensions and continued after Index Serge Lang Elliptic Functions Second Edition Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Serge Lang Department of Mathematics Yale University New Haven, CT 06520 U.S.A Editorial Board F W Gehring Department of Mathematics University of Michigan Ann Arbor, MI 48109 U.S.A P R Halmos Department of Mathematics Santa Clara University Santa Clara, CA 95053 U.S.A AMS Classifications: lOD05, 12B25 Library of Congress Cataloging in Publication Data Lang, Serge Elliptic functions (Graduate texts in mathematics; 112) Bibliography: p Functions, Elliptic I Title QA343.L35 1987 515.9'83 87-4514 The first edition of this book was published by Addison-Wesley Publishing Company, Inc., Reading, MA, in 1973 © 1987 by Springer-Verlag New York Inc Softcover reprint of the hardcover 1st edition 1987 All rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag, 175 Fifth Avenue, New York, New York 10010, U.S.A.), 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 543 ISBN-13: 978-1-4612-9142-8 e-ISBN-13: 978-1-4612-4752-4 DOl: 10.1007/978-1-4612-4752-4 Preface Elliptic functions parametrize elliptic curves, and the intermingling of the analytic and algebraic-arithmetic theory has been at the center of mathematics since the early part of the nineteenth century Some new techniques and outlooks have recently appeared on these old subjects, continuing in the tradition of Kronecker, Weber, Fricke, Hasse, Deuring Shimura's book Introduction to the arithmetic theory of automorphic functions is a splendid modern reference, which I found very helpful myself to learn some aspects of elliptic curves It emphasizes the direction of the HasseWeil zeta function, Hecke operators, and the generalizations due to him to the higher dimensional case (abelian varieties, curves of higher genus coming from an arithmetic group operating on the upper half plane, bounded symmetric domains with a discrete arithmetic group whose quotient is algebraic) I refer the interested reader to his book and the bibliography therein I have placed a somewhat different emphasis in the present exposition First, I assume less of the reader, and start the theory of elliptic functions from scratch I not discuss Hecke operators, but include several topics not covered by Shimura, notably the Deuring theory of t -adic and p-adic representations; the application to Ihara's work; a discussion of elliptic curves with non-integral invariant, and the Tate parametrization, with the applications to Serre's work on the Galois group of the division points over number fields, and to the isogeny theorem; and finally the Kronecker limit formula and the discussion of values of special modular functions constructed as quotients of theta functions, which are better than values of the Weierstrass function because they are units when properly normalized, and behave in a specially good way with respect to the action of the Galois group Thus the present book has a very different flavor from Shimura's It was unavoidable that there should be some non-empty overlapping, and I have chosen to redo the complex multiplication theory, following Deuring's algebraic method, and reproducing some ofShimura's contributions in this line (with some v VI PREFACE simplifications, e.g to his reciprocity law at fixed points, and with another proof for the theorem concerning the automorphisms of the modular function field) I not emphasize elliptic curves in characteristic p, except as they arise by reduction from characteristic O Thus I have omitted most of the theory proper to characteristic p, especially the finer theory of supersingular invariants The reader should be warned, however, that this theory is important for the deeper analysis of the arithmetic theory of elliptic curves The two appendices should help the reader get into the literature I thank Shimura for his patience in explaining to me some facts about his research; Eli Donkar for his notes of a course which provided the basis for the present book; Swinnerton-Dyer and Walter Hill for their careful reading of the manuscript New Haven, Connecticut SERGE LANG Note for the Second Edition I thank Springer-Verlag for keeping the book in print It is unchanged except for the corrections of some misprints, and two items: John Coates pointed out to me a mistake in Chapter 21, dealing with the L-functions for an order Hence I have eliminated the reference to orders at that point, and deal only with the absolute class group I have renormalized the functions in Chapter 19, following Kubert-Lang Thus I use the Klein forms and Siegel functions as in that reference Actually, the final formulation of Kronecker's Second Limit Formula comes out neater under this renormalization S L November 1986 Contents PART ONE GENERAL THEORY Chapter Elliptic Functions The Liouville Theorems The Weierstrass Function The Addition Theorem Isomorphism Classes of Elliptic Curves Endomorphisms and Automorphisms Chapter 23 25 28 Points of Finite Order The Modular Function The Modular Group Automorphic Functions of Degree 2k The Modular Function j Chapter 12 14 19 Homomorphisms Isogenies The Involution Chapter 29 32 39 Fourier Expansions Expansion for Gk, g2, g3, A andj Expansion for the Weierstrass Function Bernoulli Numbers Vll 43 45 48 CONTENTS VI11 Chapter The Modular Equation Integral Matrices with Positive Determinant The Modular Equation Relations with Isogenies 51 54 58 Chapter Higher Levels Congruenc~ Subgroups The Field of Modular Functions Over C The Field of Modular Functions Over Q Subfields of the Modular Function Field 61 62 65 72 Chapter Automorphisms of the Modular Function Field Rational Adeles of GL • Operation of the Rational Adeles on the Modular Function Field The Shimura Exact Sequence PART TWO 75 77 83 COMPLEX MULTIPLICATION ELLIPTIC CURVES WITH SINGULAR INVARIANTS Chapter Results from Algebraic Number Theory Chapter Lattices in Quadratic Fields Completions The Decomposition Group and Frobenius Automorphism Summary of Class Field Theory 89 98 101 107 Reduction of Elliptic Curves Non-degenerate Reduction, General Case Reduction of Homomorphisms Coverings of Level N Reduction of Differential Forms 111 112 113 117 Chapter 10 Complex Multiplication Generation of Class Fields, Deuring's Approach Idelic Formulation for Arbitrary Lattices Generation of Class Fields by Singular Values of Modular Functions The Frobenius Endomorphism Appendix A Relation of Kronecker 123 129 132 136 143 CONTENTS Chapter 11 ix Shimura's Reciprocity Law Relation Between Generic and Special Extensions Application to Quotients of Modular Forms 149 153 Chapter 12 The Function A(cx't)/ A('t) Behavior Under the Artin Automorphism Prime Factorization of its Values Analytic Proof for the Congruence Relation of j 161 163 168 Chapter 13 The {-adic and p-adic Representations of Deuring The r-adic Spaces Representations in Characteristic p Representations and Isogenies Reduction of the Ring of Endomorphisms The Deilring Lifting Theorem 172 174 178 181 184 Chapter 14 Ihara's Theory Deuring Representatives The Generic Situation Special Situations 187 190 191 PART THREE ELLIPTIC CURVES WITH NON-INTEGRAL INVARIANT Chapter 15 The Tate Parametrization Elliptic Curves with Non-integral Invariants Elliptic Curves Over a Complete Local Ring 197 202 Chapter 16 The Isogeny Theorems Chapter 17 The Galois p-adic Representations Results of Kummer Theory The Local Isogeny Theorems Supersingular Reduction The Global Isogeny Theorems 205 208 211 213 216 Division Points Over Number Fields A Theorem of Shafarevic The Irreducibility Theorem 221 225 314 [APP 2, §4) THE TRACE OF FROBENIUS we conclude that = 0, whence w = df The second is somewhat harder to prove, and amounts to showing that there is some z E K such that for any derivation D of K over the constants, we have