Graduate Texts in Mathematics S Axler 97 Editorial Board F.W Gehring K.A Ribet Springer-Science+Business Media, LLC Neal Koblitz Introduction to Elliptic Curves and Modular Forms Second Edition With 24 Illustrations , Springer Neal Koblitz Department of Mathematics University of Washington Seattle, WA 98195 USA Editorial Board S Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA F W Gehring Mathematics Department East Hali University of Michigan Ann Arbor, MI 48109 USA K.A Ribet Mathematics Department University of California at Berkeley Berkeley, CA 94720-3840 USA Mathematics Subject Classification (2000): 11-01, llDxx, IIGxx, 11Rxx, 14H45 Library of Congress Cataloging-in-Publieation Data Koblitz, Neal Introduetion to elliptie eurves and modular forms I Neal Koblitz - 2nd ed p em - (Graduate texts in mathematies; 97) ISBN 978-1-4612-6942-7 ISBN 978-1-4612-0909-6 (eBook) DOI 10.1007/978-1-4612-0909-6 Curves, Elliptie Forms, Modular Number Theory Title II Series QA567.2E44K63 1993 516.3 '52-de20 92-41778 Printed on aeid-free paper © 1984,1993 Springer Scienee+Business Media New York Originally published by Springer-Verlag New York in 1993 Softcover reprint ofthe hardcover 2nd edition 1993 AII rights reserved This work may not be translated or copied in whole or in part without the written pennissiori of the publisher (Springer-Science+Business 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 deve10ped is forbidden The use of general descriptive names, trade names, trademarks etc., in this publication, even if the former are noi especially identified, is noi to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone Typeset by Aseo Trade Typesetting Ltd., Hong Kong 987 654 Preface to the First Edition This textbook covers the basic properties of elliptic curves and modular forms, with emphasis on certain connections with number theory The ancient "congruent number problem" is the central motivating example for most of the book My purpose is to make the subject accessible to those who find it hard to read more advanced or more algebraically oriented treatments At the same time I want to introduce topics which are at the forefront of current research Down-to-earth examples are given in the text and exercises, with the aim of making the material readable and interesting to mathematicians in fields far removed from the subject of the book With numerous exercises (and answers) included, the textbook is also intended for graduate students who have completed the standard first-year courses in real and complex analysis and algebra Such students would learn applications of techniques from those courses thereby solidifying their understanding of some basic tools used throughout mathematics Graduate students wanting to work in number theory or algebraic geometry would get a motivational, example-oriented introduction In addition, advanced undergraduates could use the book for independent study projects, senior theses, and seminar work This book grew out of lecture notes for a course I gave at the University of Washington in 1981-1982, and from a series of lectures at the Hanoi Mathematical Institute in April, 1983 I would like to thank the auditors of both courses for their interest and suggestions My special gratitude is due to Gary Nelson for his thorough reading of the manuscript and his detailed comments and corrections I would also like to thank Professors Buhler, B Mazur, B H Gross, and Huynh Mui for their interest, advice and encouragement VI Preface to the First Edition The frontispiece was drawn by Professor A T Fomenko of Moscow State University to illustrate the theme of this book It depicts the family of elliptic curves (tori) that arises in the congruent number problem The elliptic curve corresponding to a natural number n has branch points at 0, 00, nand -no In the drawing we see how the elliptic curves interlock and deform as the branch points ± n go to infinity Note: References are given in the form [Author year]; in case of multiple works by the same author in the same year, we use a, b, after the date to indicate the order in which they are listed in the Bibliography Seatt/c Washington NEAL KOBLITZ Preface to the Second Edition The decade since the appearance of the first edition has seen some major progress in the resolution of outstanding theoretical questions concerning elliptic curves The most dramatic of these developments have been in the direction of proving the Birch and Swinnerton-Dyer conjecture Thus, one of the changes in the second edition is to update the bibliography and the discussions of the current state of knowledge of elliptic curves It was also during the 1980s that, for the first time, several important practical applications were found for elliptic curves In the first place, the algebraic geometry of elliptic curves (and other algebraic curves, especially the curves that parametrize modular forms) were found to provide a source of new error-correcting codes which sometimes are better in certain respects than all previously known ones (see [van Lint 1988]) In the second place, H.W Lenstra's unexpected discovery of an improved method of factoring integers based on elliptic curves over finite fields (see [Lenstra 1987]) led to a sudden interest in elliptic curves among researchers in cryptography Further cryptographic applications arose as the groups of elliptic curves were used as the "site" of so-called "public key" encryption and key exchange schemes (see [Koblitz 1987], [Miller 1986], [Menezes and Vanstone 1990]) Thus, to a much greater extent than I would have expected when I wrote this book, readers of the first edition came from applied areas of the mathematical sciences as well as the more traditional fields for the study of elliptic curves, such as algebraic geometry and algebraic number theory I would like to thank the many readers who suggested corrections and improvements that have been incorporated into the second edition Contents Preface to the First Edition Preface to the Second Edition v vii CHAPTER I From Congruent Numbers to Elliptic Curves I Congruent numbers A certain cubic equation Elliptic curves Doubly periodic functions The field of elliptic functions Elliptic curves in Weierstrass form The addition law Points of finite order Points over finite fields, and the congruent number problem 14 18 22 29 36 43 CHAPTER II The Hasse- Weil L-Function of an Elliptic Curve 51 I 51 56 The congruence zeta-function The zeta-function of En Varying the prime p The prototype: the Riemann zeta-function The Hasse-Weil L-function and its functional equation The critical value 64 70 79 90 x Contents CHAPTER III Modular forms I SL (Z) and its congruence subgroups Modular forms for SLiZ) Modular forms for congruence subgroups Transformation formula for the theta-function The modular interpretation and Hecke operators 98 98 108 124 147 153 CHAPTER IV Modular Forms of Half Integer Weight 176 I 177 185 202 Definitions and examples Eisenstein series of half integer weight for t o( 4) Hecke operators on forms of half integer weight The theorems of Shimura Waldspurger, Tunnell, and the congruent number problem 212 Answers, Hints, and References for Selected Exercises 223 Bibliography 240 Index 245 238 Answers, Hints, and References for Selected Exercises pyp-l = (C-'!Vb -~N), '11 N 1/4 (J)c;; \/a( -l/Nz) + b( -iN 1/412» = «-~b -~N), '11 (J)C;;\/ - Nbz + a) (b) We have YI = (C-~b -~N), (-~b)c;;\/ -Nbz + a) Since ad - be = and 41e, we have a == d mod 4, and so Ca = Cd' Thus, it remains to compare ( -~b) with '11 (J) We suppose e :2': (an analogous argument gives the same result if e < 0) Then (N"b)(IYf) = (l~t) = I, and (-.1) = (-;l ), so that (-~b) = (-~b) Now ( -~b )(J) = (~)e dad) = sgn a = (~)'11' Thus, Yl = PYP-l«6 ~), (~» Again }il and Pyp-l are equal if N is a perfect square; otherwise there exist y for which the two differ by (I, -I) At 00, P = I, h = I, t = I At 0, take p = «~ -6),12), so pool = ((_? 6), -i12), and we need f o(4)3p«6 ~), t)p-l = «~I/),tJZ+h)(U 6), -i12)=«.!h ~), -it-J-I/z+h·12)= «_\ ~), - it.jllZ=T) This is in ["0(4) if h = 4, t = I Finally, at the cusp -1 take C( = (-1 7), p = «-1 7),.J -2z + 1), so pool = W~), J2Z+i) Since C(b DC(-l Ero(4), we can take h = I To find t, compute P«6 D, t)p-l = «-1 -D, t.J -4z - I), which isj«_l -D, z) provided that t = i (a) sk/2(fo(4» = if k < For k :2': 9, it consists of elements of the form 0F(l6F - 4)P(0, F), where P is a polynomial of pure weight (k - 9)/2 Thus, for k:2': 5, dim Sk/2(["o(4» = [(k - 5)/4] (b) Since t = at the cusps 00 and 0, there are always those two regular cusps Since t = i at the cusp -1, that cusp is k-regular if and only if ik = I, i.e., 41k But I + [k/4J - [(k - 5)/4J = if k:2': 5, 4{k and = if k :2': 5, 41k, as can be verified by checking for k = 5, 6, 7, and then using induction to go from k to k + (c) 0F(0 - 16F)(04 - 2F) (a) If d' == d mod Nand N/4 is odd, then (~) = (lif(-) = (_I)(N/4-1)(d-l)/2(N14) = (_I)(N/4-1)(d'-I)/2U/~) = If N/4 is even, then d' == d mod 8, and the proof works the same way, with the additional observation that (i) = U) (b) The proof that the cusp condition holds for II [p Jk/2 is just like the analogous part of the proof of Proposition 17 in §III.3 Now let y = (~ ~)Ero(N), and let 1'1 = (-~N -~N) By Problem above, we have pyp-l = (I, XN(d»Yl' Thus, UI[p Jk/2) I[YJk/2 = UI[PYP- 1Jk/2)I[pJk/2 = x~(d)UI[YIJk/2)I[pJk/2 = x~(d)x(a)II[pJk/2' But since ad == mod N, we have x(a) = xed) Thus, UI[p Jk/2) I[YJk/2 = xx~(d)II [p Jk/2' as desired 0(00) = I, 0( -1/2) = 0, 0(0) = (1 - 0/2 m (n §IV.2 Ek/2(00) = I, Ek/2(0) = Ek/2( -1/2) = 0; ~/2(0) = (iJ2.)"k; Fk/2(00) = Fk/2( -1/2) = O If one uses (2.16) and (2.19) rather than (2.17) and (2.20), then the solutions C( and f3 to the resulting x equations involve X, which depends on I By Proposition 8, it suffices to show this for I squarefree In that case use (2.16) and (2.19) to evaluate the tooth coefficient of Ek/2 + (1 + ik) 2" k/2 Fk/2' Hk/2 = W - 2Je) + W - Je)q + ; W - Je) = if and only if A :2': is odd Use Problem above and Problem of §IV.I to find a and b so that k - aEk/2 - bFic/2 vanishes at the cusps a = I, b = (1 + il2- k/2 = E5/2 - (I + i)/.j2Fs/2, = E7/2 + (I - i)/.j2F7/2 §IV.3 The computation is almost identical to that in Problem I in §IV.1 (c) By the lemma in Proposition 43 in §III.5, right coset representatives for r l (N) modulo rl(N)n C(- l r 1(N)C(, where Ct = (b :v), are C(b = (~ :v), O:s b