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J.S. Milne: Elliptic Curves OTHER BOOKS BY THE AUTHOR Etale Cohomology Princeton Mathematical Series 33, Princeton University Press, 1980, 323+xiii pages, ISBN 0-691-08238-3 Hodge Cycles, Motives, and Shimura Varieties (with Pierre Deligne, Arthur Ogus, and Kuang-yen Shih) Lecture Notes in Math. 900, Springer-Verlag, 1982, 414 pages, ISBN 3-540- 11174-3 and 0-387-11174-3 Arithmetic Duality Theorems Academic Press, 1986, 421+x pages, ISBN 0-12-498040-6 Second corrected TeXed edition (paperback) BookSurge Publishing 2006, 339+viii pages, ISBN 1-4196-4274-X Elliptic Curves J.S. Milne Copyright c 2006 J.S. Milne. Single unbound paper copies for noncommercial personal use may be made without explicit permission from the copyright holder. All other rights reserved. A paperback version of this work is available from booksellers worldwide, in- cluding www.amazon.com and www.amazon.co.uk, and from the publisher: BookSurge Publishing, www.booksurge.com. Your purchase of this book will encourage the writing of more works like this. BibTeX information @book{milne2006, author={J.S. Milne}, title={Elliptic Curves}, year={2006}, publisher={BookSurge Publishers}, pages={238+viii}, isbn={1-4196-5257-5} } Library of Congress data available (www.jmilne.org/math/pBooks/) Mathematics Subject Classification (MSC2000): 11G, 11D, 14G. R The illustrations were written directly in PostScript code (by the author, ex- cept for the flying tori on the back cover, which use code written by W. Cassel- man). The Kea is a friendly intelligent parrot found only in the mountains of New Zealand. Preface In early 1996, I taught a course on elliptic curves. Since this was not long after Wiles had proved Fermat’s Last Theorem and I promised to explain some of the ideas underlying his proof, the course attracted an unusually large and diverse audience. As a result, I attempted to make the course accessible to all students with a knowledge only of the standard first-year graduate courses. When it was over, I collected the notes that I had handed out during the course into a single file, made a few corrections, and posted them on the Web, where they have since been downloaded tens of thousands of times. The appearance of publishers willing to turn pdf files into books quickly and cheaply and make them available worldwide while allowing the author to retain full control of the content and appearance of the work has prompted me to rewrite the notes and make them available as a paperback. J.S. Milne, October 30, 2006. Contents Contents vii IPlaneCurves 5 1 Basicdefinitions;Bezout’stheorem............... 5 2 Rationalpointsonplanecurves.................. 17 3 Thegrouplawonacubiccurve.................. 25 4 Regularfunctions;theRiemann-Rochtheorem......... 29 5 Definingalgebraiccurvesoversubfields............. 42 II Basic Theory of Elliptic Curves 45 1 Definition of an elliptic curve . .................. 45 2 The Weierstrass equation for an elliptic curve . ....... 50 3 Reduction of an elliptic curve modulo p ............. 54 4 Elliptic curves over Qp ...................... 61 5 Torsionpoints........................... 64 6N´eronmodels........................... 69 7 Algorithms for elliptic curves . .................. 76 III Elliptic Curves over the Complex Numbers 81 1 Latticesandbases......................... 81 2 Doubly periodic functions . .................. 82 3 Elliptic curves as Riemann surfaces . .............. 89 IV The Arithmetic of Elliptic Curves 101 1 Group cohomology . ......................102 2 The Selmer and Tate-Shafarevich groups . .......108 3 ThefinitenessoftheSelmergroup................110 4 Heights;completionoftheproof.................117 5 The problem of computing the rank of E.Q/ ...........126 6TheN´eron-Tatepairing......................131 7 Geometric interpretation of the cohomology groups .......133 8 FailureoftheHasse(local-global)principle...........143 9 Elliptic curves over finite fields ..................147 vii viii 10 TheconjectureofBirchandSwinnerton-Dyer..........160 11 Elliptic curves and sphere packings ................168 V Elliptic curves and modular forms 173 1 The Riemann surfaces X0.N / ..................173 2 X0.N / as an algebraic curve over Q ...............181 3 Modular forms . ........................189 4 Modular forms and the L-series of elliptic curves . .....193 5 Statementofthemaintheorems.................208 6 How to get an elliptic curve from a cusp form . .........210 7WhytheL-Series of Ef agrees with the L-Series of f .....215 8 Wiles’sproof...........................222 9 Fermat,atlast...........................226 Bibliography 229 Index 235 1 Introduction An elliptic curve over a field k is a nonsingular complete curve of genus 1 with a distinguished point. When the characteristic of k is not 2 or 3, it can be realized as a plane projective curve Y 2Z D X 3 C aXZ2 C bZ3;4a3 C 27b2 ¤ 0; and every such equation defines an elliptic curve over k. The distinguished point is .0 W 1 W 0/. For example, the following pictures show the real points (except the point at infinity) of two elliptic curves: Y 2 D X 3 C 1 Y 2 D X.X2 1/ Although the problem of computing the points on an elliptic curve E with rational numbers as coordinates has fascinated mathematicians since the time of the ancient Greeks, it was not until 1922 that it was proved that it is possible to construct all the points starting from a finite number by drawing chords and tangents. This is the famous theorem of Mordell, which shows more precisely that the rational points form a finitely generated group E.Q/. There is a sim- ple algorithm for computing the torsion subgroup of E.Q/, but there is still no proven algorithm for computing the rank. In one of the earliest applications of computers to mathematics, Birch and Swinnerton-Dyer discovered experimen- tally a relation between the rank and the numbers Np of the points on the curve read modulo the different prime numbers p. The problem of proving this rela- tion (the conjecture of Birch and Swinnerton-Dyer) is one of the most important in mathematics. Chapter IV of the book proves Mordell’s theorem and explains the conjecture of Birch and Swinnerton-Dyer. In 1955, Taniyama noted that it was plausible that the Np attached to a given elliptic curve always arise in a simple way from a modular form (in modern terminology, that the curve is modular). However, in 1985 Frey observed that this didn’t appear to be true for the elliptic curve attached to a nontrivial solution of the Fermat equation X p C Y p D Zp, p>2. His observation prompted Serre to revisit some old conjectures implying this, and Ribet proved enough of his conjectures to deduce that Frey’s observation is correct: the elliptic curve 2 attached to a nontrivial solution of the Fermat equation is not modular. Finally, in 1994 Wiles (with the help of Taylor) proved that every elliptic curve in a large class is modular. Since the class would contain any curve attached to a nontrivial solution of the Fermat equation, this proves that no such solution exists. Chapter V of the book is devoted to explaining these results. The first three chapters of the book develop the basic theory of elliptic curves. Elliptic curves have been used to shed light on some important problems that, at first sight, appear to have nothing to do with elliptic curves. I mention three such problems. Fast factorization of integers There is an algorithm for factoring integers that uses elliptic curves and is in many respects better than previous algorithms. People have been factoring in- tegers for centuries, but recently the topic has become of practical significance: given an integer n that is the product n D pq of two (large) primes p and q, there is a secret code for which anyone who knows n can encode a message, but only those who know p; q can decode it. The security of the code depends on no unauthorized person being able to factor n. See Koblitz 1987, VI 4, or Silverman and Tate 1992, IV 4. Lattices and the sphere packing problem The sphere packing problem is that of finding an arrangement of n-dimensional unit balls in Euclidean space that covers as much of the space as possible without overlaps. The arrangement is called a lattice packing if the centres of the spheres are the points of a lattice in n-space. The best packing in the plane is a lattice packing. Elliptic curves have been used to find lattice packings in many dimensions that are denser than any previously known (see IV, 11). Congruent numbers A natural number n is said to be congruent if it occurs as the area of a right triangle whose sides have rational length. If we denote the lengths of the sides of the triangle by x;y;z,thenn will be congruent if and only if the equations 2 2 2 1 x C y D z ;nD 2 xy 3 have simultaneous solutions in Q. The problem was of interest to the ancient Greeks, and was discussed systematically by Arab scholars in the tenth cen- tury. Fibonacci showed that 5 and 6 are congruent, Fermat that 1; 2; 3; are not congruent, and Euler proved that 7 is congruent, but it appeared hopeless to find a simple criterion for deciding whether a given n is congruent until Tunnell showed that the conjecture of Birch and Swinnerton-Dyer implies the following critertion: An odd square-free n is congruent if and only if the number of triples of integers .x;y;z/satisfying 2x2 C y2 C 8z2 D n is equal to twice the number of triples satisfying 2x2 C y2 C 32z2 D n. See Koblitz 1984. Among the many works on the arithmetic of elliptic curves, I mention here only the survey article Cassels 1966, which gave the first modern exposition of the subject, Tate’s Haverford lectures (reproduced in Silverman and Tate 1992), which remain the best elementary introduction, and the two volumes Silverman 1986, 1994, which have become the standard reference.
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