J.S. Milne: Elliptic Curves

1 Milne: Elliptic CurvesOTHERBOOKS BY THEAUTHORE tale CohomologyPrinceton Mathematical Series 33, Princeton University Press, 1980, 323+xiiipages, ISBN 0-691-08238-3 Hodge Cycles, Motives, and Shimura Varieties(with Pierre Deligne, ArthurOgus, 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 TheoremsAcademic 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-XElliptic MilneCopyrightc 2006 unbound paper copies for noncommercial personal use may be madewithout explicit permission from the copyright other rights paperback version of this work is available from booksellers worldwide, in-cluding and , and from the publisher:BookSurge Publishing, purchase of this book will encourage the writing of more works like Milne},title={ Elliptic Curves },year={2006},publisher={BookSurge Publishers},pages={238+viii},isbn={1-419 6-5257-5}}Library of Congress data available ( )Mathematics Subject Classification (MSC2000).

2 11G, 11D, illustrations were written directly in PostScriptR 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 early 1996, I taught a course on Elliptic Curves . Since this was not long afterWiles had proved Fermat s Last Theorem and I promised to explain some of theideas underlying his proof, the course attracted an unusually large and diverseaudience. As a result, I attempted to make the course accessible to all studentswith a knowledge only of the standard first-year graduate it was over, I collected the notes that I had handed out during thecourse into a single file, made a few corrections, and posted them on the Web,where they have since been downloaded tens of thousands of appearance of publishers willing to turn pdf files into books quicklyand cheaply and make them available worldwide while allowing the author toretain full control of the content and appearance of the work has prompted meto rewrite the notes and make them available as a Milne,October 30, Basicdefinitions;Bezout stheorem.

3 52 173 254 Regularfunctions;theRiemann-Rochtheorem .. 295 42II Basic Theory of Elliptic Curves451 Definition of an Elliptic curve .. 452 The Weierstrass equation for an Elliptic curve .. 503 Reduction of an Elliptic curve 544 Elliptic Curves 615 646N 697 Algorithms for Elliptic Curves .. 76 III Elliptic Curves over the Complex Numbers811 812 Doubly periodic functions .. 823 Elliptic Curves as Riemann surfaces .. 89IV The Arithmetic of Elliptic Curves1011 Group cohomology ..1022 The Selmer and Tate-Shafarevich groups ..1083 Heights; The problem of computing the rank Geometric interpretation of the cohomology FailureoftheHasse(local-global)principle ..1439 Elliptic Curves over finite Elliptic Curves and sphere Elliptic Curves and modular forms1731 The Riemann an algebraic curve Modular forms ..1894 Modular forms and theL-series of Elliptic Curves ..1935 Statementofthemaintheorems.

4 2086 How to get an Elliptic curve from a cusp form ..2107 WhytheL-Series ofEfagrees with theL-Series Wiles sproof ..2229 Fermat, Elliptic curve over a fieldkis a nonsingular complete curve of genus 1 with adistinguished point. When the characteristic ofkis not2or3, it can be realizedas a plane projective curveY2 ZDX3 CaXZ2 CbZ3;4a3C27b2 0;and every such equation defines an Elliptic curve overk. The distinguished For example, the following pictures show the real points (exceptthe point at infinity) of two Elliptic 1/Although the problem of computing the points on an Elliptic curveEwithrational numbers as coordinates has fascinated mathematicians since the time ofthe ancient Greeks, it was not until 1922 that it was proved that it is possibleto construct all the points starting from a finite number by drawing chords andtangents. This is the famous theorem of Mordell, which shows more preciselythat the rational points form a finitely generated There is a sim-ple algorithm for computing the torsion subgroup , but there is still noprovenalgorithm for computing the rank.

5 In one of the earliest applications ofcomputers to mathematics, Birch and Swinnerton-Dyer discovered experimen-tally a relation between the rank and the numbersNpof the points on the curveread modulo the different prime numbersp. The problem of proving this rela-tion (the conjecture of Birch and Swinnerton-Dyer) is one of the most importantin mathematics. Chapter IV of the book proves Mordell s theorem and explainsthe conjecture of Birch and 1955, Taniyama noted that it was plausible that theNpattached to a givenelliptic curve always arise in a simple way from a modular form (in modernterminology, that the curve is modular). However, in 1985 Frey observed thatthis didn t appear to be true for the ellipticcurve attached to a nontrivial solutionof the Fermat equationXpCYpDZp,p>2. His observation promptedSerre to revisit some old conjectures implying this, and Ribet proved enough ofhis conjectures to deduce that Frey s observation is correct: the Elliptic curve2attached to a nontrivial solution of the Fermat equation is not modular.

6 Finally,in 1994 Wiles (with the help of Taylor) proved that every Elliptic curve in a largeclass is modular. Since the class would contain any curve attached to a nontrivialsolution of the Fermat equation, this proves that no such solution exists. ChapterV of the book is devoted to explaining these first three chapters of the book develop the basic theory of Curves have been used to shed light on some important problemsthat, at first sight, appear to have nothing to do with Elliptic Curves . I mentionthree such factorization of integersThere is an algorithm for factoring integers that uses Elliptic Curves and is inmany respects better than previous algorithms. People have been factoring in-tegers for centuries, but recently the topic has become of practical significance:given an integernthat is the productnDpqof two (large) primespandq,there is a secret code for which anyone who knowsncan encode a message,but only those who knowp;qcan decode it. The security of the code dependson no unauthorized person being able to factorn.

7 See Koblitz 1987, VI 4, orSilverman and Tate 1992, IV and the sphere packing problemThe sphere packing problem is that of finding an arrangement ofn-dimensionalunit balls in Euclidean space that covers as much of the space as possible withoutoverlaps. The arrangement is called a lattice packing if the centres of the spheresare the points of a lattice best packing inthe plane is a Curves have been used to find lattice packings in many dimensions thatare denser than any previously known (see IV, 11).Congruent numbersA natural numbernis said to becongruentif it occurs as the area of a righttriangle whose sides have rational length. If we denote the lengths of the sidesof the triangle byx;y;z,thennwill be congruent if and only if the equationsx2Cy2Dz2;nD12xy3have simultaneous solutions inQ. The problem was of interest to the ancientGreeks, and was discussed systematically by Arab scholars in the tenth cen-tury. Fibonacci showed that5and6are congruent, Fermat that1;2;3;are notcongruent, and Euler proved that7is congruent, but it appeared hopeless tofind a simple criterion for deciding whether a givennis congruent until Tunnellshowed that the conjecture of Birch and Swinnerton-Dyer implies the followingcritertion:An odd square-freenis congruent if and only if the number oftriples of ;y;z/satisfying2x2Cy2C8z2 Dnis equalto twice the number of triples Koblitz the many works on the arithmetic of Elliptic Curves , I mention hereonly the survey article Cassels 1966, which gave the first modern exposition ofthe subject, Tate s Haverford lectures (reproduced in Silverman and Tate 1992),which remain the best elementary introduction, and the two volumes Silverman1986, 1994, which have become the standard knowledge of the basic algebra, analysis, and topology usually taught in ad-vanced undergraduate or beginning graduate courses.

8 Some knowledge of alge-braic geometry and algebraic number theory will be useful but not use the standard notations:Nis the set of natural numbersf0;1;2;:::g,Zthe ring of integers,Qthe field of rational numbers,Rthe field of real numbers,Cthe field of complex numbers, andFpthe field withpelements. A numberfield is a finite extension the book,kis a field andkalis an algebraic closure is a field containingk, and a homomorphism ofk-fields is a homomor-phism of fields acting as the identity map rings will be commutative with1, and homomorphisms of rings are re-quired to ,A is the group of units inA:A Dfa2 Ajthere exists ab2 Asuch thatabD1g:For an abelian groupX,XnDfx2 XjnxD0g. For a finite setS,#Sor(occasionally) S denotes the number of elements ofS. For an elementaof aset with an equivalence relation, we sometimes use a to denote the equivalenceclass defined to beY, or equalsYby definition;X YXis a subset ofY(not necessarily proper, ,Xmay equalY);X YXandYare isomorphic;X'YXandYare canonically isomorphic, or there is agiven or unique isomorphism from one to the addition to the references listed at the end, I refer to the following of mycourse notes (available at ).

9 ANTA lgebraic Number Theory (August 31, 1998).AGAlgebraic Geometry (February 20, 2005).CFTC lass Field Theory (May 6, 1997).FTFields and Galois Theory (February 19, 2005).MFModular Functions and Modular Forms (May 22, 1997).ACKNOWLEDGEMENTSI thank the following for providing corrections and comments for earlier ver-sions of this work: Alan Bain, Leen Bleijenga, Keith Conrad, Jean Cougnard,Mark Faucette, Michael M uller, Holger Partsch, Jasper Scholten, and IPlane Curves1 Basic definitions; Bezout s theoremIn this section we review part of the theory of plane Curves . Omitted details (andmuch more) can be found in Fulton 1969 and Walker ringsWe shall make frequent use of the fact that polynomial rings over fields areunique factorization domains: for the field itself, there is nothing to prove, andthe general case follows by induction from the statement that ifAis a uniquefactorization domain, then so also isA X (AG, 1). Thus, ink X1;:::;Xn ,every polynomialfcan be written as a productfDfm11 fmrrof powers of irreducible polynomialsfiwith nofibeing a constant multiple ofanother, and the factorization is unique up to replacing thefiwith constant mul-tiples.

10 Therepeated factorsoffare thosefiwithmi> irreducible,then the /that it generates is plane curvesTheaffine k. A nonconstant polynomialf2k X;Y , assumed to have no repeated factor inkal X;Y ,definesanaffineplane curveCfoverkwhose points with coordinates in any fieldK karethe zeros offinK2 ; ;y/D0g:56 CHAPTER I. PLANE CURVESFor anyc2k , the curvesCfandCcfhave the same points in every fieldK,and so we don t distinguish curveCfis said to beirreducibleiffis irreducible ink X;Y ,andit is said begeometrically irreducibleiffremains irreducible curveCf, we can writefDf1f2 frwith thefidistinct irreduciblepolynomials ink X;Y , [ [ theCfiirreducible Curves . TheCfiare called theirreducible often writeCWfD0to mean thatCis the curveCf,orevenCWf1Df2to mean thatCis the curveCf1 f2 ;Y/be an irreducible polynomial inQ p2 X;Y ,noconstant multiple of which lies inQ X;Y , ;Y/be its conjugateoverQ(obtained by replacing eachp2inf1with p2). ; ; ;Y/lies inQ X;Y because it is fixed by the Galois group ofQ p2 = irreducible but not geometrically irreducible.]]