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Undergraduate Texts in Mathematics Joseph H. Silverman John T. Tate Rational Points on Elliptic Curves Second Edition Undergraduate Texts in Mathematics Undergraduate Texts in Mathematics SeriesEditors: SheldonAxler SanFranciscoStateUniversity,SanFrancisco,CA,USA KennethRibet UniversityofCalifornia,Berkeley,CA,USA AdvisoryBoard: ColinAdams,WilliamsCollege DavidA.Cox,AmherstCollege PamelaGorkin,BucknellUniversity RogerE.Howe.YaleUniversity MichaelOrrison,HarveyMuddCollege JillPipher,BrownUniversity FadilSantosa,UniversityofMinnesota Undergraduate Texts in Mathematics are generally aimed at third- and fourth- yearundergraduatemathematicsstudentsatNorthAmericanuniversities.Thesetexts strivetoprovidestudentsandteacherswithnewperspectivesandnovelapproaches. The books include motivation that guides the reader to an appreciation of interre- lationsamongdifferentaspectsofthesubject.Theyfeatureexamplesthatillustrate keyconceptsaswellasexercisesthatstrengthenunderstanding. Moreinformationaboutthisseriesathttp://www.springer.com/series/666 Joseph H. Silverman • John T. Tate Rational Points on Elliptic Curves Second Edition 123 JosephH.Silverman JohnT.Tate DepartmentofMathematics DepartmentofMathematics BrownUniversity HarvardUniversity Providence,RI,USA Cambridge,MA,USA ISSN0172-6056 ISSN2197-5604 (electronic) UndergraduateTextsinMathematics ISBN978-3-319-18587-3 ISBN978-3-319-18588-0 (eBook) DOI10.1007/978-3-319-18588-0 LibraryofCongressControlNumber:2015940539 SpringerChamHeidelbergNewYorkDordrechtLondon ©SpringerInternationalPublishingSwitzerland1992,2015 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsorthe editorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforanyerrors oromissionsthatmayhavebeenmade. Printedonacid-freepaper SpringerInternationalPublishingAGSwitzerlandispartofSpringerScience+BusinessMedia(www. springer.com) Preface Preface to the Original 1992 Edition In1961thesecondauthordeliveredaseriesoflecturesatHaverfordCollege onthesubjectof“RationalPointsonCubicCurves.”Theselectures,intended for junior and senior mathematics majors, were recorded, transcribed, and printed in mimeograph form. Since that time, they have been widely dis- tributed as photocopies of ever-decreasing legibility, and portions have appeared in various textbooks (Husemo¨ller [25], Chahal [9]), but they have never appeared in their entirety. In view of the recent interest in the theory of elliptic curves for subjects ranging from cryptography (Lenstra [30], Koblitz [27]) to physics (Luck–Moussa–Waldschmidt [31]), as well as the tremendous amount of purely mathematical activity in this area, it seems a propitious time to publish an expanded version of those original notes suit- ableforpresentationtoanadvancedundergraduateaudience. We have attempted to maintain much of the informality of the original Haverfordlecturers.Ourmaingoalindoingthishasbeentowriteatextbook inatechnicallydifficultfieldthatis“readable”bytheaverageundergraduate mathematics major. We hope that we have succeeded in this goal. The most obvious drawback to such an approach is that we have not been entirely rig- orous in all of our proofs. In particular, much of the foundational material on elliptic curves presented in Chapter 1 is meant to explain and convince, rather than to rigorously prove. Of course, the necessary algebraic geometry canmostlybedevelopedinonemoderatelylongchapter,aswehavedonein AppendixA.Buttheemphasisofthisbookisonnumbertheoreticaspectsof ellipticcurves,sowefeelthataninformalapproachtotheunderlyinggeom- etryispermissible,sinceitallowsusmorerapidaccesstothenumbertheory. Forthosewhowishtodelvemoredeeplyintothegeometry,thereareseveral good books on the theory of algebraic curves suitable for an undergraduate v vi Preface course, such as Reid [37], Walker [57], and Brieskorn–Kno¨rrer [8]. In the later chapters we have generally provided all of the details for the proofs of themaintheorems. The original Haverford lectures make up Chapters 1, 2, 3, and the first twosections of Chapter 4.Inafewplaces we haveadded asmallamountof explanatory material, references have been updated to include some discov- eriesmadesince1961,andalargenumberofexerciseshavebeenadded.But thosewhohaveseentheoriginalmimeographednoteswillrecognizethatthe changes have been kept to a minimum. In particular, the emphasis is still on proving (special cases of) the fundamental theorems in the subject: (1) the Nagell–Lutz theorem, which gives a precise procedure for finding all of the rational points of finite order on an elliptic curve; (2) Mordell’s theorem, which says that the group of rational points on an elliptic curve is finitely generated; (3) a special case of Hasse’s theorem, due to Gauss, which de- scribesthenumberofpointsonanellipticcurvedefinedoverafinitefield. InSection4.4wehavedescribedLenstra’sellipticcurvealgorithmforfac- toring large integers. This is one of the recent applications of elliptic curves tothe“realworld,”towit,theattempttobreakcertainwidelyusedpublickey ciphers.Wehaverestrictedourselvestodescribingthefactorizationalgorithm itself, since there have been many popular descriptions of the corresponding ciphers.1 Chapters 5 and 6 are new. Chapter 5 deals with integer points on elliptic curves.Section5.2islooselybasedonanIAPundergraduatelecturegivenby thefirstauthoratMITin1983.TheremainingsectionsofChapter5containa proofofaspecialcaseofSiegel’stheorem,whichassertsthatanellipticcurve hasonlyfinitelymanyintegralpoints.Theproof,basedonThue’smethodof Diophantineapproximation,iselementary,butintricate.However,inviewof Vojta’s[56]andFaltings’[15]recentspectacularapplicationsofDiophantine approximation techniques, it seems appropriate to introduce this subject at anundergraduatelevel.Chapter6givesanintroductiontothetheoryofcom- plexmultiplication.Ellipticcurveswithcomplexmultiplicationariseinmany different contexts in number theory and in other areas of mathematics. The goal of Chapter 6 is to explain how points of finite order on elliptic curves with complex multiplication can be used to generate extension fields with Abelian Galois groups, much as roots of unity generate Abelian extensions oftherationalnumbers.ForChapter6only,wehaveassumedthatthereader isfamiliarwiththerudimentsoffieldtheoryandGaloistheory. 1Thatwaswhatwesaidinthefirstedition,butinthissecondedition,wehaveincludeda discussionofellipticcurvecryptography;seeSection4.5. Preface vii Finally, we have included an appendix giving an introduction to projective geometry, with an especial emphasis on curves in the projective plane. The first three sections of Appendix A provide the background needed for reading the rest of the book. In Section A.4 of the appendix we give an ele- mentaryproofofBezout’stheorem,andinSectionA.5,weprovidearigorous discussion of the reduction modulo p map and explain why it induces a ho- momorphismontherationalpointsofanellipticcurve. The contents of this book should form a leisurely semester course, with sometimeleftoverforadditionaltopicsineitheralgebraicgeometryornum- ber theory. The first author has also used this material as a supplementary special topic at the end of an undergraduate course in modern algebra, cov- ering Chapters 1, 2, and 4 (excluding Section 4.3) in about four weeks of class. We note that the last five chapters are essentially independent of one another (except Section 4.3 depends on the Nagell–Lutz theorem, proven in Chapter 2). This gives the instructor maximum freedom in choosing topics if time is short. It also allows students to read portions of the book on their own,e.g.,asasuitableprojectforareadingcourseorhonorsthesis.Wehave included many exercises, ranging from easy calculations to published theorems. An exercise marked with a (∗) is likely to be somewhat challenging. An exercise marked with (∗∗) is either extremely difficult to solve with the materialthatwecoverorisacurrentlyunsolvedproblem. Ithasbeensaidthat“itispossibletowriteendlesslyonellipticcurves.”2 Weheartilyagreewiththissentiment,buthaveattemptedtoresistsuccumb- ing to its blandishments. This is especially evident in our frequent decision to prove special cases of general theorems, even when only a few additional pageswouldberequiredtoproveamoregeneralresult.Ourgoalthroughout has been to illuminate the coherence and the beauty of the arithmetic the- oryofellipticcurves;wehappilyleavethetaskofbeingencyclopedictothe authorsofmoreadvancedmonographs. Preface to the 2015 Edition Themostimportantchangetotheneweditionistheadditionoftwonewsec- tions.InSection4.5webrieflydiscusshowandwhyellipticcurvesareusedin moderncryptography,andinSection6.6,wegiveanoverviewofhowelliptic 2FromtheintroductiontoEllipticCurves:DiophantineAnalysis,SergeLang,Springer- Verlag,NewYork,1978.ProfessorLangfollowshisassertionwiththestatementthat“Thisis notathreat,”indicatingthathe,too,hasavoidedthetemptationtowriteabookofindefinite length. viii Preface curvesplayakeyroleinWiles’proofofFermat’sLastTheorem.Wehavealso takentheopportunitytomakenumerouscorrections,bothtypographicaland mathematical, to add a few new problems, and to update historical material toreflectsomeoftheexcitingadvancesofthepast25years. Electronic Resources The interested reader will find additional material and a list of errata on the RationalPointsonEllipticCurveshomepage: www.math.brown.edu/˜jhs/RPECHome.html Thiswebpageincludessomeofthenumericalexercisesinthebook,allowing the reader to cut and paste them into other programs, rather than having to retypethem. Therearenowmanycommercialandfreecomputerpackagesthatperform calculationsofvaryinglevelsofsophisticationonellipticcurves,3 including, forexample, Sage: http://www.sagemath.org Pari/GP: http://pari.math.u-bordeaux.fr Nobookiseverfreefromerrororincapableofbeingimproved.Wewould be delighted to receive comments, good or bad, and corrections from our readers.Youcansendmailtousat [email protected] Acknowledgments First Edition, First Printing: The authors would like to thank Rob Gross, Emma Previato, Michael Rosen, Seth Padowitz, Chris Towse, Paul van Mulbregt, Eileen O’Sullivan, and the students of Math 153 (especially Jeff Achter and Jeff Humphrey) for reading and providing corrections to the original draft. They would also like to thank Davide Cervone for producing beautifulillustrationsfromtheiroriginaljaggeddiagrams. The first author owes a tremendous debt of gratitude to Susan for her patienceandunderstanding,toDebbyforherfluorescentattirebrighteningup 3Thiswasnotthecasewhenthefirsteditionofthisbookappearedin1992,atwhichtime the first author had created a small stand-alone application for Macintosh computers and a somewhatmorehighlyfeaturedsetofroutinesforMathematica.Theseantiquepackagesare nolongeravailable. Preface ix the days, to Danny for his unfailing good humor, and to Jonathan for taking timelynapsduringcriticalstagesinthepreparationofthismanuscript. The second author would like to thank Louis Solomon for the invitation todeliverthePhilipsLecturesatHaverfordCollegeintheSpringof1961. Providence,USA JosephH.Silverman Cambridge,USA JohnT.Tate March27,1992 First Edition (Second Printing) and Second Edition: We, the authors, would like the thank the following individuals for sending comments and corrections: G. Allison, T. Anderson, P. Berman, D. Appleby, K.Bender, G.Bender,A.Berkovich,J.Blumenstein,P.deBoor,J.Brillhart,D.Clausen, S.Datta,Z.Fang,D.Freeman,L.Goldberg,F.Goldstein,A.Guth,D.Gupta, A.Granville,R.Hoibakk,I.Igusic,M.Kida,P.Kahn,J.Kraft,C.Levesque, B. Levin, J. Lipman, R. Lipes, A. Mazel-Gee, M. Mossinghoff, K. Nolish, B. Pelz, R. Pennington, R. Pries, A. Rajan, K. Ribet, M. Reid, H. Rose, L. Go´mez-Sańchez, R. Schwartz, D. Schwein J.-P. Serre, M. Szydlo, L.Tartar,J.Tobey,R.Urian,C.R.Videla,J.Wendel,A.Ziv. Providence,USA JosephH.Silverman Cambridge,USA JohnT.Tate March27,2015