December13,2011 Time: 06:03pm fm.tex Elliptic Tales i This page intentionally left blank December13,2011 Time: 06:03pm fm.tex Elliptic Tales Curves, Counting, and Number Theory • • • • • Avner Ash Robert Gross PrincetonUniversityPress PrincetonandOxford iii Copyright©2012byPrincetonUniversityPress PublishedbyPrincetonUniversityPress,41WilliamStreet,Princeton,NewJersey08540 IntheUnitedKingdom:PrincetonUniversityPress,6OxfordStreet,Woodstock, OxfordshireOX201TW press.princeton.edu AllRightsReserved LibraryofCongressCataloging-in-PublicationData Ash,Avner,1949– Elliptictales:curves,counting,andnumbertheory/AvnerAsh,RobertGross. p.cm. Includesbibliographicalreferencesandindex. ISBN978-0-691-15119-9(hardcover) 1. Ellipticfunctions. 2. Curves,Elliptic. 3. Numbertheory. I. Gross,Robert,1959– II. Title. QA343.A972012 515(cid:1).983—dc23 2011044712 BritishLibraryCataloging-in-PublicationDataisavailable ThisbookhasbeencomposedinMinionPro Printedonacid-freepaper.∞ TypesetbySRNovaPvtLtd,Bangalore,India PrintedintheUnitedStatesofAmerica 13579108642 iv ForEdieandRosemary v This page intentionally left blank Thanlongenfolktogononpilgrimages, Andpalmeresfortosekenstrangestrondes. —GeoffreyChaucer, TheCanterburyTales • • • • • ThescienceofPureMathematics,initsmoderndevelopments, mayclaimtobethemostoriginalcreationofthehumanspirit.... Theoriginalityofmathematicsconsistsinthefactthatin mathematicalscienceconnectionsbetweenthingsareexhibited which,apartfromtheagencyofhumanreason,areextremely unobvious.Thustheideas,nowinthemindsofcontemporary mathematicians,lieveryremotefromanynotionswhichcanbe immediatelyderivedbyperceptionthroughthesenses. —AlfredNorthWhitehead, ScienceandtheModernWorld vii This page intentionally left blank December13,2011 Time: 06:03pm fm.tex Contents • • • • • Preface xiii Acknowledgments xix Prologue 1 PARTI.DEGREE Chapter1.DegreeofaCurve 13 1. GreekMathematics 13 2. Degree 14 3. ParametricEquations 20 4. OurTwoDefinitionsofDegreeClash 23 Chapter2.AlgebraicClosures 26 1. SquareRootsofMinusOne 26 2. ComplexArithmetic 28 3. RingsandFields 30 4. ComplexNumbersandSolvingEquations 32 5. Congruences 34 6. ArithmeticModuloaPrime 38 7. AlgebraicClosure 38 Chapter3.TheProjectivePlane 42 1. PointsatInfinity 42 2. ProjectiveCoordinatesonaLine 46 3. ProjectiveCoordinatesonaPlane 50