椭圆曲线的算术理论

　　The preface to a textbook frequently contains the author's justification for offering the public "another book" on the given subject. For our chosen topic， the arithmetic of elliptic curves， there is little need for such an apologia.Considering the vast amount of research currently being done in this area，the paucity of introductory texts is somewhat surprising. Parts of the theory are contained in various books of Lang （especially [La 3] and [La 5]）; and there are books of Koblitz （[Ko]） and Robert （[Rob]， now out of print） which concentrate mostly on the analytic and modular theory. In addition， survey articles have been written by Cassels （[Ca 7]， really a short book） and Tate （[Ta 5]. which is beautifully written， but includes no proofs）. Thus the author hopes that this volume will fill a real need， both for the serious student who wishes to learn the basic facts about the arithmetic of elliptic curves; and for the research mathematician who needs a reference source for those same basic facts.本书为英文版。

暂缺《椭圆曲线的算术理论》作者简介

Preface
Introduction
CHAPTERI
AlgebraicVarieties
1.AffineVarieties
2.ProjectiveVarieties
3.MapsbetweenVarieties
CHAPTERII
AlgebraicCurves
1.Curves
2.MapsbetweenCurves
3.Divisors
4.Differentials
5.TheRiemann-RochTheorem
CHAPTERIII
TheGeometryofEllipticCurves
1.WeierstrassEquations
2.TheGroupLaw
3.EllipticCurves
4.Isogenies
5.TheInvariantDifferential
6.TheDualIsogeny
7.TheTateModule
8.TheWeilPairing
9.TheEndomorphismRing
10.TheAutomorphismGroup
CHAPTERIV
TheFormalGroupofanEllipticCurve
1.ExpansionaroundO
2.FormalGroups
3.GroupsAssociatedtoFormalGroups
4.TheInvariantDifferential
5.TheFormalLogarithm
6.FormalGroupsoverDiscreteValuationRings
7.FormalGroupsinCharacteristicp
CHAPTERV
EllipticCurvesoverFiniteFields
1.NumberofRationalPoints
2.TheWeilConjectures
3.TheEndomorphismRing
4.CalculatingtheHasseInvariant
CHAPTERVI
EllipticCurvesoverC
1.EllipticIntegrals
2.EllipticFunctions
3.ConstructionofEllipticFunctions
4.Maps-AnalyticandAlgebraic
5.Uniformization
6.TheLefschetzPrinciple
CHAPTERVII
EllipticCurvesoverLocalFields
1.MinimalWeierstrassEquations
2.ReductionModulo
3.PointsofFiniteOrder
4.TheActionofInertia
5.GoodandBadReduction
6.TheGroupE/Eo
7.TheCriterionofNeron-Ogg-Shafarevich
CHAPTERVllI
EllipticCurvesoverGlobalFields
1.TheWeakMordell-WeilTheorem
2.TheKummerPairingviaCohomology
3.TheDescentProcedure
4.TheMordell-WeilTheoremoverQ
5.HeightsonProjectiveSpace
6.HeightsonEllipticCurves
7.TorsionPoints
8.TheMinimalDiscriminant
9.TheCanonicalHeight
10.TheRankofanEllipticCurve
CHAPTERIX
IntegralPointsonEllipticCurves
1.DiophantineApproximation
2.DistanceFunctions
3.Siegel'sTheorem
4.TheS-UnitEquation
5.EffectiveMethods
6.Shafarevich'sTheorem
7.TheCurveY2=X3+D
8.Roth'sTheorem--AnOverview
CHAPTERX
ComputingtheMordell-WeilGroup
1.AnExample
2.Twisting--GeneralTheory
3.HomogeneousSpaces
4.TheSelmerandShafarevich-TateGroups
5.Twisting--EllipticCurves
6.TheCurveY2=X3+DX
APPENDIXA
EllipticCurvesinCharacteristics2and3
APPENDIXB
GroupCohomology(HOandH1)
1.CohomologyofFiniteGroups
2.GaloisCohomology
3.Non-AbelianCohomology
APPENDIXC
FurtherTopics:AnOverview
11.ComplexMultiplication
12.ModularFunctions
13.ModularCurves
14.TateCurves
15.NeronModelsandTate'sAlgorithm
16.L-Series
17.DualityTheory
18.LocalHeightFunctions
19.TheImageofGalois
20.FunctionFieldsandSpecializationTheorems
NotesonExercises
Bibliography
ListofNotation
Index