椭圆曲线上的有理点

内容简介

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　　The theory of elliptic curves involves a blend of algebra，geometry， analysis，and number theory.This book stresses this interplay as it develops the basic theory，providing an opportunity for readers to appreciate the unity of modern mathematics.The book s accessibility，the informal writing style，and a wealth of exercises make it an ideal introduction for those interested in learning about Diophantine equations and arithmetic geometry.

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目录

PrefaceComputer PackagesAcknowledgmentsIntroductionCHAPTER 1Geometry and Arithmetic1.Rational Points on Conics2.The Geometry of Cubic Curves3.Weierstrass Normal Form4.Explicit Formulas for the Group LawExercisesCHAPTER 2Points of Finite Order1.Points of Order Two and Three2.Real and Complex Points on Cubic Curves3.The Discriminant4.Points of Finite Order Have Integer Coordinates5.The Nagell—Lutz Theorem and Further DevelopmentsExercisesCHAPTER 3The Group of Rational Points1.Heights and Descent2.The Height of P + P03.The Height of 2P4.A Useful Homomorphism5.Mordell’s Tneorem6.Examples and Further Developments7.Singular Cubic CurvesExercisesCHAPTER 4Cubic Curves over Finite Fields1.Rational Points over Finite Fields2.A Theorem of Gauss3.Points of Finite Order Revisited4.A Factorization Algorithm Using Elliptic CurvesExercisesCHAPTER 5Integer Points on Cubic Curves1.How Many Integer Points？2.Taxicabs and Sums of Two Cubes3.Thue’s Theorem and Diophantine Approximation4.Construction of an Auxiliary Polynomial5.The Auxiliary Polynomialls Small6.The Auxiliary Polynomial Does Not Vanish7.Proof of the Diophantine Approximation Theorem8.Further DevelopmentsExercisesCHAPTER 6Complex Multiplication1.Abelian Extensions of Q2.Algebraic Points on Cubic Curves3.A Galois Representation4.Complex Multiplication5.Abelian Extensions of Q（i）ExercisesAPPENDIX AProjective Geometry1.Homogeneous Coordinates and the Projective Plane2.Curves in the Projective Plane3.Intersections of Projective Curves4.Intersection Multiplicities and a Proof of Bezout’s Theorem5.Reduction Modulo pExercisesBibliographyList of NotationIndex

封面

书名:椭圆曲线上的有理点

作者:西尔弗曼

页数:258

定价:¥59.0

出版社:世界图书出版公司

出版日期:2015-01-01

ISBN:9787510086328

PDF电子书大小:42MB 高清扫描完整版