椭圆曲线理论是代数、几何、分析和数论的混合体，西尔弗曼所著的《椭圆曲线上的有理点(英文版)》在讲述基本理论的同时强调各部分之间的相互作用，以便读者更好的学习现代数学的精髓。

本书的可读性强，写作风格自由，配合大量的练习使得本书成为对Diophantine方程和算术几何感兴趣的读者的理想选择。

Preface

Computer Packages

Acknowledgments

Introduction

CHAPTER 1

 Geometry and Arithmetic

 1.Rational Points on Conics

 2.The Geometry of Cubic Curves

 3.Weierstrass Normal Form

 4.Explicit Formulas for the Group Law

 Exercises

CHAPTER 2

 Points of Finite Order

 1.Points of Order Two and Three

 2.Real and Complex Points on Cubic Curves

 3.The Discriminant

 4.Points of Finite Order Have Integer Coordinates

 5.The Nagell—Lutz Theorem and Further Developments

 Exercises

CHAPTER 3

 The Group of Rational Points

 1.Heights and Descent

 2.The Height of P + P0

 3.The Height of 2P

 4.A Useful Homomorphism

 5.Mordell's Tneorem

 6.Examples and Further Developments

 7.Singular Cubic Curves

 Exercises

CHAPTER 4

 Cubic Curves over Finite Fields

 1.Rational Points over Finite Fields

 2.A Theorem of Gauss

 3.Points of Finite Order Revisited

 4.A Factorization Algorithm Using Elliptic Curves

 Exercises

CHAPTER 5

 Integer Points on Cubic Curves

 1.How Many Integer Points？

 2.Taxicabs and Sums of Two Cubes

 3.Thue's Theorem and Diophantine Approximation

 4.Construction of an Auxiliary Polynomial

 5.The Auxiliary Polynomialls Small

 6.The Auxiliary Polynomial Does Not Vanish

 7.Proof of the Diophantine Approximation Theorem

 8.Further Developments

 Exercises

CHAPTER 6

 Complex Multiplication

 1.Abelian Extensions of Q

 2.Algebraic Points on Cubic Curves

 3.A Galois Representation

 4.Complex Multiplication

 5.Abelian Extensions of Q（i）

 Exercises

APPENDIX A

 Projective Geometry

 1.Homogeneous Coordinates and the Projective Plane

 2.Curves in the Projective Plane

 3.Intersections of Projective Curves

 4.Intersection Multiplicities and a Proof of Bezout's Theorem

 5.Reduction Modulo p

 Exercises

Bibliography

List of Notation

Index