椭圆曲线上的有理点

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

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.