椭圆曲线上的有理点

椭圆曲线上的有理点

(美) 西尔弗曼 (Silverman,J.H.) , 著

出版社:世界图书出版公司北京公司

年代:2014

定价:59.0

书籍简介:

椭圆曲线理论是代数、几何、分析和数论的混合体,书中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。目次:几何和算术;有限阶点;有理点群;有限域上的三次曲线;三次曲线上的整数点;复数乘法;射影几何。读者对象:数学专业的本科生、研究生和相关的读者。

书籍目录:

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

内容摘要:

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.

书籍规格:

书籍详细信息
书名椭圆曲线上的有理点站内查询相似图书
9787510086328
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出版地北京出版单位世界图书出版公司北京公司
版次影印本印次1
定价(元)59.0语种英文
尺寸23 × 15装帧平装
页数印数

书籍信息归属:

椭圆曲线上的有理点是世界图书出版公司北京公司于2014.9出版的中图分类号为 O187.1 的主题关于 椭圆曲线-英文 的书籍。