[内容简介]《椭圆曲线算术中的高等论题(英文版)》内容简介：美国哈佛大学从1977年开始，曾多次举办”椭圆曲线” 班，《椭圆曲线算术中的高等论题(英文版)》作者是该讨论班成员之一。椭圆曲线是一个古老的数学课题，最近由于代数数论和代数几何等现代数学的进展，使它得到了新的活力。《椭圆曲线算术中的高等论题(英文版)》是以1986年版的《椭圆曲线的算术理论》为蓝本，但在知识体系上做了较大的改动形成了这不教程，讲述上也更加专业，但在思想上是作者前《椭圆曲线算术中的高等论题(英文版)》的延续。包括椭圆和模型函数；复乘方法；椭圆曲线；Néron模型；复域上的椭圆曲线等内容。每章末都配有大量习题。目次：椭圆和模型函数；复乘方法；椭圆曲线；Néron模型；复域上的椭圆曲线。
[目次]CHAPTER Ⅰ

　　Elliptic and Modular Functions

　　The Modular Group

　　The Modular Curve X(1)

　　Modular Functions

　　Uniformization and Fields of Moduli

　　Elliptic Functions Revisited

　　q-Expansions of Elliptic Functions

　　q-Expansions of Modular Functions

　　Jacobi's Product Formula for A(T)

　　Hecke Operators

　　Hecke Operators Acting on Modular Forms

　　L-Series Attached to Modular Forms

　　Exercises

　　CHAPTER Ⅱ

　　Complex Multiplication

　　Complex Multiplication over C

　　Rationality Questions

　　Class Field Theory —— A Brief Review

　　The Hilbert Class Field

　　The Maximal Abelian Extension

　　Integrality of j

　　Cyclotomic Class Field Theory

　　The Main Theorem of Complex Multiplication

　　The Associated GrSssencharacter

　　The L-Series Attached to a CM Elliptic Curve

　　Exercises

　　CHAPTER Ⅲ

　　Elliptic Surfaces

　　Elliptic Curves over Function Fields

　　The Weak Mordell-Weil Theorem

　　Elliptic Surfaces

　　Heights on Elliptic Curves over Unction Fields

　　Split Elliptic Surfaces and Sets of Bounded Height

　　The Mordell-Weil Theorem for Fhnction Fields

　　The Geometry of Algebraic Surfaces

　　The Geometry of Fibered Surfaces

　　The Geometry of Elliptic Surfaces

　　Heights and Divisors on Varieties

　　Specialization Theorems for Elliptic Surfaces

　　Integral Points on Elliptic Curves over Function Fields

　　Exercises

　　CHAPTER Ⅳ

　　The N6ron Model

　　Group Varieties

　　Schemes and S-Schemes

　　Group Schemes

　　Arithmetic Surfaces

　　N6ron Models

　　Existence of N6ron Models

　　Intersection Theory, Minimal Models, and Blowing-Up

　　The Special Fiber of a N6ron Model

　　Tate's Algorithm to Compute the Special Fiber

　　The Conductor of an Elliptic Curve

　　Ogg's Formula

　　Exercises

　　CHAPTER Ⅴ

　　Elliptic Curves over Complete Fields

　　Elliptic Curves over C

　　Elliptic Curves over R

　　The Tate Curve

　　The Tate Map Is Surjective

　　Elliptic Curves over p-adic Fields

　　Some Applications of p-adic Uniformization

　　Exercises

　　CHAPTER Ⅵ

　　Local Height Functions

　　Existence of Local Height Functions

　　Local Decomposition of the Canonical Height

　　Archimedean Absolute Values —— Explicit Formulas

　　Non-Archimedean Absolute Values —— Explicit Formulas

　　Exercises

　　APPENDIX A

　　Some Useful Tables

　　Bernoulli Numbers and (2k)

　　Fourier Coefficients of A(T) and j(T)

　　Elliptic Curves over Q with Complex Multiplication

　　Notes on Exercises

　　References

　　List of Notation

　　Index