美国哈佛大学从1977年以来曾多次举办“椭圆曲线”班，《椭圆曲线算术(第2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。本书则是以上述观点处理椭圆函数的算术理论，包括椭圆曲线的几何背景，椭圆曲线的形式群，有限域上的椭圆函数、复数、局部域和整体域等基本内容，最后两章讨论整数和有理数。本书由希尔弗曼著。

Preface to the Second Edition

Preface to the First Edition

Introduction

CHAPTER 1

Algebraic Varieties

1. Affine Varieties

2. Projective Varieties

3. Maps Between Varieties

Exercises

CHAPTER 2

Algebraic Curves

1. Curves

2. Maps Between Curves

3. Divisors

4. Differentials

5. The Riemarm-Roch Theorem

Exercises

CHAPTER 3

The Geometry of Elliptic Curves

1. Weierstrass Equations

2. The Group Law

3. Elliptic Curves

4. Isogenies

5. The Invariant Differential

6. The Dual Isogeny

7. The Tate Module

8. The Weil Pairing

9. The Endomorphism Ring

10. The Automorphism Group

Exercises

CHAPTER 4

The Formal Group of an Elliptic Curve

1. Expansion Around O

2. Formal Groups

3. Groups Associated to Formal Groups

4. The Invariant Differential

5. The Formal Logarithm

6. Formal Groups over Discrete Valuation Rings

7. Formal Groups in Characteristic p

Exercises

CHAPTER 5

Elliptic Curves over Finite Fields

1. Number of Rational Points

2. The Weil Conjectures

3. The Endomorphism Ring

4. Calculating the Hasse Invariant

Exercises

CHAPTER 6

Elliptic Curves over C

1. Elliptic Integrals

2. Elliptic Functions

3. Construction of Elliptic Functions

4. Maps Analytic and Maps Algebraic

5. Uniformization

6. The Lefschetz Principle

Exercises

CHAPTER 7

Computing the Mordell-Weil Group

CHAPTER 8

Algorithmic Aspects of Elliptic Curves

APPENDIX