本书在没有运用大量现代代数几何知识的前提下，几乎自成体系地对代数曲线理论进行了很好的阐释。这样的讲解方式使得非专业的人员对这门重要而且系统的学科非常容易理解。同样，对于专业人士来说，从本书中也可学习到不少新颖的内容，如Tate留数理论、高阶导数和特征p中的Weiertrass点、Riemann假设的Stohr-Voloch证明和不可分留数域扩张。本书基于单变量函数域理论，其独特之处包括了射影曲线：奇性及部分平面曲线。每章末都附有练习，可以帮助读者理解所学内容。

Preface

Introduction

1 Background

 1.1 Valuations

 1.2 Completions

 1.3 Difrerential Fomls

 1.4 Residues

 1.5 Exercises

2 Function Fields

 2.1 Divisors and Adeles

 2.2 Weil Differentials

 2.3 Elliptic Functions

 2.4 GeonleIric Function Fields

 2.5 ResidIles and Duality

 2.6 Excrcises

3 Finite Extenslons

 3.1 Norm and Conorm

 3.2 Scalar Extensions

 3.3 The Different

 3.4 SingularPrimeDiVisors

 3.5 Galois Extensions

 3.6 Hyperelliptic Functtions

 3.7 Exercises

4 Projective Curves

 4.1 Proiective Varieties

 4.2 Maps to

 4.3 Projective Embeddings

 4.4 Weierstrass Points

 4.5 Plane Curves

 4.6 Exerciscs

5 Zeta Functions

 5.1 The Euler Product

 5.2 The Functional Equation

 5.3 The Riemann Hypothesis

 5.4 Exercises

A Elementary Field Theory

References

Index