[内容简介]
本书以统一的观点概述数论的现状及其不同分支的发展趋势，由基本问题出发，揭示现代数论的中心思想。主要论题包括类域论的非-Abel-般化、递归计算、丢番图方程、Zeta-函数和L-函数。
　　本书新版作了大量修订，内容上也作了扩充，增加了一些新的章节，如怀尔斯对费马大定理的证明，综合不同理论而得到的现代数论的相关技巧。此外，作者还专门增加一章，讲述算术上同调和非交换几何，关于具有多个有理点的簇中点的计数问题的一个报告，质数判定中的多项式时间算法以及其他论题。
[目次]
Part I Problems and Tricks
1 Elementary Number Theory
1.1 Problems About Primes. Divisibility and Primality
1.2 Diophantine Equations of Degree One and Two
1.3 Cubic Diophantine Equations
1.4 Approximations and Continued Fractions
1.5 Diophantine Approximation and the Irrationality
2 Some Applications of Elementary Number Theory
2.1 Factorization and Public Key Cryptosystems
2.2 Deterministic Primality Tests
2.3 Factorization of Large Integers
Part II Ideas and Theories
3 Induction and Recursion
3.1 Elementary Number Theory From the Point of View of Logic
3.2 Diophantine Sets
3.3 Partially Recursive Functions and Enumerable Sets
3.4 Diophantineness of a Set and algorithmic Undecidability
4 Arithmetic of algebraic numbers
4.1 Algebraic Numbers: Their Realizations and Geometry
4.2 Decomposition of Prime Ideals, Dedekind Domains, and Valuations
4.3 Local and Global Methods
4.4 Class Field Theory
4.5 Galois Group in Arithetical Problems
5 Arithmetic of algebraic varieties
5.1 Arithmetic Varieties and Basic Notions of Algebraic Geometry
5.2 Geometric Notions in the Study of Diophantine equations
5.3 Elliptic curves, Abelian Varieties, and Linear Groups
5.4 Diophantine Equations and Galois Repressentations
5.5 The Theorem of Faltings and Finiteness Problems in Diophantine Geometry
6 Zeta Functions and Modular Forms
6.1 Zeta Functions of Arithmetic Schemes
6.2 L-Functions, the Theory of Tate and Explicite Formulae
6.3 Modular Forms and Euler Products
6.4 Modular Forms and Galois Representations
6.5 Automorphic Forms and The Langlands Program
7 Fermat's Last Theorem and Families of Modular Forms
7.1 Shimura-Taniyama-Weil Conjecture and Reciprocity Laws
7.2 Theorem of Langlands-Tunnell and Modularity Modulo 3
7.3 Modularity of Galois representations and Universal Deformation Rings
7.4 Wiles' Main Theorem and Isomorphism Criteria for Local Rings
7.5 Wiles' Induction Step: Application of the Criteria and Galois Cohomology
7.6 The Relative Invariant, the Main Inequality and The Minimal Case
7.7 End of Wiles' Proof and Theorem on Absolute Irreducibility
Part III Analogies and Visions
III-0 Introductory survey to part III: motivations and description
III.1 Analogies and differences between numbers and functions: 8-point, Archimedean properties etc.
III.2 Arakelov geometry, fiber over 8, cycles, Green functions (d'apres Gillet-Soule)
III.3 -functions, local factors at 8, Serre's T-factors
III.4 A guess that the missing geometric objects are noncommutative spaces
8 Arakelov Geometry and Noncommutative Geometry
8.1 Schottky Uniformization and Arakelov Geometry
8.2 Cohomological Constructions
8.3 Spectral Triples, Dynamics and Zeta Functions
8.4 Reduction mod 8
References
Index