现代数论导引(第2版影印版)(精)/国外数学名著系列

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。

　　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。

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

要使我国的数学事业更好地发展起来，需要数学家淡泊名利并付出更艰苦地努力。另一方面，我们也要从客观上为数学家创造更有利的发展数学事业的外部环境，这主要是加强对数学事业的支持与投资力度，使数学家有较好的工作与生活条件，其中也包括改善与加强数学的出版工作。... 从出版方面来讲，除了较好较快地出版我们自己的成果外，引进国外的先进出版物无疑也是十分重要与必不可少的。从数学来说，施普林格(Springer)出版社至今仍然是世界上最具权威的出版社。科学出版社影印一批他们出版的好的新书，使我国广大数学家能以较低的价格购买，特别是在边远地区工作的数学家能普遍见到这些书，无疑是对推动..