中文名: 現代數論導引
原名: Introduction to Modern Number Theory
作者: (俄)Yu.I.Manin,Alexei A.Panchishkin
資源格式: PDF
版本: 第2版
出版社: Springer Berlin Heidelberg
書號: 3642057977
發行時間: 2010年
地區: 美國
語言: 英文
簡介:

內容簡介:
本書以統一的觀點概述數論的現狀及其不同分支的發展趨勢，由基本問題出發，揭示現代數論的中心思想。主要論題包括類域論的非—bel一般化、遞歸計算、丟番圖方程、Zeta—函數和L-函數。
本書新版作了大量修訂，內容上也作了擴充，增加了一些新的章節，如懷爾斯對費馬大定理的證明，綜合不同理論而得到的現代數論的相關技巧。此外，作者還專門增加一章，講述算術上同調和非交換幾何，關於具有多個有理點的簇中點的計數問題的一個報告，質數判定中的多項式時間算法以及其他論題。
內容截圖:

目錄:
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