現(xiàn)代數(shù)論導(dǎo)引

內(nèi)容概要

本書(shū)以統(tǒng)一的觀點(diǎn)概述數(shù)論的現(xiàn)狀及其不同分支的發(fā)展趨勢(shì)，由基本問(wèn)題出發(fā)，揭示現(xiàn)代數(shù)論的中心思想。主要論題包括類域論的非-Abel-般化、遞歸計(jì)算、丟番圖方程、Zeta-函數(shù)和L-函數(shù)。    本書(shū)新版作了大量修訂，內(nèi)容上也作了擴(kuò)充，增加了一些新的章節(jié)，如懷爾斯對(duì)費(fèi)馬大定理的證明，綜合不同理論而得到的現(xiàn)代數(shù)論的相關(guān)技巧。此外，作者還專門(mén)增加一章，講述算術(shù)上同調(diào)和非交換幾何，關(guān)于具有多個(gè)有理點(diǎn)的簇中點(diǎn)的計(jì)數(shù)問(wèn)題的一個(gè)報(bào)告，質(zhì)數(shù)判定中的多項(xiàng)式時(shí)間算法以及其他論題。

書(shū)籍目錄

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 IntegersPart 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 IrreducibilityPart 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 8ReferencesIndex

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