国外数学名著系列 3:现代数论导引

图书标准信息

• 作者 [俄罗斯]马宁（Manin Yu.I.） 著
• 出版社 科学出版社
• 出版时间 2006-09
• 版次 1
• ISBN 9787030166876
• 定价 78.00元
• 装帧 精装
• 开本 16开
• 纸张 胶版纸
• 页数 536页
• 字数 630千字

【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。

【目录】

PartIProblemsandTricks
1ElementaryNumberTheory
1.1ProblemsAboutPrimes.DivisibilityandPrimality
1.2DiophantineEquationsofDegreeOneandTwo
1.3CubicDiophantineEquations
1.4ApproximationsandContinuedFractions
1.5DiophantineApproximationandtheIrrationality
2SomeApplicationsofElementaryNumberTheory
2.1FactorizationandPublicKeyCryptosystems
2.2DeterministicPrimalityTests
2.3FactorizationofLargeIntegers
PartIIIdeasandTheories
3InductionandRecursion
3.1ElementaryNumberTheoryFromthePointofViewofLogic
3.2DiophantineSets
3.3PartiallyRecursiveFunctionsandEnumerableSets
3.4DiophantinenessofaSetandalgorithmicUndecidability
4Arithmeticofalgebraicnumbers
4.1AlgebraicNumbers:TheirRealizationsandGeometry
4.2DecompositionofPrimeIdeals,DedekindDomains,andValuations
4.3LocalandGlobalMethods
4.4ClassFieldTheory
4.5GaloisGroupinAritheticalProblems
5Arithmeticofalgebraicvarieties
5.1ArithmeticVarietiesandBasicNotionsofAlgebraicGeometry
5.2GeometricNotionsintheStudyofDiophantineequations
5.3Ellipticcurves,AbelianVarieties,andLinearGroups
5.4DiophantineEquationsandGaloisRepressentations
5.5TheTheoremofFaltingsandFinitenessProblemsinDiophantineGeometry
6ZetaFunctionsandModularForms
6.1ZetaFunctionsofArithmeticSchemes
6.2L-Functions,theTheoryofTateandExpliciteFormulae
6.3ModularFormsandEulerProducts
6.4ModularFormsandGaloisRepresentations
6.5AutomorphicFormsandTheLanglandsProgram
7FermatsLastTheoremandFamiliesofModularForms
7.1Shimura-Taniyama-WeilConjectureandReciprocityLaws
7.2TheoremofLanglands-TunnellandModularityModulo3
7.3ModularityofGaloisrepresentationsandUniversalDeformationRings
7.4WilesMainTheoremandIsomorphismCriteriaforLocalRings
7.5WilesInductionStep:ApplicationoftheCriteriaandGaloisCohomology
7.6TheRelativeInvariant,theMainInequalityandTheMinimalCase
7.7EndofWilesProofandTheoremonAbsoluteIrreducibility
PartIIIAnalogiesandVisions
III-0IntroductorysurveytopartIII:motivationsanddescription
III.1Analogiesanddifferencesbetweennumbersandfunctions:8-point,Archimedeanpropertiesetc.
III.2Arakelovgeometry,fiberover8,cycles,Greenfunctions(dapresGillet-Soule)
III.3-functions,localfactorsat8,SerresT-factors
III.4Aguessthatthemissinggeometricobjectsarenoncommutativespaces
8ArakelovGeometryandNoncommutativeGeometry
8.1SchottkyUniformizationandArakelovGeometry
8.2CohomologicalConstructions
8.3SpectralTriples,DynamicsandZetaFunctions
8.4Reductionmod8
References
Index