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
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