PartIProblemsandTricks.
ElementaryNumberTheory
1.1ProblemsAboutPrimes.DivisibilityandPrimality
1.1.1ArithmeticalNotation
1.1.2Primesandcompositenumbers
1.1.3TheFactorizationTheoremandtheEuclidean
Algorithm
1.1.4CalculationswithResidueClasses
1.1.5TheQuadraticReciprocityLawandItsUse
1.1.6TheDistributionofPrimes
1.2DiophantineEquationsofDegreeOneandTwo
1.2.1TheEquationax+by=c
1.2.2LinearDiophantineSystems
1.2.3EquationsofDegreeTwo
1.2.4TheMinkowski-HassePrincipleforQuadraticForms
1.2.5Pell'sEquation
1.2.6RepresentationofIntegersandQuadraticFormsby
QuadraticForms
1.2.7AnalyticMethods
1.2.8EquivalenceofBinaryQuadraticForms
1.3CubicDiophantineEquations
1.3.1TheProblemoftheExistenceofaSolution
1.3.2AdditionofPointsonaCubicCurve
1.3.3TheStructureoftheGroupofRationalPointsofa
Non-SingularCubicCurve
1.3.4CubicCongruencesModuloaPrime
1.4ApproximationsandContinuedFractions
1.4.1BestApproximationstoIrrationalNumbers
1.4.2FareySeries
1.4.3ContinuedFractions
1.4.4SL2-Equivalence
1.4.5PeriodicContinuedFractionsandPell'sEquation
1.5DiophantineApproximationandtheIrrationality
1.5.1IdeasintheProofthat(3)isIrrational
1.5.2TheMeasureofIrrationalityofaNumber
1.5.3TheThue-Siegel-RothTheorem,Transcendental
Numbers,andDiophantineEquations
1.5.4ProofsoftheIdentities(1.5.1)and(1.5.2)
1.5.5TheRecurrentSequencesanandbn
1.5.6TranscendentalNumbersandtheSeventhHilbert
Problem
1.5.7WorkofYu.V.Nesterenkoon.e.[Nes99]
2SomeApplicationsofElementaryNumberTheory
2.1FactorizationandPublicKeyCryptosystems
2.1.1FactorizationisTime-Consuming
2.1.2One-WayFunctionsandPublicKeyEncryption
2.1.3APublicKeyCryptosystem
2.1.4StatisticsandMassProductionofPrimes
2.1.5ProbabilisticPrimalityTests
2.1.6TheDiscreteLogarithmProblemandThe
Diffie-HellmanKeyExchangeProtocol
2.1.7ComputingoftheDiscreteLogarithmonElliptic
CurvesoverFiniteFields(ECDLP)
2.2DeterministicPrimalityTests
2.2.1Adleman-Pomerance-RumelyPrimalityTest:BasicIdeas
2.2.2GaussSumsandTheirUseinPrimalityTesting.
2.2.3DetailedDescriptionofthePrimalityTest2.2.4PrimesisinP
2.2.5ThealgorithmofM.Agrawal,N.KayalandN.Saxena
2.2.6PracticalandTheoreticalPrimalityProving.The
ECPP(EllipticCurvePrimalityProvingbyF.Morain,see[AtMo93b])
2.2.7PrimesinArithmeticProgression
2.3FactorizationofLargeIntegers
2.3.1ComparativeDifficultyofPrimalityTestingand
Factorization
2.3.2FactorizationandQuadraticForms
2.3.3TheProbabilisticAlgorithmCLASNO
2.3.4TheContinuedFractionsMethod(CFRAC)andReal
QuadraticFields
2.3.5TheUseofEllipticCurves
PartIIIdeasandTheories
3InductionandRecursion
3.1ElementaryNumberTheoryFromthePointofViewofLogic
3.1.1ElementaryNumberTheory
3.1.2Logic
3.2DiophantineSets
3.2.1EnumerabilityandDiophantineSets
3.2.2Diophantinenessofenumerablesets
3.2.3FirstpropertiesofDiophantinesets
3.2.4DiophantinenessandPell'sEquation
3.2.5TheGraphoftheExponentisDiophantine
3.2.6DiophantinenessandBinomialcoefficients
3.2.7Binomialcoefficientsasremainders
3.2.8DiophantinenessoftheFactorial
3.2.9FactorialandEuclideanDivision
3.2.10SupplementaryResults
3.3PartiallyRecursiveFunctionsandEnumerableSets
3.3.1PartialFunctionsandComputableFunctions
3.3.2TheSimpleFunctions
3.3.3ElementaryOperationsonPartialfunctions
3.3.4PartiallyRecursiveDescriptionofaFunction
3.3.5OtherRecursiveFunctions
3.3.6FurtherPropertiesofRecursiveFunctions
3.3.7LinkwithLevelSets
3.3.8LinkwithProjectionsofLevelSets
3.3.9Matiyasevich'sTheorem
3.3.10Theexistenceofcertainbijections
3.3.11Operationsonprimitivelyenumerablesets
3.3.12GSdel'sfunction
3.3.13DiscussionofthePropertiesofEnumerableSets
3.4DiophantinenessofaSetandalgorithmicUndecidability
3.4.1Algorithmicundecidabilityandunsolvability
3.4.2SketchProofoftheMatiyasevichTheorem
Arithmeticofalgebraicnumbers
4.1AlgebraicNumbers:TheirRealizationsandGeometry
4.1.1AdjoiningRootsofPolynomials
4.1.2GaloisExtensionsandFrobeniusElements
4.1.3TensorProductsofFieldsandGeometricRealizations
ofAlgebraicNumbers
4.1.4Units,theLogarithmicMap,andtheRegulator
4.1.5LatticePointsinaConvexBody
4.1:6DeductionofDirichlet'sTheoremFromMinkowski'sLemma
4.2DecompositionofPrimeIdeals,DedekindDomains,andValuations
4.2.1PrimeIdealsandtheUniqueFactorizationProperty
4.2.2FinitenessoftheClassNumber..
4.2.3DecompositionofPrimeIdealsinExtensions
4.2.4Decompositionofprimesincyslotomicfields
4.2.5PrimeIdeals,ValuationsandAbsoluteValues
4..3LocalandGlobalMethods
4.3.1p-adicNumbers
4.3.2Applicationsofp-adicNumberstoSolvingCongruenca
4.3.3TheHilbertSymbol
4.3.4AlgebraicExtensionsofQp,andtheTateField
4.3.5NormalizedAbsoluteValues
4.3.6PlacesofNumberFieldsandtheProductFormula
4.3.7AdelesandIdeles
TheRingofAdeles
TheIdeleGroup
4.3.8TheGeometryofAdelesandIdeles
4.4ClassFieldTheory
4.4.1AbelianExtensionsoftheFieldofRationalNumbers
4.4.2FrobeniusAutomorphismsofNumberFieldsand
Artin'sReciprocityMap
4.4.3TheChebotarevDensityTheorem
4.4.4TheDecompositionLawand
theArtinReciprocityMap
4.4.5TheKerneloftheReciprocityMap
4.4.6TheArtinSymbol
4.4.7GlobalPropertiesoftheArtinSymbol
4.4.8ALinkBetweentheArtinSymbolandLocalSymbols
4.4.9PropertiesoftheLocalSymbol
4.4.10AnExplicitConstructionofAbelianExtensionsofa
LocalField,andaCalculationoftheLocalSymbol
4.4.11AbelianExtensionsofNumberFields
4.5GaloisGroupinArithetical'Problems
4.5.1Dividingacircleintonequalparts
4.5.2KummerExtensionsandthePowerResidueSymbol
4.5.3GaloisCohomology
4.5.4ACohomologicalDefinitionoftheLocalSymbol
4.5.5TheBrauerGroup,theReciprocityLawandthe
Minkowski-HassePrinciple 5Arithmeticofalgebraicvarieties 5.1ArithmeticVarietiesandBasicNotionsofAlgebraicGeometry
5.1.1EquationsandRings
5.1.2Thesetofsolutionsofasystem
5.1.3Example:TheLanguageofCongruences
5.1.4EquivalenceofSystemsofEquations
5.1.5SolutionsasK-algebraHomomorphisms
5.1.6TheSpectrumofARing
5.1.7RegularFunctions
5.1.8ATopologyonSpec(A)
5.1.9Schemes
5.1.10Ring-ValuedPointsofSchemes
5.1.11SolutionstoEquationsandPointsofSchemes
5.1.12Chevalley'sTheorem
5.1.13SomeGeometricNotions
5.2GeometricNotionsintheStudyofDiophantineequations
5.2.1BasicQuestions
5.2.2Geometricclassification
5.2.3ExistenceofRationalPointsandObstructionstothe
HassePrinciple
5.2.4FiniteandInfiniteSetsofSolutions
5.2.5Numberofpointsofboundedheight
5.2.6HeightandArakelovGeometry
5.3Ellipticcurves,AbelianVarieties,andLinearGroups
5.3.1AlgebraicCurvesandRiemannSurfaces
5.3.2EllipticCurves
5.3.3TareCurveandItsPointsofFiniteOrder
5.3.4TheMordell-WeilTheoremandGaloisCohomology
5.3.5AbelianVarietiesandJacobians
5.3.6TheJacobianofanAlgebraicCurve
5.3.7Siegel'sFormulaandTamagawa'Measure
5.4DiophantineEquationsandGaloisRepresentations
5.4.1TheTareModuleofanEllipticCurve
5.4.2TheTheoryofComplexMultiplication
5.4.3Charactersofl-adicRepresentations
5.4.4RepresentationsinPositiveCharacteristic
5.4.5TheTateModuleofaNumberField
5.5TheTheoremofFairingsandFinitenessProblemsinDiophantineGeometry
5.5.1ReductionoftheMordellConjecturetothefinitenessConjecture
5.5.2TheTheoremofShafarevichonFinitenessforEllipticCurves
5.5.3PassagetoAbelianvarieties
5.5.4FinitenessproblemsandTatesconjecture
5.5.5ReductionoftheconjecturesofTatetothefinitenesspropertiesorisogenies5.5.6TheFaltings-ArakelovHeight
5.5.7HeightsunderisogeniesandConjectureTZetaFunctionsandModularForms
6.1ZetaFunctionsofArithmeticSchemes
6.1.1ZetaFunctionsofArithmeticSchemes
6.1.2AnalyticContinuationoftheZetaFunctions
6.1.3SchemesoverFiniteFieldsandDeligne'sTheorem
6.1.4ZetaFunctionsandExponentialSums
6.2L-Functions,theTheoryofTateandExpliciteFormulae
6.2.1L-FunctionsofRationalGaloisRepresentations
6.2.2TheFormalismofArtin
6.2.3Example:TheDedekindZetaFunction
6.2.4HeckeCharactersandtheTheoryofTare
6.2.5ExplicitFormulae
6.2.6TheWeilGroupanditsRepresentations
6.2.7ZetaFunctions,L-FunctionsandMotives
6.3ModularFormsandEulerProducts
6.3.1ALinkBetweenAlgebraicVarietiesandL-Functions
6.3.2Classicalmodularforms
6.3.3Application:TateCurveandSemistableEllipticCurves
6.3.4Analyticfamiliesofellipticcurvesandcongruence
subgroups
6.3.5Modularformsforcongruencesubgroups
6.3.6HeckeTheory
6.3.7PrimitiveForms
6.3.8Weil'sInverseTheorem
6,4ModularFormsandGaloisRepresentations
6.4.1Ramanujan'scongruenceandGaloisRepresentations
6.4.2ALinkwithEichler-Shimura'sConstruction
6.4.3TheShimura-Taniyama-WeilConjecture
6.4.4TheConjectureofBirchandSwinnerton-Dyer
6.4.5TheArtinConjectureandCuspForms
TheArtinconductor
6.4.6ModularRepresentationsoverFiniteFields
6.5AutomorphicFormsandTheLanglandsProgram
6.5.1ARelationBetweenClassicalModularFormsand
RepresentationTheory
6.5.2AutomorphicL-Functions
FurtheranalyticpropertiesofautomorphicL-functions
6.5.3TheLanglandsFunctorialityPrinciple
6.5.4AutomorphicFormsandLanglandsConjectures
Fermat'sLastTheoremandFamiliesofModularForms
7.1Shimura-Taniyama-WeilConjectureandReciprocityLaws
7.1.1ProblemofPierredeFermat(1601-1665)
7.1.2G.Lam6'sMistake
7.1.3AshortoverviewofWiles'MarvelousProof
7.1.4TheSTWConjecture
7.1.5AconnectionwiththeQuadraticReciprocityLaw
7.1.6AcompleteproofoftheSTWconjecture
7.1.7Modularityofsemistableellipticcurves
7.1.8Structureoftheproofoftheorem7.13(Semistable
STWConjecture)
7.2TheoremofLanglands-Tunnelland
ModularityModulo3
7.2.1Galoisrepresentations:preparation
7.2.2Modularitymodulop
7.2.3Passagefromcuspformsofweightonetocuspforms
ofweighttwo..
7.2.4PreliminaryreviewofthestagesoftheproofofTheorem7.13onmodularity
7.3ModularityofGaloisrepresentationsandUniversalDeformationRings
7.3.1GaloisRepresentationsoverlocalNoetherianalgebras
7.3.2DeformationsofGaloisRepresentations
7.3.3ModularGaloisrepresentations
7.3.4AdmissibleDeformationsandModularDeformations
7.3.5UniversalDeformationRings
7.4Wiles'MainTheoremandIsomorphismCriteriaforLocalRings
7.4.1StrategyoftheproofoftheMainTheorem7.33
7.4.2Surjectivityof
7.4.3Constructionsoftheuniversaldeformationring
7.4.4AsketchofaconstructionoftheuniversalmodulardeformationringT2
7.4.5UniversalityandtheChebotarevdensitytheorem
7.4.6IsomorphismCriteriaforlocalrings
7.4.7J-structuresandthesecondcriterionofisomorphismoflocalrings
7.5Wiles'InductionStep:ApplicationoftheCriteriaandGaloisCohomology
7.5.1Wiles'inductionstepintheproofofMainTheorem7.33
7.5.2Aformularelatingpreparation
7.5.3TheSelmergroupand
7.5.4Infinitesimaldeformations
7.5.5Deformationsoftype
7.6TheRelativeInvariant,theMainInequalityandTheMinimalCase
7.6.1TheRelativeinvariant
7.6.2TheMainInequality
7.6.3TheMinimalCase
7.7EndofWiles'ProofandTheoremonAbsoluteIrreducibility
7.7.1TheoremonAbsoluteIrreducibility
7.7.2Fromp=3top=5
7.7.3FamiliesofellipticcurveswithfixedE
7.7.4TheendoftheproofThemostimportantinsightsPartIIIAnalogiesandVisionsIII-0IntroductorysurveytopartIII:motivationsanddescription
III.1Analogiesanddifferencesbetweennumbersandfunctions:point,Archimedeanpropertiesetc
III.1.1Cauchyresidueformulaandtheproductformula
III.l.2Arithmeticvarieties
III.l.3Infinitesimalneighborhoodsoffibers
III,.2Arakelovgeometry,fiberovercycles,Greenfunctions
(d'apresGillet-Soule)
III.2.1ArithmeticChowgroups
III.2.2Arithmeticintersectiontheoryandarithmetic
Riemann-Rochtheorem
III.2.3Geometricdescriptionoftheclosedfibersatinfinity
III.3-functions,localfactorsatSerre'sF-factors
III.3.1ArchimedeanL-factors
III.3.2Deninger'sformulae
III.4Aguessthatthemissinggeometricobjectsare
noncommutativespaces
III.4.1Typesandexamplesofnoncommutativespaces,and
howtoworkwiththem.Noncommutativegeometry
andarithmetic
IsomorphismofnoncommutativespacesandMoritaequivalence
Thetoolsofnoncommutativegeometry
III.4.2Generalitiesonspectraltriples
III.4.3ContentsofPartIII:descriptionofpartsofthisprogram
8ArakelovGeometryandNoncommutativeGeometry
8.1SchottkyUniformizationandArakelovGeometry
8.1.1Motivationsandthecontextoftheworkof
Consani-Marcolli
8.1.2Analyticconstructionofdegeneratingcurvesover
completelocalfields
8.1.3SchottkygroupsandnewperspectivesinArakelov
geometry
SchottkyuniformizationandSchottkygroups
FuchsianandSchottkyuniformization
8.1.4Hyperbolichandlebodies
Geodesicsinr
8.1.5Arakelovgeometryandhyperbolicgeometry
ArakelovGreenfunction
Crossratioandgeodesics
DifferentialsandSchottkyuniformization
Greenfunctionandgeodesics
8.2CohomologicalConstructions
8.2.1Archimedeancohomology
Operators
SL(2,R)representatious
8.2.2LocalfactorandArchimedeancohomology
8.2.3Cohomologicalconstructions
8.2.4ZetafunctionofthespecialfiberandReidemeister
torsion
8.3SpectralTriples,DynamicsandZetaFunctions
8.3.1Adynamicaltheoryatinfinity
8.3.2Homotopyquotion
8.3.3Filtration
8.3.4Hilbertspaceandgrading
8.3.5Cuntz-Kriegeralgebra
SpectraltriplesforSchottkygroups
8.3.6Arithmeticsurfaces:homologyandcohomology
8.3.7Archimedeanfactorsfromdynamics
8.3.8ADynamicaltheoryforMumfordcurves
Genustwoexample
8.3.9Cohomologyof
8.3.10SpectraltriplesandMumfordcurves
8.4Reductionmod
8.4.1Homotopyquotientsand"reductionmodinfinity"
8.4.2Baum-Connesmap
References
Index...
