Introduction to Modern Number Theory
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1 Encyclopaedia of Mathematical Sciences 49 Introduction to Modern Number Theory Fundamental Problems, Ideas and Theories Bearbeitet von Yu. I Manin, Alexei A Panchishkin Neuausgabe Buch. xvi, 514 S. Hardcover ISBN Format (B x L): 15,5 x 23,5 cm Gewicht: 2020 g Weitere Fachgebiete > EDV, Informatik > Datenbanken, Informationssicherheit, Geschäftssoftware > Informations- und Kodierungstheorie Zu Leseprobe schnell und portofrei erhältlich bei Die Online-Fachbuchhandlung beck-shop.de ist spezialisiert auf Fachbücher, insbesondere Recht, Steuern und Wirtschaft. Im Sortiment finden Sie alle Medien (Bücher, Zeitschriften, CDs, ebooks, etc.) aller Verlage. Ergänzt wird das Programm durch Services wie Neuerscheinungsdienst oder Zusammenstellungen von Büchern zu Sonderpreisen. Der Shop führt mehr als 8 Millionen Produkte.
2 Contents Part I Problems and Tricks 1 Elementary Number Theory Problems About Primes. Divisibility and Primality ArithmeticalNotation Primesandcompositenumbers The Factorization Theorem and the Euclidean Algorithm CalculationswithResidueClasses TheQuadraticReciprocityLawandItsUse TheDistributionofPrimes DiophantineEquationsofDegreeOneandTwo The Equation ax + by = c LinearDiophantineSystems EquationsofDegreeTwo The Minkowski Hasse Principle for Quadratic Forms Pell s Equation Representation of Integers and Quadratic Forms by QuadraticForms AnalyticMethods EquivalenceofBinaryQuadraticForms CubicDiophantineEquations TheProblemoftheExistenceofaSolution AdditionofPointsonaCubicCurve The Structure of the Group of Rational Points of a Non SingularCubicCurve CubicCongruencesModuloaPrime ApproximationsandContinuedFractions BestApproximationstoIrrationalNumbers FareySeries ContinuedFractions... 51
3 VIII Contents SL 2 Equivalence Periodic Continued Fractions and Pell s Equation DiophantineApproximationandtheIrrationality Ideas in the Proof that ζ(3) isirrational TheMeasureofIrrationalityofaNumber The Thue Siegel Roth Theorem, Transcendental Numbers,andDiophantineEquations ProofsoftheIdentities(1.5.1)and(1.5.2) The Recurrent Sequences a n and b n Transcendental Numbers and the Seventh Hilbert Problem Work of Yu.V. Nesterenko on e π,[nes99] Some Applications of Elementary Number Theory FactorizationandPublicKeyCryptosystems FactorizationisTime-Consuming One WayFunctionsandPublicKeyEncryption APublicKeyCryptosystem StatisticsandMassProductionofPrimes Probabilistic Primality Tests The Discrete Logarithm Problem and The Diffie-HellmanKeyExchangeProtocol Computing of the Discrete Logarithm on Elliptic CurvesoverFiniteFields(ECDLP) DeterministicPrimalityTests Adleman Pomerance Rumely Primality Test: Basic Ideas GaussSumsandTheirUseinPrimalityTesting DetailedDescriptionofthePrimalityTest PrimesisinP The algorithm of M. Agrawal, N. Kayal and N. Saxena Practical and Theoretical Primality Proving. The ECPP (Elliptic Curve Primality Proving by F.Morain, see [AtMo93b]) PrimesinArithmeticProgression FactorizationofLargeIntegers Comparative Difficulty of Primality Testing and Factorization FactorizationandQuadraticForms The Probabilistic Algorithm CLASNO The Continued Fractions Method (CFRAC) and Real QuadraticFields The Use of Elliptic Curves
4 Contents IX Part II Ideas and Theories 3 Induction and Recursion Elementary Number Theory From the Point of View of Logic ElementaryNumberTheory Logic DiophantineSets Enumerability and Diophantine Sets Diophantinenessofenumerablesets FirstpropertiesofDiophantinesets Diophantineness and Pell s Equation TheGraphoftheExponentisDiophantine DiophantinenessandBinomialcoefficients Binomialcoefficientsasremainders DiophantinenessoftheFactorial FactorialandEuclideanDivision Supplementary Results PartiallyRecursiveFunctionsandEnumerableSets Partial Functions and Computable Functions TheSimpleFunctions ElementaryOperationsonPartialfunctions Partially Recursive Description of a Function OtherRecursiveFunctions FurtherPropertiesofRecursiveFunctions LinkwithLevelSets LinkwithProjectionsofLevelSets Matiyasevich stheorem Theexistenceofcertainbijections Operationsonprimitivelyenumerablesets Gödel sfunction DiscussionofthePropertiesofEnumerableSets Diophantineness of a Set and algorithmic Undecidability Algorithmic undecidability and unsolvability SketchProofoftheMatiyasevichTheorem Arithmetic of algebraic numbers Algebraic Numbers: Their Realizations and Geometry AdjoiningRootsofPolynomials GaloisExtensionsandFrobeniusElements Tensor Products of Fields and Geometric Realizations ofalgebraicnumbers Units, the Logarithmic Map, and the Regulator LatticePointsinaConvexBody...123
5 X Contents Deduction of Dirichlet s Theorem From Minkowski s Lemma Decomposition of Prime Ideals, Dedekind Domains, and Valuations Prime Ideals and the Unique Factorization Property FinitenessoftheClassNumber DecompositionofPrimeIdealsinExtensions Decompositionofprimesincyslotomicfields PrimeIdeals,ValuationsandAbsoluteValues LocalandGlobalMethods p adicnumbers Applications of p adic Numbers to Solving Congruences TheHilbertSymbol Algebraic Extensions of Q p,andthetatefield NormalizedAbsoluteValues Places of Number Fields and the Product Formula AdelesandIdeles TheRingofAdeles TheIdeleGroup TheGeometryofAdelesandIdeles ClassFieldTheory Abelian Extensions of the Field of Rational Numbers Frobenius Automorphisms of Number Fields and Artin sreciprocitymap TheChebotarevDensityTheorem The Decomposition Law and theartinreciprocitymap TheKerneloftheReciprocityMap TheArtinSymbol GlobalPropertiesoftheArtinSymbol A Link Between the Artin Symbol and Local Symbols PropertiesoftheLocalSymbol An Explicit Construction of Abelian Extensions of a Local Field, and a Calculation of the Local Symbol AbelianExtensionsofNumberFields GaloisGroupinAritheticalProblems Dividing a circle into n equalparts Kummer Extensions and the Power Residue Symbol GaloisCohomology ACohomologicalDefinitionoftheLocalSymbol The Brauer Group, the Reciprocity Law and the Minkowski HassePrinciple...184
6 Contents XI 5 Arithmetic of algebraic varieties Arithmetic Varieties and Basic Notions of Algebraic Geometry EquationsandRings Thesetofsolutionsofasystem Example:TheLanguageofCongruences EquivalenceofSystemsofEquations Solutions as K-algebraHomomorphisms TheSpectrumofARing RegularFunctions A Topology on Spec(A) Schemes Ring-ValuedPointsofSchemes Solutions to Equations and Points of Schemes Chevalley stheorem SomeGeometricNotions Geometric Notions in the Study of Diophantine equations BasicQuestions Geometricclassification Existence of Rational Points and Obstructions to the HassePrinciple Finite and Infinite Sets of Solutions Number of points of bounded height HeightandArakelovGeometry Elliptic curves, Abelian Varieties, and Linear Groups AlgebraicCurvesandRiemannSurfaces Elliptic Curves TateCurveandItsPointsofFiniteOrder The Mordell Weil Theorem and Galois Cohomology AbelianVarietiesandJacobians TheJacobianofanAlgebraicCurve Siegel sformulaandtamagawameasure DiophantineEquationsandGaloisRepresentations The Tate Module of an Elliptic Curve TheTheoryofComplexMultiplication Characters of l-adicrepresentations RepresentationsinPositiveCharacteristic TheTateModuleofaNumberField The Theorem of Faltings and Finiteness Problems in DiophantineGeometry Reduction of the Mordell Conjecture to the finiteness Conjecture The Theorem of Shafarevich on Finiteness for Elliptic Curves PassagetoAbelianvarieties FinitenessProblemsandTate sconjecture...252
7 XII Contents Reduction of the conjectures of Tate to the finiteness propertiesforisogenies TheFaltings ArakelovHeight Heights under isogenies and Conjecture T Zeta Functions and Modular Forms ZetaFunctionsofArithmeticSchemes ZetaFunctionsofArithmeticSchemes AnalyticContinuationoftheZetaFunctions Schemes over Finite Fields and Deligne s Theorem ZetaFunctionsandExponentialSums L-Functions,theTheoryofTateandExplicitFormulae L-FunctionsofRationalGaloisRepresentations TheFormalismofArtin Example:TheDedekindZetaFunction HeckeCharactersandtheTheoryofTate ExplicitFormulae TheWeilGroupanditsRepresentations Zeta Functions, L-Functions andmotives ModularFormsandEulerProducts A Link Between Algebraic Varieties and L Functions Classicalmodularforms Application: Tate Curve and Semistable Elliptic Curves Analytic families of elliptic curves and congruence subgroups Modularformsforcongruencesubgroups HeckeTheory PrimitiveForms Weil s Inverse Theorem ModularFormsandGaloisRepresentations Ramanujan s congruence and Galois Representations ALinkwithEichler Shimura sconstruction TheShimura Taniyama WeilConjecture TheConjectureofBirchandSwinnerton Dyer TheArtinConjectureandCuspForms The Artin conductor ModularRepresentationsoverFiniteFields AutomorphicFormsandTheLanglandsProgram A Relation Between Classical Modular Forms and RepresentationTheory Automorphic L-Functions Further analytic properties of automorphic L-functions TheLanglandsFunctorialityPrinciple AutomorphicFormsandLanglandsConjectures...339
8 Contents XIII 7 Fermat s Last Theorem and Families of Modular Forms Shimura Taniyama Weil Conjecture and Reciprocity Laws Problem of Pierre de Fermat ( ) G.Lamé smistake A short overview of Wiles Marvelous Proof TheSTWConjecture A connection with the Quadratic Reciprocity Law AcompleteproofoftheSTWconjecture Modularity of semistable elliptic curves Structure of the proof of theorem 7.13 (Semistable STWConjecture) Theorem of Langlands-Tunnell and ModularityModulo Galoisrepresentations:preparation Modularity modulo p Passage from cusp forms of weight one to cusp forms ofweighttwo Preliminary review of the stages of the proof of Theorem7.13onmodularity Modularity of Galois representations and Universal DeformationRings Galois Representations over local Noetherian algebras DeformationsofGaloisRepresentations ModularGaloisrepresentations Admissible Deformations and Modular Deformations UniversalDeformationRings Wiles Main Theorem and Isomorphism Criteria for Local Rings Strategy of the proof of the Main Theorem Surjectivity of ϕ Σ Constructions of the universal deformation ring R Σ A sketch of a construction of the universal modular deformation ring T Σ Universality and the Chebotarev density theorem IsomorphismCriteriaforlocalrings J structures and the second criterion of isomorphism oflocalrings Wiles Induction Step: Application of the Criteria and Galois Cohomology Wiles induction step in the proof of MainTheorem A formula relating #Φ RΣ and #Φ RΣ : preparation The Selmer group and Φ RΣ Infinitesimal deformations Deformations of type D Σ...377
9 XIV Contents 7.6 The Relative Invariant, the Main Inequality and The Minimal Case TheRelativeinvariant TheMainInequality TheMinimalCase End of Wiles Proof and Theorem on Absolute Irreducibility Theorem on Absolute Irreducibility From p =3to p = Families of elliptic curves with fixed ρ 5,E Theendoftheproof Themostimportantinsights Part III Analogies and Visions III-0 Introductory survey to part III: motivations and description III.1 Analogies and differences between numbers and functions: -point,archimedeanpropertiesetc III.1.1 Cauchyresidueformulaandtheproductformula III.1.2 Arithmeticvarieties III.1.3 Infinitesimal neighborhoods of fibers III.2 Arakelov geometry, fiber over, cycles, Green functions (d après Gillet-Soulé) III.2.1 ArithmeticChowgroups III.2.2 Arithmetic intersection theory and arithmetic Riemann-Rochtheorem III.2.3 Geometric description of the closed fibers at infinity III.3 ζ-functions, local factors at, Serre s Γ -factors III.3.1 Archimedean L-factors III.3.2 Deninger sformulae III.4 A guess that the missing geometric objects are noncommutativespaces III.4.1 Types and examples of noncommutative spaces, and how to work with them. Noncommutative geometry andarithmetic Isomorphism of noncommutative spaces and Morita equivalence Thetoolsofnoncommutativegeometry III.4.2 Generalitiesonspectraltriples III.4.3 Contents of Part III: description of parts of this program412
10 Contents XV 8 Arakelov Geometry and Noncommutative Geometry SchottkyUniformizationandArakelovGeometry Motivations and the context of the work of Consani-Marcolli Analytic construction of degenerating curves over completelocalfields Schottky groups and new perspectives in Arakelov geometry Schottky uniformization and Schottky groups FuchsianandSchottkyuniformization Hyperbolichandlebodies Geodesics in X Γ Arakelovgeometryandhyperbolicgeometry ArakelovGreenfunction Crossratioandgeodesics DifferentialsandSchottkyuniformization Greenfunctionandgeodesics CohomologicalConstructions Archimedeancohomology Operators SL(2, R) representations LocalfactorandArchimedeancohomology Cohomologicalconstructions Zeta function of the special fiber and Reidemeister torsion SpectralTriples,DynamicsandZetaFunctions A dynamical theory at infinity Homotopyquotion Filtration Hilbertspaceandgrading Cuntz Kriegeralgebra SpectraltriplesforSchottkygroups Arithmetic surfaces: homology and cohomology Archimedeanfactorsfromdynamics ADynamicaltheoryforMumfordcurves Genustwoexample Cohomology of W(Δ/Γ ) T SpectraltriplesandMumfordcurves Reduction mod Homotopy quotients and reduction mod infinity Baum-Connesmap References Index...503
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