.

0 (hodnocen0 x )

BK

2nd ed.

Heidelberg : Springer, c2005

XV,514 s.

ISBN 3-540-20364-8 (váz.)

Number theory ; 1

Encyclopaedia of mathematical sciences ; 49

Obsahuje předmluvu, rejstřík

Bibliografie: s. 461-502

Čísla - teorie čísel - příručky

000041371

Part I Problems and Tricks // 1 Elementary Number Theory 9 // 1.1 Problems About Primes. Divisibility and Primality 9 // 1.1.1 Arithmetical Notation 9 // 1.1.2 Primes and composite numbers 10 // 1.1.3 The Factorization Theorem and the Euclidean // Algorithm 12 // 1.1.4 Calculations with Residue Classes 13 // 1.1.5 The Quadratic Reciprocity Law and Its Use 15 // 1.1.6 The Distribution of Primes 17 // 1.2 Diophantine Equations of Degree One and Two 22 // 1.2.1 The Equation ax + by = c 22 // 1.2.2 Linear Diophantine Systems 22 // 1.2.3 Equations of Degree Two 24 // 1.2.4 The Minkowski-Hasse Principle for Quadratic Forms. .. 26 // 1.2.5 Pell’s Equation 28 // 1.2.6 Representation of Integers and Quadratic Forms by // Quadratic Forms 29 // 1.2.7 Analytic Methods 33 // 1.2.8 Equivalence of Binary Quadratic Forms 35 // 1.3 Cubic Diophantine Equations 38 // 1.3.1 The Problem of the Existence of a Solution 38 // 1.3.2 Addition of Points on a Cubic Curve 38 // 1.3.3 The Structure of the Group of Rational Points of a // Non-Singular Cubic Curve 40 // 1.3.4 Cubic Congruences Modulo a Prime 47 // 1.4 Approximations and Continued Fractions 50 // 1.4.1 Best Approximations to Irrational Numbers 50 // 1.4.2 Farey Series 50 // 1.4.3 Continued Fractions 51 // Vili Contents // 1.4.4 SL2-Equivalence 53 // 1.4.5 Periodic Continued Fractions and Pell’s Equation 53 // 1.5 Diophantine Approximation and the Irrationality 55 // 1.5.1 Ideas in the Proof that Ł(3) is Irrational 55 // 1.5.2 The Measure of Irrationality of a Number 56 // 1.5.3 The Thue-Siegel-Roth Theorem, Transcendental // Numbers, and Diophantine Equations 57 // 1.5.4 Proofs of the Identities (1.5.1) and (1.5.2) 58 // 1.5.5 The Recurrent Sequences an and bn 59 // 1.5.6 Transcendental Numbers and the Seventh Hilbert // Problem 61 // 1.5.7 Work of Yu.V. Nesterenko on e71", [Nes99] 61 //

2 Some Applications of Elementary Number Theory 63 // 2.1 Factorization and Public Key Cryptosystems 63 // 2.1.1 Factorization is Time-Consuming 63 // 2.1.2 One-Way Functions and Public Key Encryption 63 // 2.1.3 A Public Key Cryptosystem 64 // 2.1.4 Statistics and Mass Production of Primes 66 // 2.1.5 Probabilistic Primality Tests 66 // 2.1.6 The Discrete Logarithm Problem and The , // Diffie-Hellman Key Exchange Protocol 67 // 2.1.7 Computing of the Discrete Logarithm on Elliptic // Curves over Finite Fields (ECDLP) 68 // 2.2 Deterministic Primality Tests 69 // 2.2.1 Adleman-Pomerance-Rumely Primality Test: Basic // Ideas 69 // 2.2.2 Gauss Sums and Their Use in Primality Testing 71 // 2.2.3 Detailed Description of the Primality Test 75 // 2.2.4 Primes is in P 78 // 2.2.5 The algorithm of M. Agrawal, N. Kayal and N. Saxena . 81 // 2.2.6 Practical and Theoretical Primality Proving. The ECPP (Elliptic Curve Primality Proving by F.Morain, see [AtMo93b]) 81 // 2.2.7 Primes in Arithmetic Progression 82 // 2.3 Factorization of Large Integers 84 // 2.3.1 Comparative Difficulty of Primality Testing and // Factorization 84 // 2.3.2 Factorization and Quadratic Forms 84 // 2.3.3 The Probabilistic Algorithm CLASNO 85 // 2.3.4 The Continued Fractions Method (CFRAC) and Real // Quadratic Fields 87 // 2.3.5 The Use of Elliptic Curves 90 // Contents // IX // part II Ideas and Theories // 3 induction and Recursion 95 // 3 i Elementary Number Theory From the Point of View of Logic . 95 // 3.1.1 Elementary Number Theory 95 // 3.1.2 Logic 96 // 3.2 Diophantine Sets 98 // 3.2.1 Enumerability and Diophantine Sets 98 // 3.2.2 Diophantineness of enumerable sets 98 // 3.2.3 First properties of Diophantine sets 98 // 3.2.4 Diophantineness and Pell’s Equation 99 // 3.2.5 The Graph of the Exponent is Diophantine 100 // 3.2.6 Diophantineness and Binomial coefficients 100 //

3.2.7 Binomial coefficients as remainders 101 // 3.2.8 Diophantineness of the Factorial 101 // 3.2.9 Factorial and Euclidean Division 101 // 3.2.10 Supplementary Results 102 // 3.3 Partially Recursive Functions and Enumerable Sets 103 // 3.3.1 Partial Functions and Computable Functions 103 // 3.3.2 The Simple Functions 103 // 3.3.3 Elementary Operations on Partial functions 103 // 3.3.4 Partially Recursive Description of a Function 104 // 3.3.5 Other Recursive Functions 106 // 3.3.6 Further Properties of Recursive Functions 108 // 3.3.7 Link with Level Sets 108 // 3.3.8 Link with Projections of Level Sets 108 // 3.3.9 Matiyasevich’s Theorem 109 // 3.3.10 The existence of certain bijections 109 // 3.3.11 Operations on primitively enumerable sets Ill // 3.3.12 Gôdels function ? // 3.3.13 Discussion of the Properties of Enumerable Sets 112 // 3.4 Diophantineness of a Set and algorithmic Undecidability 113 // 3.4.1 Algorithmic undecidability and unsolvability 113 // 3.4.2 Sketch Proof of the Matiyasevich Theorem 113 // 4 Arithmetic of algebraic numbers 115 // 4.1 Algebraic Numbers: Their Realizations and Geometry 115 // 4.1.1 Adjoining Roots of Polynomials 115 // 4.1.2 Galois Extensions and Frobenius Elements 117 // 4.1.3 Tensor Products of Fields and Geometric Realizations // of Algebraic Numbers 119 // 4.1.4 Units, the Logarithmic Map, and the Regulator 121 // 4.1.5 Lattice Points in a Convex Body 123 // X Contents // 4.1.6 Deduction of Dirichlets Theorem Prom Minkowski’s // Lemma 125 // 4.2 Decomposition of Prime Ideals, Dedekind Domains, and // Valuations 126 // 4.2.1 Prime Ideals and the Unique Factorization Property 126 // 4.2.2 Finiteness of the Class Number 128 // 4.2.3 Decomposition of Prime Ideals in Extensions 129 // 4.2.4 Decomposition of primes in cyslotomic fields 131 // 4.2.5 Prime Ideals, Valuations and Absolute Values 132 //

4.3 Local and Global Methods 134 // 4.3.1 p-adic Numbers 134 // 4.3.2 Applications of p-adic Numbers to Solving Congruences 138 // 4.3.3 The Hilbert Symbol 139 // 4.3.4 Algebraic Extensions of Qp, and the Tate Field 142 // 4.3.5 Normalized Absolute Values 143 // 4.3.6 Places of Number Fields and the Product Formula 145 // 4.3.7 Adeles and Ideles 146 // The Ring of Adeles 146 // The Idele Group 149 // 4.3.8 The Geometry of Adeles and Ideles 149 // 4.4 Class Field Theory 155 // 4.4.1 Abelian Extensions of the Field of Rational Numbers .. 155 // 4.4.2 Frobenius Automorphisms of Number Fields and // Artin’s Reciprocity Map 157 // 4.4.3 The Chebotarev Density Theorem 159 // 4.4.4 The Decomposition Law and // the Art in Reciprocity Map 159 // 4.4.5 The Kernel of the Reciprocity Map 160 // 4.4.6 The Artin Symbol 161 // 4.4.7 Global Properties of the Artin Symbol 162 // 4.4.8 A Link Between the Artin Symbol and Local Symbols.. 163 // 4.4.9 Properties of the Local Symbol 164 // 4.4.10 An Explicit Construction of Abelian Extensions of a Local Field, and a Calculation of the Local Symbol 165 // 4.4.11 Abelian Extensions of Number Fields 168 // 4.5 Galois Group in Arithetical Problems 172 // 4.5.1 Dividing a circle into n equal parts 172 // 4.5.2 Kummer Extensions and the Power Residue Symbol 175 // 4.5.3 Galois Cohomology 178 // 4.5.4 A Cohomological Definition of the Local Symbol 182 // 4.5.5 The Brauer Group, the Reciprocity Law and the // Minkowski-Hasse Principle 184 // Arithmetic of algebraic varieties 191 // 5 1 Arithmetic Varieties and Basic Notions of Algebraic Geometry 191 // 5.1.1 Equations and Rings 191 // 5 1.2 The set of solutions of a system 191 // 5.1.3 Example: The Language of Congruences 192 // 5.1.4 Equivalence of Systems of Equations 192 // 5.1.5 Solutions as iGalgebra Homomorphisms 192 // 5.1.6 The Spectrum of A Ring 193 // 5.1.7 Regular Functions 193 //

5.1.8 A Topology on Spec(A) 193 // 5.1.9 Schemes 196 // 5.1.10 Ring-Valued Points of Schemes 197 // 5.1.11 Solutions to Equations and Points of Schemes 198 // 5.1.12 Chevalley’s Theorem 199 // 5.1.13 Some Geometric Notions 199 // 5.2 Geometric Notions in the Study of Diophantine equations 202 // 5.2.1 Basic Questions 202 // 5.2.2 Geometric classification 203 // 5.2.3 Existence of Rational Points and Obstructions to the // Hasse Principle 204 // 5.2.4 Finite and Infinite Sets of Solutions 206 // 5.2.5 Number of points of bounded height 208 // 5.2.6 Height and Arakelov Geometry 211 // 5.3 Elliptic curves, Abelian Varieties, and Linear Groups 213 // 5.3.1 Algebraic Curves and Riemann Surfaces 213 // 5.3.2 Elliptic Curves 213 // 5.3.3 Tate Curve and Its Points of Finite Order 219 // 5.3.4 The Mordeil - Weil Theorem and Galois Cohomology 221 // 5.3.5 Abelian Varieties and Jacobians 226 // 5.3.6 The Jacobian of an Algebraic Curve 228 // 5.3.7 Siegel’s Formula and Tamagawa Measure 231 // 5.4 Diophantine Equations and Galois Representations 238 // 5.4.1 The Tate Module of an Elliptic Curve 238 // 5.4.2 The Theory of Complex Multiplication 240 // 5.4.3 Characters of r-adic Representations 242 // 5.4.4 Representations in Positive Characteristic 243 // 5.4.5 The Tate Module of a Number Field 244 // 5.5 The Theorem of Faltings and Finiteness Problems in // Diophantine Geometry 247 // 5.5.1 Reduction of the Mordell Conjecture to the finiteness // Conjecture 247 // 5.5.2 The Theorem of Sbafarevich on Finiteness for Elliptic // Curves 249 // 5.5.3 Passage to Abelian varieties 250 // 5.5.4 Finiteness Problems and Tate’s Conjecture 252 // XII Contents // 5.5.5 Reduction of the conjectures of Tate to the finiteness // properties for isogenies 253 // 5.5.6 The Faltings-Arakelov Height 255 // 5.5.7 Heights under isogenies and Conjecture T 257 //

6 Zeta Functions and Modular Forms 261 // 6.1 Zeta Functions of Arithmetic Schemes 261 // 6.1.1 Zeta Functions of Arithmetic Schemes 261 // 6.1.2 Analytic Continuation of the Zeta Functions 263 // 6.1.3 Schemes over Finite Fields and Deligne’s Theorem 263 // 6.1.4 Zeta Functions and Exponential Sums 267 // 6.2 L-Functions, the Theory of Tate and Explicit Formulae 272 // 6.2.1 L-Functions of Rational Galois Representations 272 // 6.2.2 The Formalism of Artin 274 // 6.2.3 Example: The Dedekind Zeta Function 276 // 6.2.4 Hecke Characters and the Theory of Tate 278 // 6.2.5 Explicit Formulae 285 // 6.2.6 The Weil Group and its Representations 288 // 6.2.7 Zeta Functions, L-Functions and Motives 290 // 6.3 Modular Forms and Euler Products 296 // 6.3.1 A Link Between Algebraic Varieties and L-Functions. . 296 // 6.3.2 Classical modular forms 296 // 6.3.3 Application: Tate Curve and Semistable Elliptic Curves 299 // 6.3.4 Analytic families of elliptic curves and congruence // subgroups 301 // 6.3.5 Modular forms for congruence subgroups 302 // 6.3.6 Hecke Theory 304 // 6.3.7 Primitive Forms 310 // 6.3.8 Weil’s Inverse Theorem 312 // 6.4 Modular Forms and Galois Representations 317 // 6.4.1 Ramanujan’s congruence and Galois Representations 317 // 6.4.2 A Link with Eichler-Shimura’s Construction 319 // 6.4.3 The Shimura-Taniyama-Weil Conjecture 320 // 6.4.4 The Conjecture of Birch and Swinnerton-Dyer 321 // 6.4.5 The Artin Conjecture and Cusp Forms 327 // The Artin conductor 329 // 6.4.6 Modular Representations over Finite Fields 330 // 6.5 Automorphic Forms and The Langlands Program 332 // 6.5.1 A Relation Between Classical Modular Forms and Representation Theory 332 // 6.5.2 Automorphic L-Functions 335 // Further analytic properties of automorphic L-functions. 338 // 6.5.3 The Langlands Functoriality Principle 338 // 6.5.4 Automorphic Forms and Langlands Conjectures 339 //

7.1 t’s Last Theorem and Families of Modular Forms---341 // ... Shimura-Taniyama-Weil Conjecture and Reciprocity Laws 341 // 7 11 problem of Pierre de Fermat (1601-1665) 341 // G.Lamé’s Mistake 342 // A short overview of Wiles’ Marvelous Proof 343 // The STW Conjecture 344 // A connection with the Quadratic Reciprocity Law 345 // A complete proof of the STW conjecture 345 // Modularity of semistable elliptic curves 348 // Structure of the proof of theorem 7.13 (Semistable STW Conjecture) 349 // 7.2 Theorem of Langlands-Tunnell and Modularity Modulo 3 // 7.2.1 Galois representations: preparation // 7.2.2 Modularity modulo p // 7.2.3 Passage from cusp forms of weight one to cusp forms // of weight two // 7.2.4 Preliminary review of the stages of the proof of Theorem 7.13 on modularity // 7.3 Modularity of Galois representations and Universal // Deformation Rings 357 // 7.3.1 Galois Representations over local Noetherian algebras .. 357 // 7.3.2 Deformations of Galois Representations 357 // 7.3.3 Modular Galois representations 359 // 7.3.4 Admissible Deformations and Modular Deformations 361 // 7.3.5 Universal Deformation Rings 363 // 7.4 Wiles’ Main Theorem and Isomorphism Criteria for Local // Rings 365 // 7.4.1 Strategy of the proof of the Main Theorem 7.33 365 // 7.4.2 Surjectivity of </?Ł• 365 // 7.4.3 Constructions of the universal deformation ring Re 367 // 7.4.4 A sketch of a construction of the universal modular deformation ring 368 // 7.4.5 Universality and the Chebotarev density theorem 369 // 7.4.6 Isomorphism Criteria for local rings 370 // 7.4.7 J-structures and the second criterion of isomorphism // of local rings 371 // 7.5 Wiles’ Induction Step: Application of the Criteria and Galois // Cohomology 373 // 7.5.1 Wiles’ induction step in the proof of // Main Theorem 7.33 373 // 7.5.2 A formula relating and #???, preparation 374 //

7.5.3 The Selmer group and ?? 375 // 7.5.4 Infinitesimal deformations 375 // 7.5.5 Deformations of type T>z 377 // 7.6 The Relative Invariant, the Main Inequality and The Minimal // Case 382 // 7.6.1 The Relative invariant 382 // 7.6.2 The Main Inequality 383 // 7.6.3 The Minimal Case 386 // 7.7 End of Wiles’ Proof and Theorem on Absolute Irreducibility .. 388 // 7.7.1 Theorem on Absolute Irreducibility 388 // 7.7.2 From p = 3 to p = ? 390 // 7.7.3 Families of elliptic curves with fixed p5 E 391 // 7.7.4 The end of the proof 392 // The most important insights 393 // Part III Analogies and Visions // III-O Introductory survey to part III: motivations and description 397 // IH.l Analogies and differences between numbers and functions: oo-point, Archimedean properties etc 397 // III. 1.1 Cauchy residue formula and the product formula 397 // III.1.2 Arithmetic varieties 398 // III. 1.3 Infinitesimal neighborhoods of fibers 398 // III.2 Arakelov geometry, fiber over oo, cycles, Green functions (d’aprčs Gillet-Soulé) 399 // 111.2.1 Arithmetic Chow groups 400 // 111.2.2 Arithmetic intersection theory and arithmetic // Riemann-Roch theorem 401 // HI.2.3 Geometric description of the closed fibers at infinity 402 // 111.3  -functions, local factors at oo, Serre’s T-factors 404 // 111.3.1 Archimedean L-factors 405 // 111.3.2 Deninger’s formulae 406 // 111.4 A guess that the missing geometric objects are // noncommutative spaces 407 // 111.4.1 Types and examples of noncommutative spaces, and how to work with them. Noncommutative geometry // and arithmetic 407 // Isomorphism of noncommutative spaces and Morita // equivalence 409 // The tools of noncommutative geometry 410 // 111.4.2 Generalities on spectral triples 411 // 111.4.3 Contents of Part III: description of parts of this program412 // Contents XV // ? lov Geometry and Noncommutative Geometry 415 //

8 L1 Motivations and the context of the work of // Consani-Marcolli 415 // 8 12 Analytic construction of degenerating curves over // complete local fields 416 // 8 13 Schottky groups and new perspectives in Arakelov // geometry 420 // Schottky uniformization and Schottky groups 421 // Fuchsian and Schottky uniformization 424 // 8.1.4 Hyperbolic handlebodies 425 // Geodesics in 42 // 8.1.5 Arakelov geometry and hyperbolic geometry 427 // Arakelov Green function 427 // Cross ratio and geodesics 428 // Differentials and Schottky uniformization 428 // Green function and geodesics 430 // 8.2 Cohomological Constructions 431 // 8.2.1 Archimedean cohomology 431 // Operators 433 // SL(2,R) representations 434 // 8.2.2 Local factor and Archimedean cohomology 435 // 8.2.3 Cohomological constructions 436 // 8.2.4 Zeta function of the special fiber and Reidemeister // torsion 437 // 8.3 Spectral Triples, Dynamics and Zeta Functions 440 // 8.3.1 A dynamical theory at infinity 442 // 8.3.2 Homotopy quotion 443 // 8.3.3 Filtration 444 // 8.3.4 Hilbert space and grading 446 // 8.3.5 Cuntz-Krieger algebra 446 // Spectral triples for Schottky groups 448 // 8.3.6 Arithmetic surfaces: homology and cohomology 449 // 8.3.7 Archimedean factors from dynamics 450 // 8.3.8 A Dynamical theory for Mumford curves 451 // Genus two example 452 // 8.3.9 Cohomology of ?\\?(?/?)? 454 // 8.3.10 Spectral triples and Mumford curves 456 // 8.4 Reduction mod oo 458 // 8.4.1 Homotopy quotients and deduction mod infinity” 458 // 8.4.2 Baum-Connes map 460 // References 461 // Index 503