Contents 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 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 oflntegers 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 1.4.3 Continued Fractions 51 1.4.4 SL2-Equivalence 53 1.4.5 Periodic Continued FYactions and Pell's Equation 53 1.5 Diophantine Approximation and the Irrationality 55 1.5.1 Ideas in the Proof thatζ (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 1.5.7 Work of Yu.V Nesterenko on e7, [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 0ne-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 Dime-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 2.2.2 Gauss Sums and Their Use in Primality Testing 71 2.2.3 Detailed Description of the Primality Test 75 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 Dimculty 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 FYactions Method (CFRAC) and Real Quadratic Fields 87 2.3.5 The Use of Elliptic Curves 90 Part II Ideas and Theories 3 Induction and Recursion 95 3.1 Elementary Number Theory From the Point of View of Logic 95 3.1.1 Elementary Number Theory 95 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 3.3.3 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 3.3.8 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 111 3.3.12 Godel's function 111 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 4.1.6 Deduction of Dirichlet's Theorem From 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 4.3.2 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 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 Artin 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 Cohoruological Definition of the Local Symbol 182 4.5.5 The Brauer Group, the Reciprocity Law and the Minkowski-Hasse Principle 184 5 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 K -algebra Homomorphisms 192 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 Mordell - 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 Forruula 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 Coruplex Multiplication 240 5.4.3 Characters of L-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 Shafarevich on Finiteness for Elliptic Curves 249 5.5.3 Passage to Abelian varieties 250 5.5.4 Finiteness problems and Tate's conjecture 252 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 Explicite Formulae 272 6.2.1 LFunctions 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 faruilies of elliptic curves and congruence 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 Fermat's Last Theorem and Families of Modular Forms....341 7.1 Shimura-Taniyama-Weil Conjecture and Reciprocity Laws 341 7.1.1 Problem of Pierre de Fermat (1601-1665) 341 7.1.2 G.Lame's Mistake 342 7.1.3 A short overview of Wiles Marvelous Proof 343 7.1.4 The STW Conjecture 344 7.1.5 A connection with the Quadratic Reciprocity Law 345 7.1.6 A complete proof of the STW conjecture 345 7.1.7 Modularity of semistable elliptic curves 348 7.1.8 Structure of the proof of theorem 7.13 (Semistable STW Conjecture) 349 7.2 Theorem of Langlands-Tunnell and Modularity Modul0 3 352 7.2.1 Galois representations: preparation 352 7.2.2 Modularity ruodulo p 354 7.2.3 Passage from cusp forms of weight one to cusp forms 7.2.4 Preliminary review of the stages of the proof of Theorem 7.13 0n modularity 356 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 cpz 365 7.4.3 Constructions of the universal deformation ring Ry 367 7.4.4 A sketch of a construction of the universal modular deformation ring T∑ 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 ofisomorphisru 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 #qRz and #qRz: preparation 374 7.5.3 The Selmer group and qRr 375 7.5.4 Infinitesimal deformations 375 7.5.5 Deformations of type DE 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 7.7 End of Wiles Proof and Theorem on Absolute Irreducibility 388 7.7.1 Theorem on Absolute Irreducibility 388 7.7.2 Froru p=3 to p=5 390 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-0 Introductory survey to part III: motivations and Description 397 III.1 Analogies and differences between numbers and functions: ∞-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 neighborh∞ds of fibers 398 IIl.2 Arakelov geometry, fiber over ∞, cycles, Green functions (d’apres Gillet-Soule) 399 III.2.1 Arithmetic Chow groups 400 III.2.2 Arithmetic intersection theory and arithmetic Riemann-Roch theorem 401 III.2.3 Geometric description of the closed fibers at infinity 402 III.3 ζ-functions, local factors at ∞, Serre's，Γ-factors 404 III.3.1 Archimedean L-factors 405 III.3.2 Deninger's formulae 406 III.4 A guess that the missing geometric objects are noncommutative spaces 407 III.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 III.4.2 Generalities on spectral triples 411 III.4.3 Contents of Part III: description of parts of this program412 8 Arakelov Geometry and Noncommutative Geometry 415 8.1 Schottky Uniformization and Arakelov Geometry 415 8.1.1 Motivations and the context of the work of Consani-Marcolli 415 8.1.2 Analytic construction of degenerating curves over complete local fields 416 8.1.3 Schottky groups and new perspectives in Arakelov Schottky uniformization and Schottky groups 421 Fuchsian and Schottky uniformization 424 8.1.4 Hyperbolic handlebodies 425 Geodesics in Γ 427 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 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 8.3.3 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 450 8.3.9 Cohomology of W(△/Γ）T 454 8.3.10 Spectral triples and Mumford curves 456 8.4 Reduction mod ∞ 458 8.4.1 Homotopy quotients and "reduction mod infinity" 458 8.4.2 Baum-Connes map 460 References 461 Index 503
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