Introduction to Modern Number Theory (eBook)

Preface 5
Contents 7
Introduction 16
Problems and Tricks 22
1 Number Theory 23
1.1 Problems About Primes. Divisibility and Primality 23
1.2 Diophantine Equations of Degree One and Two 36
1.3 Cubic Diophantine Equations 52
1.4 The Structure of the Continuum. Approximations and Continued Fractions 64
1.5 Diophantine Approximation and the Irrationality of .(3) 69
2 Some Applications of Elementary Number Theory 76
2.1 Factorization and Public Key Cryptosystems 76
2.2 Deterministic Primality Tests 82
2.3 Factorization of Large Integers 97
Ideas and Theories 105
3 Induction and Recursion 106
3.1 Elementary Number Theory From the Point of View of Logic 106
3.2 Diophantine Sets 109
3.3 Partially Recursive Functions and Enumerable Sets 114
3.4 Diophantineness of a Set and algorithmic Undecidability 124
4 Arithmetic of algebraic numbers 126
4.1 Algebraic Numbers: Their Realizations and Geometry 126
4.2 Decomposition of Prime Ideals, Dedekind Domains, and Valuations 137
4.3 Local and Global Methods 145
4.4 Class Field Theory 166
4.5 Galois Group in Arithetical Problems 183
5 Arithmetic of algebraic varieties 201
5.1 Arithmetic Varieties and Basic Notions of Algebraic Geometry 201
5.2 Geometric Notions in the Study of Diophantine equations 212
5.3 Elliptic curves, Abelian Varieties, and Linear Groups 223
5.4 Diophantine Equations and Galois Representations 248
5.5 The Theorem of Faltings and Finiteness Problems in Diophantine Geometry 257
6 Zeta Functions and Modular Forms 270
6.1 Zeta Functions of Arithmetic Schemes 270
6.2 L-Functions, the Theory of Tate and Explicite Formulae 281
6.3 Modular Forms and Euler Products 305
6.4 Modular Forms and Galois Representations 326
6.5 Automorphic Forms and The Langlands Program 341
7 Fermat’s Last Theorem and Families of Modular Forms 350
7.1 The Shimura–Taniyama–Weil Conjecture and Higher Reciprocity Laws 350
7.2 Theorem of Langlands-Tunnell and Modularity Modulo 3 361
7.3 Modularity of Galois representations and Universal Deformation Rings 366
7.4 Wiles’ Main Theorem and Isomorphism Criteria for Local Rings 374
7.5 Wiles’ Induction Step: Application of the Criteria and Galois Cohomology 382
7.6 The Relative Invariant, the Main Inequality and The Minimal Case 391
7.7 End of Wiles’ Proof and Theorem on Absolute Irreducibility 397
Analogies and Visions 403
III-0 Introductory survey to part III: motivations and description 404
III.1 Analogies and differences between numbers and functions: 8-point, Archimedean properties etc. 404
III.2 Arakelov geometry, fiber over 8, cycles, Green functions (d’après Gillet-Soulé) 406
III.3 Theory of .-functions, local factors at 8, Serre’s G-factors and generally an interpretation of zeta functions as determinants of the arithmetical Frobenius: Deninger’s program
III.4 A guess that the missing geometric objects are noncommutative spaces 414
8 Arakelov Geometry and Noncommutative Geometry ( d’après C. Consani and M. Marcolli, [ CM]) 421
8.1 Schottky Uniformization and Arakelov Geometry 421
8.2 Cohomological Constructions, Archimedean Frobenius and Regularized Determinants 437
8.3 Spectral Triples, Dynamics and Zeta Functions 446
8.4 Reduction mod 8 464
References 467
Index 509