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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
411
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
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