Introduction to Modern Number Theory - Fundamental Problems, Ideas and Theories

Preface

Contents

Introduction

Problems and Tricks

1 Number Theory

1.1 Problems About Primes. Divisibility and Primality

1.2 Diophantine Equations of Degree One and Two

1.3 Cubic Diophantine Equations

1.4 The Structure of the Continuum. Approximations and Continued Fractions

1.5 Diophantine Approximation and the Irrationality of .(3)

2 Some Applications of Elementary Number Theory

2.1 Factorization and Public Key Cryptosystems

2.2 Deterministic Primality Tests

2.3 Factorization of Large Integers

Ideas and Theories

3 Induction and Recursion

3.1 Elementary Number Theory From the Point of View of Logic

3.2 Diophantine Sets

3.3 Partially Recursive Functions and Enumerable Sets

3.4 Diophantineness of a Set and algorithmic Undecidability

4 Arithmetic of algebraic numbers

4.1 Algebraic Numbers: Their Realizations and Geometry

4.2 Decomposition of Prime Ideals, Dedekind Domains, and Valuations

4.3 Local and Global Methods

4.4 Class Field Theory

4.5 Galois Group in Arithetical Problems

5 Arithmetic of algebraic varieties

5.1 Arithmetic Varieties and Basic Notions of Algebraic Geometry

5.2 Geometric Notions in the Study of Diophantine equations

5.3 Elliptic curves, Abelian Varieties, and Linear Groups

5.4 Diophantine Equations and Galois Representations

5.5 The Theorem of Faltings and Finiteness Problems in Diophantine Geometry

6 Zeta Functions and Modular Forms

6.1 Zeta Functions of Arithmetic Schemes

6.2 L-Functions, the Theory of Tate and Explicite Formulae

6.3 Modular Forms and Euler Products

6.4 Modular Forms and Galois Representations

6.5 Automorphic Forms and The Langlands Program

7 Fermat’s Last Theorem and Families of Modular Forms

7.1 The Shimura–Taniyama–Weil Conjecture and Higher Reciprocity Laws

7.2 Theorem of Langlands-Tunnell and Modularity Modulo 3

7.3 Modularity of Galois representations and Universal Deformation Rings

7.4 Wiles’ Main Theorem and Isomorphism Criteria for Local Rings

7.5 Wiles’ Induction Step: Application of the Criteria and Galois Cohomology

7.6 The Relative Invariant, the Main Inequality and The Minimal Case

7.7 End of Wiles’ Proof and Theorem on Absolute Irreducibility

Analogies and Visions

III-0 Introductory survey to part III: motivations and description

III.1 Analogies and differences between numbers and functions: 8-point, Archimedean properties etc.

III.2 Arakelov geometry, fiber over 8, cycles, Green functions (d’après Gillet-Soulé)

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

8 Arakelov Geometry and Noncommutative Geometry ( d’après C. Consani and M. Marcolli, [ CM])

8.1 Schottky Uniformization and Arakelov Geometry

8.2 Cohomological Constructions, Archimedean Frobenius and Regularized Determinants

8.3 Spectral Triples, Dynamics and Zeta Functions

8.4 Reduction mod 8

References

Index