Invitation to the mathematics of Fermat-Wiles / Yves Hellegouarch.

Format:

Book

Author/Creator:

Hellegouarch, Yves.

Standardized Title:

Invitation aux mathématiques de Fermat-Wiles. English

Language:

English

French

Subjects (All):

Fermat's last theorem.

Fermat's theorem.

Algebraic number theory.

Physical Description:

xi, 381 pages : illustrations ; 25 cm

Place of Publication:

San Diego, Calif. : Academic, [2002]

Summary:

Assuming only modest knowledge of undergraduate level math, Invitation to the Mathematics of Fermat-Wiles presents diverse concepts required to comprehend Wiles' extraordinary proof. Furthermore, it places these concepts in their historical context. This book can be used in introduction to mathematics theories courses and in special topics courses on Fermat's last theorem. It contains themes suitable for development by students as an introduction to personal research as well as numerous exercises and problems. However, the book will also appeal to the inquiring and mathematically informed reader intrigued by the unraveling of this fascinating puzzle. Key Features* Rigorously presents the concepts required to understand Wiles' proof, assuming only modest undergraduate level math * Sets the math in its historical context* Contains several themes that could be further developed by student research and numerous exercises and problems* Written by Yves Hellegouarch, who himself made an important contribution to the proof of Fermat's last theorem* Written by Yves Hellegouarch, who himself made an important contribution to the proof of Fermat's last theorem.

Contents:

1 Paths 1

1.1 Diophantus and his Arithmetica 2

1.2 Translations of Diophantus 2

1.3 Fermat 3

1.4 Infinite descent 4

1.5 Fermat's "theorem" in degree 4 7

1.6 The theorem of two squares 9

1.6.1 A modern proof 10

1.6.2 "Fermat-style" proof of the crucial theorem 12

1.6.3 Representations as sums of two squares 13

1.7 Euler-style proof of Fermat's last theorem for n = 3 16

1.8 Kummer, 1847 18

1.8.1 The ring of integers of Q([zeta]) 18

1.8.2 A lemma of Kummer on the units of Z[[zeta]] 23

1.8.3 The ideals of Z[[zeta]] 25

1.8.4 Kummer's proof (1847) 26

1.8.5 Regular primes 31

1.9 The current approach 33

2 Elliptic functions 68

2.1 Elliptic integrals 68

2.2 The discovery of elliptic functions in 1718 71

2.3 Euler's contribution (1753) 75

2.4 Elliptic functions: structure theorems 77

2.5 Weierstrass-style elliptic functions 80

2.6 Eisenstein series 85

2.7 The Weierstrass cubic 87

2.8 Abel's theorem 89

2.9 Loxodromic functions 92

2.10 The function [rho] 95

2.11 Computation of the discriminant 97

2.12 Relation to elliptic functions 99

3 Numbers and groups 118

3.1 Absolute values on Q 118

3.2 Completion of a field equipped with an absolute value 123

3.3 The field of p-adic numbers 127

3.4 Algebraic closure of a field 131

3.5 Generalities on the linear representations of groups 134

3.6 Galois extensions 140

3.6.1 The Galois correspondence 141

3.6.2 Questions of dimension 143

3.6.3 Stability 146

3.7 Resolution of algebraic equations 149

3.7.1 Some general principles 149

3.7.2 Resolution of the equation of degree three 152

4 Elliptic curves 172

4.1 Cubics and elliptic curves 172

4.2 Bezout's theorem 179

4.3 Nine-point theorem 183

4.4 Group laws on an elliptic curve 185

4.5 Reduction modulo p 189

4.6 N-division points of an elliptic curve 192

4.6.1 2-division points 192

4.6.2 3-division points 193

4.6.3 n-division points of an elliptic curve defined over Q 194

4.7 A most interesting Galois representation 195

4.8 Ring of endomorphisms of an elliptic curve 197

4.9 Elliptic curves over a finite field 202

4.10 Torsion on an elliptic curve defined over Q 205

4.11 Mordell-Weil theorem 211

4.12 Back to the definition of elliptic curves 211

4.13 Formulae 215

4.14 Minimal Weierstrass equations (over Z) 218

4.15 Hasse-Weil L-functions 223

4.15.1 Riemann zeta function 223

4.15.2 Artin zeta function 224

4.15.3 Hasse-Weil L-function 226

5 Modular forms 255

5.1 Brief historical overview 255

5.2 The theta functions 260

5.3 Modular forms for the modular group SL[subscript 2](Z)/{I, -I} 274

5.3.1 Modular properties of the Eisenstein series 274

5.3.2 The modular group 280

5.3.3 Definition of modular forms and functions 287

5.4 The space of modular forms of weight k for SL[subscript 2](Z) 289

5.5 The fifth operation of arithmetic 294

5.6 The Petersson Hermitian product 297

5.7 Hecke forms 299

5.7.1 Hecke operators for SL[subscript 2](Z) 300

5.8 Hecke's theory 304

5.8.1 The Mellin transform 306

5.8.2 Functional equations for the functions L(f, s) 307

5.9 Wiles' theorem 308

6 New paradigms, new enigmas 325

6.1 A second definition of the ring Z[subscript p] of p-adic integers 326

6.2 The Tate module T[subscript l](E) 328

6.3 A marvellous result 330

6.4 Tate loxodromic functions 331

6.5 Curves E[subscript A,B,C] 332

6.5.1 Reduction of certain curves E[subscript A,B,C] 333

6.5.2 Property of the field K[subscript p] associated to E[subscript a[superscript p],b[superscript p],c[superscript p] 335

6.5.3 Summary of the properties of E[subscript a[superscript p],b[superscript p],c[superscript p] 335

6.6 The Serre conjectures 336

6.7 Mazur-Ribet's theorem 339

6.7.1 Mazur-Ribet's theorem 340

6.7.2 Other applications 341

6.8 Szpiro's conjecture and the abc conjecture 343

6.8.1 Szpiro's conjecture 343

6.8.2 abc conjecture 344

6.8.3 Consequences 344

Appendix The origin of the elliptic approach to Fermat's last theorem 359.

Notes:

Includes bibliographical references (pages 371-374) and index.

ISBN:

0123392519

OCLC:

47726300