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Invitation to the Mathematics of Fermat-Wiles - Yves Hellegouarch

Invitation to the Mathematics of Fermat-Wiles


Published: 1st October 2001
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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.

"This text provides a sweeping introduction to all those mathematical topics, concepts, methods, techniques, and classical results that are necessary to understand Andrew Wiles's theory culminating in the first complete proof of Fermat's last theorem. The text is accessible, without compromising the rigor of its mathematical exposition, to reasoned undergraduate students, at least so for the most part it can serve as the basis for various teaching courses. It sets the whole discussion in a fascinating, generally educating historical context, thereby travelling - metaphorically speaking - through the centuries of mathematical history. No doubt, it is a true blessing that the English translation of this unique book is now at hand for a much wider public." Werner Kleinert (Berlin) in Zentralblatt MATH 1036

Forewordp. viii
Pathsp. 1
Diophantus and his Arithmeticap. 2
Translations of Diophantusp. 2
Fermatp. 3
Infinite descentp. 4
Fermat's "theorem" in degree 4p. 7
The theorem of two squaresp. 9
A modern proofp. 10
"Fermat-style" proof of the crucial theoremp. 12
Representations as sums of two squaresp. 13
Euler-style proof of Fermat's last theorem for n = 3p. 16
Kummer, 1847p. 18
The ring of integers of Q([zeta])p. 18
A lemma of Kummer on the units of Z[[zeta]]p. 23
The ideals of Z[[zeta]]p. 25
Kummer's proof (1847)p. 26
Regular primesp. 31
The current approachp. 33
Exercises and problemsp. 35
Elliptic functionsp. 68
Elliptic integralsp. 68
The discovery of elliptic functions in 1718p. 71
Euler's contribution (1753)p. 75
Elliptic functions: structure theoremsp. 77
Weierstrass-style elliptic functionsp. 80
Eisenstein seriesp. 85
The Weierstrass cubicp. 87
Abel's theoremp. 89
Loxodromic functionsp. 92
The function [rho]p. 95
Computation of the discriminantp. 97
Relation to elliptic functionsp. 99
Exercises and problemsp. 101
Numbers and groupsp. 118
Absolute values on Qp. 118
Completion of a field equipped with an absolute valuep. 123
The field of p-adic numbersp. 127
Algebraic closure of a fieldp. 131
Generalities on the linear representations of groupsp. 134
Galois extensionsp. 140
The Galois correspondencep. 141
Questions of dimensionp. 143
Stabilityp. 146
Conclusionsp. 146
Resolution of algebraic equationsp. 149
Some general principlesp. 149
Resolution of the equation of degree threep. 152
Exercises and problemsp. 155
Elliptic curvesp. 172
Cubics and elliptic curvesp. 172
Bezout's theoremp. 179
Nine-point theoremp. 183
Group laws on an elliptic curvep. 185
Reduction modulo pp. 189
N-division points of an elliptic curvep. 192
2-division pointsp. 192
3-division pointsp. 193
n-division points of an elliptic curve defined over Qp. 194
A most interesting Galois representationp. 195
Ring of endomorphisms of an elliptic curvep. 197
Elliptic curves over a finite fieldp. 202
Torsion on an elliptic curve defined over Qp. 205
Mordell-Weil theoremp. 211
Back to the definition of elliptic curvesp. 211
Formulaep. 215
Minimal Weierstrass equations (over Z)p. 218
Hasse-Weil L-functionsp. 223
Riemann zeta functionp. 223
Artin zeta functionp. 224
Hasse-Weil L-functionp. 226
Exercises and problemsp. 228
Modular formsp. 255
Brief historical overviewp. 255
The theta functionsp. 260
Modular forms for the modular group SL[subscript 2](Z)/{I, -I}p. 274
Modular properties of the Eisenstein seriesp. 274
The modular groupp. 280
Definition of modular forms and functionsp. 287
The space of modular forms of weight k for SL[subscript 2](Z)p. 289
The fifth operation of arithmeticp. 294
The Petersson Hermitian productp. 297
Hecke formsp. 299
Hecke operators for SL[subscript 2](Z)p. 300
Hecke's theoryp. 304
The Mellin transformp. 306
Functional equations for the functions L(f, s)p. 307
Wiles' theoremp. 308
Exercises and problemsp. 313
New paradigms, new enigmasp. 325
A second definition of the ring Z[subscript p] of p-adic integersp. 326
The Tate module T[subscript l](E)p. 328
A marvellous resultp. 330
Tate loxodromic functionsp. 331
Curves E[subscript A,B,C]p. 332
Reduction of certain curves E[subscript A,B,C]p. 333
Property of the field K[subscript p] associated to E[subscript a[superscript p],b[superscript p],c[superscript p]p. 335
Summary of the properties of E[subscript a[superscript p],b[superscript p],c[superscript p]p. 335
The Serre conjecturesp. 336
Mazur-Ribet's theoremp. 339
Mazur-Ribet's theoremp. 340
Other applicationsp. 341
Szpiro's conjecture and the abc conjecturep. 343
Szpiro's conjecturep. 343
abc conjecturep. 344
Consequencesp. 344
Exercises and problemsp. 348
The origin of the elliptic approach to Fermat's last theoremp. 359
Bibliographyp. 371
Indexp. 375
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ISBN: 9780123392510
ISBN-10: 0123392519
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 400
Published: 1st October 2001
Publisher: Elsevier Science Publishing Co Inc
Country of Publication: US
Dimensions (cm): 24.4 x 17.1  x 2.54
Weight (kg): 0.91