An invitation to the mathematics of Fermat-Wiles /

An invitation to the mathematics of Fermat-Wiles /

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Bibliographic Details
Author / Creator:Hellegouarch, Yves.
Uniform title:Invitation aux mathématiques de Fermat-Wiles. English
Imprint:San Diego, Calif. : Academic Press, 2002.
Description:xi, 381 p. : ill. ; 25 cm.
Language:English
Subject:
Fermat's last theorem.
Fermat's theorem.
Algebraic number theory.
Curves, Elliptic.
Fermat's last theorem.
Forms, Modular.
Format: Print Book
URL for this record:http://pi.lib.uchicago.edu/1001/cat/bib/4557352
Hidden Bibliographic Details
ISBN:0123392519
Notes:Includes bibliographical references and index.
Translated from the French, 2nd ed.
Table of Contents:
  • Foreword
  • 1. Paths
  • 1.1. Diophantus and his Arithmetica
  • 1.2. Translations of Diophantus
  • 1.3. Fermat
  • 1.4. Infinite Descent
  • 1.5. Fermat's "Theorem" in Degree 4
  • 1.6. The Theorem of Two Squares
  • 1.6.1. A Modern Proof
  • 1.6.2. "Fermat-Style" Proof of the Crucial Theorem
  • 1.6.3. Representations as Sums of Two Squares
  • 1.7. Euler-Style Proof of Fermat's Last Theorem for n=3
  • 1.8. Kummer, 1847
  • 1.8.1. The Ring of Integers of Q(?)
  • 1.8.2. A Lemma of Kummer on the Units of Z[?]
  • 1.8.3. The Ideals of Z[?]
  • 1.8.4. Kummer's Proof (1847)
  • 1.8.5. Regular Primes
  • 1.9. The Current Approach
  • Exercises and Problems
  • 2. Elliptic Functions
  • 2.1. Elliptic Integrals
  • 2.2. The Discovery of Elliptic Functions in 1718
  • 2.3. Euler's Contribution (1753)
  • 2.4. Elliptic Functions: Structure Theorems
  • 2.5. Weierstrass-Style Elliptic Functions
  • 2.6. Eisenstein Series
  • 2.7. The Weierstrass Cubic
  • 2.8. Abel's Theorem
  • 2.9. Loxodromic Functions
  • 2.10. The Function ?
  • 2.11. Computation of the Discriminant
  • 2.12. Relation to Elliptic Functions
  • Exercises and Problems
  • 3. Numbers and groups
  • 3.1. Absolute Values on Q
  • 3.2. Completion of a Fequipped with an Absolute Value
  • 3.3. The Field of p-adic Numbers
  • 3.4. Algebraic Closure of a Field
  • 3.5. Generalities on the Linear Representations of Groups
  • 3.6. Galois Extensions
  • 3.6.1. The Galois Correspondence
  • 3.6.2. Questions of Dimension
  • 3.6.3. Stability
  • 3.6.4. Conclusions
  • 3.7. Resolution of Algebraic Equations
  • 3.7.1. Some General Principles
  • 3.7.2. Resolution of the Equation of Degree Three
  • Exercises and Problems
  • 4. Elliptic Curves
  • 4.1. Cubics and Elliptic Curves
  • 4.2. B'ezout's Theorem
  • 4.3. Nine-Point Theorem
  • 4.4. Group Laws on an Elliptic Curve
  • 4.5. Reduction Modulo p
  • 4.6. N-Division Points of an Elliptic Curve
  • 4.6.1. 2-Division Points
  • 4.6.2. 3-Division Points
  • 4.6.3. n-Division Points of an Elliptic Curve Defined Over Q
  • 4.7. A Most Interesting Galois Representation
  • 4.8. Ring of Endomorphisms of an Elliptic Curve
  • 4.9. Elliptic Curves Over a Finite Field
  • 4.10. Torsion on an Elliptic Curve Defined Over Q
  • 4.11. Mordell-Weil Theorem
  • 4.12. Back to the Definition of Elliptic Curves
  • 4.13. Formulae
  • 4.14. Minimal Weierstrass Equations (Over Z)
  • 4.15. Hasse-Weil L-Functions
  • 4.15.1. Riemann Zeta Function
  • 4.15.2. Artin Zeta Function
  • 4.15.3. Hasse-Weil L-Function
  • Exercises and Problems
  • 5. Modular Forms
  • 5.1. Brief Historical Overview
  • 5.2. The Theta Functions
  • 5.3. Modular Forms for the Modular Group SL2(Z)/(I,?I)
  • 5.3.1. Modular Properties of the Eisenstein Series
  • 5.3.2. The Modular Group
  • 5.3.3. Definition of Modular Forms and Functions
  • 5.4. The Space of Modular Forms of Weight k for SL2(Z)
  • 5.5. The Fifth Operation of Arithmetic
  • 5.6. The Petersson Hermitian Product
  • 5.7. Hecke Forms
  • 5.7.1. Hecke Operators for SL2(Z)
  • 5.8. Hecke's Theory
  • 5.8.1. The Mellin Transform
  • 5.8.2. Functional Equations for the Functions L(f,s)
  • 5.9. Wiles' Theorem
  • Exercises and Problems
  • 6. New Paradigms, New Enigmas
  • 6.1. A Second Definition of the Ring Zp of p-adic Integers
  • 6.2. The Tate Module Tl(E)
  • 6.3. A Marvellous Result
  • 6.4. Tate Loxodromic Functions
  • 6.5. Curves EA,B,C
  • 6.5.1. Reduction of Certain Curves EA,B,C
  • 6.5.2. Property of the Field Kp Associated to Eap,bp,cp
  • 6.5.3. Summary of the Properties of Eap,bp,cp
  • 6.6. The Serre Conjectures
  • 6.7. Mazur-Ribet's Theorem
  • 6.7.1. Mazur-Ribet's Theorem
  • 6.7.2. Other Applications
  • 6.8. Szpiro's Conjecture and the abc Conjecture
  • 6.8.1. Szpiro's Conjecture
  • 6.8.2. abc Conjecture
  • 6.8.3. Consequences
  • Exercises and Problems
  • Appendix: The Origin of the Elliptic Approach to Fermat's Last Theorem
  • Bibliography
  • Index

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