Saito Takeshi. Fermat’s Last Theorem: Basic Tools

Transl. from the Japan.: Masato Kuwata. — American Mathematical Society, 2013. — xvi, 200 p. — (Translations of Mathematical Monographs. Vol. 243). — ISBN 978-0-8218-9848-2.This book, together with the companion volume, Fermat's Last Theorem: The Proof, presents in full detail the proof of Fermat's Last Theorem given by Wiles and Taylor. With these two books, the reader will be able to see the whole picture of the proof to appreciate one of the deepest achievements in the history of mathematics. Crucial arguments, including the so-called 3-5 trick, R=T theorem, etc., are explained in depth. The proof relies on basic background materials in number theory and arithmetic geometry, such as elliptic curves, modular forms, Galois representations, deformation rings, modular curves over the integer rings, Galois cohomology, etc. The first four topics are crucial for the proof of Fermat's Last Theorem; they are also very important as tools in studying various other problems in modern algebraic number theory. In order to facilitate understanding the intricate proof, an outline of the whole argument is described in the first preliminary chapter of the first volume.Preface
Preface to the English EditionSynopsis
Simple paraphrase
Elliptic curves
Elliptic curves and modular forms
Conductor of an elliptic curve and level of a modular form
ℓ-torsion points of elliptic curves and modular forms
Elliptic curves
Elliptic curves over a field
Reduction mod p
Morphisms and the Tate modules
Elliptic curves over an arbitrary scheme
Generalized elliptic curves
Modular forms
The j-invariant
Moduli spaces
Modular curves and modular forms
Construction of modular curves
The genus formula
The Hecke operators
The q-expansions
Primary forms, primitive forms
Elliptic curves and modular forms
Primary forms, primitive forms, and Hecke algebras
The analytic expression
The q-expansion and analytic expression
The q-expansion and Hecke operators
Galois representations
Frobenius substitutions
Galois representations and finite group schemes
The Tate module of an elliptic curve
Modular ℓ-adic representations
Ramification conditions
Finite flat group schemes
Ramification of the Tate module of an elliptic curve
Level of modular forms and ramificationThe 3–5 trick
Proof of Theorem 2.54
Summary of the Proof of Theorem 0.1
R = T
What is R = T?
Deformation rings
Hecke algebras
Some commutative algebra
Hecke modules
Outline of the Proof of Theorem 5.22
Commutative algebra
Proof of Theorem 5.25
Proof of Theorem 5.27
Deformation rings
Functors and their representations
The existence theorem
Proof of Theorem 5.8
Proof of Theorem 7.7
Appendix A. Supplements to scheme theory
Various properties of schemes
Group schemes
Quotient by a finite group
Flat covering
G-torsor
Closed condition
Cartier divisor
Smooth commutative group schemeBibliography
Symbol Index
Subject IndexTrue PDF