Saito Takeshi. Fermat’s Last Theorem: The Proof

Transl. from the Japan.: Masato Kuwata. — American Mathematical Society, 2014. — xvi, 224 p. — (Translations of Mathematical Monographs. Vol. 245). — ISBN 978-0-8218-9849-9.This is the second volume of the book on the proof of Fermat's Last Theorem by Wiles and Taylor (the first volume is published in the same series; see TMM, Vol. 243). Here the detail of the proof announced in the first volume is fully exposed. The book also includes basic materials and constructions in number theory and arithmetic geometry that are used in the proof. In the first volume the modularity lifting theorem on Galois representations has been reduced to properties of the deformation rings and the Hecke modules. The Hecke modules and the Selmer groups used to study deformation rings are constructed, and the required properties are established to complete the proof. The reader can learn basics on the integral models of modular curves and their reductions modulo that lay the foundation of the construction of the Galois representations associated with modular forms. More background materials, including Galois cohomology, curves over integer rings, the Néron models of their Jacobians, etc., are also explained in the text and in the appendices.Preface
Preface to the English Edition
Modular curves over ℤ
Elliptic curves in characteristic p>0
Cyclic group schemes
Drinfeld level structure
Modular curves over ℤ
Modular curve Y(r)_ℤ[1/r]
Igusa curves
Modular curve Y₁(N)_ℤ
Modular curve Y₀(N)_ℤ
Compactifications
Modular forms and Galois representations
Hecke algebras with ℤ coefficients
Congruence relations
Modular mod ℓ representations and non-Eisenstein ideals
Level of modular forms and ramification of ℓ-adic representations
Old part
Néron model of the Jacobian J₀(Mp)
Level of modular forms and ramification of mod ℓ representationsHecke modules
Full Hecke algebras
Hecke modules
Proof of Proposition 10.11
Deformation rings and group rings
Family of lift ings
Proof of Proposition 10.37
Proof of Theorem 5.22
Selmer groups
Cohomology of groups
Galois cohomology
Selmer groups
Selmer groups and deformation rings
Calculation of local conditions and proof of Proposition 11.38
Proof of Theorem 11.37Appendix B. Curves over discrete valuation rings
Curves
Semistable curve over a discrete valuation ring
Dual chain complex of curves over a discrete valuation ring
Appendix C. Finite commutative group scheme over ℤ_p
Finite flat commutative group scheme over F_p
Finite flat commutative group scheme over ℤ_p
Appendix D. Jacobian of a curve and its Néron model
The divisor class group of a curve
The Jacobian of a curve
The Néron model of an abelian variety
The Néron model of the Jacobian of a curveBibliography
Symbol Index
Subject IndexФайл: отскан. стр. (b/w 600 dpi) + OCR