The works of Wiles (and Taylor, \(\dots\)). II. (Travaux de Wiles (et Taylor, \(\dots\)). II.) (French) Zbl 0957.11028

Séminaire Bourbaki. Volume 1994/95. Exposés 790-804. Paris: Société Mathématique de France, Astérisque. 237, 333-355, Exp. No. 804 (1996).
This exposé follows that of J.-P. Serre [Exposé No. 803 (1996; Zbl 0957.11025)] explaining the works of Wiles and Taylor-Wiles on the Shimura-Taniyama-Weil conjecture and Fermat’s last theorem.
Here the author gives explanations of the proof of Wiles’ main theorem on Galois representations over local rings and modular forms associated with them giving sufficient conditions for the modularity of this representation. Table of contents:
1) Semistable representations; 2) Deformations of a semistable representation; 3) Hecke algebras and associated Galois representations; 4) Hecke deformations of a semistable representation; 5) Formulation of the main theorem (Theorem 2 concerning an isomorphism of a certain universal deformation ring \(R_\Sigma\) with a Hecke local ring \(T_\Sigma\), where \(\Sigma\) is a finite set of primes); 6) Theorem 2 in the minimal case; 7) Theorem 2 in the general case.
Appendix I: Representations of a group over a local ring (after H. Carayol); Appendix II: Deformations of Galois representations; Appendix III: Complete intersection rings; Appendix IV: Selmer groups.
For the entire collection see [Zbl 0851.00039].


11G05 Elliptic curves over global fields
11F80 Galois representations
11F11 Holomorphic modular forms of integral weight
11D41 Higher degree equations; Fermat’s equation
11F85 \(p\)-adic theory, local fields
11G18 Arithmetic aspects of modular and Shimura varieties
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