This book provides a comprehensive account of a key, perhaps the most important, theory that forms the basis of Taylor-Wiles proof of Fermat's last theorem. Hida begins with an overview of the theory of automorphic forms on linear algebraic groups and then covers the basic theory and recent results on elliptic modular forms, including a substantial simplification of the Taylor-Wiles proof by Fujiwara and Diamond. He offers a detailed exposition of the representation theory of profinite groups (including deformation theory), as well as the Euler characteristic formulas of Galois cohomology groups. The final chapter presents a proof of a non-abelian class number formula.
Contents
Preface; 1. Overview of modular forms; 2. Representations of a group; 3. Representations and modular forms; 4. Galois cohomology; 5. Modular L-values and Selmer groups; Bibliography; Subject index; List of statements; List of symbols.
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