This book provides a comprehensive account of the key theory upon which the Taylor-Wiles proof of Fermat's last theorem is based. It begins with an overview of the theory of automorphic forms on linear algebraic groups and covers the basic theory and recent results on elliptic modular forms. It includes a detailed exposition of the representation theory of profinite groups and contains several new results from the author. The book will appeal to graduate students and researchers in number theory (including algebraic and analytic number theorists) and arithmetic algebraic geometry.

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