Modular Forms and Fermat’s Last Theorem This volume contains expanded versions of lectures given at an instructional conference on number theory and arithmetic geometry held August 9 through 18 1995 at Boston University. Contributors includeThe purpose of the conference and of this book is to introduce and explain the many ideas and techniques used by Wiles in his proof that every (semi stable) elliptic curve over Q is modular and to explain how Wiles result can be combined with Ribets theorem and ideas of Frey and Serre to show at long last that Fermats Last Theorem is true. The book begins with an overview of the complete proof followed by several introductory chapters surveying the basic theory of elliptic curves modular functions modular curves Galois cohomology and finite group schemes. Representation theory which lies at the core of Wiles proof is dealt with in a chapter on automorphic representations and the Langlands Tunnell theorem and this is followed by in depth discussions of Serres conjectures Galois deformations universal deformation rings Hecke algebras complete intersections and more as the reader is led step by step through Wiles proof. In recognition of the historical significance of Fermats Last Theorem the volume concludes by looking both forward and backward in time reflecting on the history of the problem while placing Wiles theorem into a more general Diophantine context suggesting future applications. Students and professional mathematicians alike will find this volume to be an indispensable resource for mastering the epoch making proof of Fermats Last Theorem