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. Contributor's 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 Ribet's theorem and ideas of Frey and Serre to show, at long last, that Fermat's 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 Serre's 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 Fermat's 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 Fermat's Last Theorem.
Kenneth A. Ribet studied mathematics at Brown University and Harvard University. He received his PhD in 1973 from Harvard, where his advisor was John Tate. After three years of teaching in Princeton and two years of research in Paris, Ribet joined the University of California, Berkeley faculty in 1978.
Ribet is known for his work in number theory and algebraic geometry. He played a prominent role in the proof of Fermat's Last Theorem by showing that this statement was a logical consequence of a conjecture about elliptic curves. (Andrew Wiles proved this conjecture in 1995, thereby obtaining Fermat's Last Theorem as a corollary.)
Ribet was elected to the American Academy of Arts and Sciences in 1997 and the National Academy of Sciences in 2000. He was awarded the Fermat Prize in 1989 and received an honorary PhD from Brown University in 1998. Ribet was inducted as a Vigneron d'honneur by the Jurade de Saint Emilion in 1988.
