Literature Guide for Mathematics 573

There are several survey papers dealing with the ideas involved in the proof of Fermat's Last Theorem. They contain further references to the literature. All were written prior to the paper of Wiles and Taylor which completes Wiles' attack on the problem. (The papers of Wiles and Taylor will appear in Annals of Math.)

A Marvelous Proof, F. Gouvea,
Amer. Math. Monthly 1994, 203-221

This survey describes the state of affairs as of late 1993 and gives a glimpse of most of the main ideas of the proof.

A report on Wiles' Cambridge lectures, K. Rubin and A. Silverberg,
Bull. Amer. Math. Soc. (N.S.), 31 (1) 1994, 15--38

This summarizes A. Wiles' lectures in Cambridge, UK in June 1993. It requires more background than the survey above. This is available from the AMS.

The Shimura-Taniyama Conjecture (d'apres Wiles), H. Darmon
CICMA Lecture Notes 1994-02

This survey gives a detailed discussion of the proof that there are an an infinite number of elliptic curves over the rationals which are not isomorphic over the complex numbers and whose L-functions are associated to modular forms. It is much more detailed than the two references above. This monograph made be obtained by sending email to cicma@abacus.concordia.ca and providing the monograph title and mailing address where you wish to receive the report.

A recent preprint by K. Ribet has extensive literature references
Galois representations and modular forms, K. Ribet
MSRI preprint 1995-032 (available from the MSRI preprint server
There are other surveys written prior to 1993 which explain the relation between modular forms and Fermat's Last Theorem.
Number theory as gadfly, B. Mazur
Am. Math. Monthly,98, 1991, 593-610

From the Taniyama-Shimura conjecture to Fermat's last Theorem, K. Ribet
Ann. Fac. Sci. Toulouse,11 (1990) 116--139

There are many open problems concerning Galois representations and modular forms. Some of the deepest are described in Serre's paper below.
Sur les representations modulaires de degre 2 de GalQ, Jean-Pierre Serre
Duke Math. J., 54, 1987, 179--230

For background on algebraic curves and rational points prior to developments of the last decade see the survey below.
Arithmetic on curves,B. Mazur
Bull. AMS 14 (2) 1986, 207--259