Andrew Wiles' recent work \ref[Ann. of Math. (2) 141 (1995), no. 3, 443--551] on the Taniyama conjecture, completed in a recent joint article with Richard Taylor \ref[Ann. of Math. (2) 141 (1995), no. 3, 553--572], has generated an enormous amount of interest in the mathematical community. As a response to this interest, a number of surveys have appeared. This article is another such survey, covering in detail the mathematical background of the proof. The author says in the introduction: "It is premature to undertake a detailed analysis of the results of Wiles and Taylor-Wiles. The aim of this survey is more modest: to present an introduction to the circle of ideas which form the background for these results. Because of the intense publicity surrounding Fermat's last theorem, a good deal of the material I have chosen has been discussed in news and expository articles which were written in connection with Wiles' 1993 announcement. In view of the burgeoning literature on the subject, I imagined these notes as a somewhat biased guide to reference works and expository articles. In the end, what I have written might best be characterized as an abbreviated survey with a disproportionately large list of references."

The author has been a major player in creating the mathematics that undergirds Wiles' work, and this account of that mathematics reflects that fact by its depth, precision, and useful insights. This survey is more detailed than many others that have appeared so far, and while it does not go into the mechanics of the proof, it does offer an excellent pathway into the background mathematics, with several useful insights along the way. The list of references is impressive, and very helpful. Mathematicians interested in learning more about this circle of ideas would do well to start here.

Reviewed by Fernando Q. Gouvêa