This paper is a survey of the proof of the Shimura-Taniyama conjecture for semistable elliptic curves obtained recently by A. Wiles \ref[Ann. of Math. (2) 141 (1995), no. 3, 443--551; MR 96d:11071] and R. Taylor and Wiles \ref[Ann. of Math. (2) 141 (1995), no. 3, 553--572; MR 96d:11072]. The application of this result to Fermat's last theorem via Frey curves is recalled but not particularly emphasized.

The author starts with the definition of an elliptic curve defined over a field $K$. Then he goes on to introduce modular forms, modular elliptic curves (via $L$-functions) and the various relevant Galois representations. The last five sections of the paper describe the problem of classifying the deformations of a Galois representation $\rho\sb0\colon{\rm Gal}({\overline Q}/ Q) \rightarrow{\rm GL}\sb2({\overline F}\sb p)$ and Wiles' method to decide whether such a deformation is modular (i.e. is isomorphic to the representation associated to a weight $2$ cusp form). In the final section the author explains how Wiles proved the modularity of a semistable elliptic curve $E$ over $Q$ by applying his methods to the Galois representation associated to the $3$-torsion of $E$ (sometimes the $5$-torsion also has to be considered).

{For the entire collection see MR 96f:11004.}

Reviewed by Andrea Mori