This paper is a survey of the proof of the ShimuraTaniyama conjecture
for semistable elliptic curves obtained recently by A. Wiles \ref[Ann.
of Math. (2) 141 (1995), no. 3, 443551; MR 96d:11071] and R. Taylor
and Wiles \ref[Ann. of Math. (2) 141 (1995), no. 3, 553572; 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
