Andrew Wiles

Modular elliptic curves and Fermat's last theorem

When Andrew John Wiles was 10 years old, he read Eric Temple Bell’s The Last Problem and was so impressed by it that he decided that he would be the first person to prove Fermat’s Last Theorem. This theorem states that there are no nonzero integers a, b, c, n with n > 2 such that an + bn = cn. The object of this paper is to prove that all semistable elliptic curves over the set of rational numbers are modular. Fermat’s Last Theorem follows as a corollary by virtue of previous work by Frey, Serre and Ribet.

Ring-Theoretic Properties of Certain Hecke Algebras

The purpose of this article is to provide a key ingredient of [W2] by establishing that certain minimal Hecke algebras considered there are complete intersections. As is recorded in [W2], a method going back to Mazur [M] allows one to show that these algebras are Gorenstein, but for the complete intersection property a new approach is required. The methods of this paper are related to those of Chapter 3 of [W2]. The methods of Section 3 of this paper are based on a previous approach of one of us (A.W.). We would like to thank Henri Darmon, Fred Diamond and Gerd Faltings for carefully reading the first version of this article. Gerd Faltings has also suggested a simplification of our argument as well as of the argument of Chapter 3 of [W2] and we would like to thank him for allowing us to reproduce these in the appendix to this paper. R. T. would like to thank A. W. for his invitation to collaborate and for sharing his many insights into the questions considered. R. T. would also like to thank Princeton University, Universite de Paris 7 and Harvard University for their hospitality during this collaboration. A. W. was supported by an NSF grant.

Base change and a problem of Serre

We establish a version of “level-lowering” for mod p Galois representations arising from the reductions of representations associated to Hilbert modular forms. In particular, we show that level-lowering can be easily achieved if one replaces the base field with a suitable solvable extension. This is often enough for applications to proving the modularity of p-adic representations.

Solvable points on genus one curves

A genus one curve defined over Q which has points over Qp for all primes p may not have a rational point. It is natural to study the classes of Q-extensions over which all such curves obtain a global point. In this article, we show that every such genus one curve with semistable Jacobian has a point defined over a solvable extension of Q.

Modular Forms, Elliptic Curves, and Fermat’s Last Theorem

The equation of Fermat has undoubtedly had a far greater influence on the development of mathematics than anyone could have imagined. After 1847 most serious mathematical approaches to the problem followed the line introduced by Kummer. This approach involved a detailed analysis of the ideal class groups of cyclotomic fields.