William Stein's homepage

Introduction

My research program reflects the essential interplay between abstract theory and explicit machine computation during the latter half of the twentieth century; it sits at the intersection of recent work of B. Mazur, K. Ribet, R. Taylor, and A. Wiles on Galois representations (see [21,24,26,28]) with work of J. Cremona, N. Elkies, and J.-F. Mestre on explicit modular forms computations (see [9,11]).

In 1969 B. Birch [4] described computations that led to the most fundamental open conjecture in the theory of elliptic curves:

I want to describe some computations undertaken by myself and Swinnerton-Dyer on EDSAC by which we have calculated the zeta-functions of certain elliptic curves. As a result of these computations we have found an analogue for an elliptic curve of the Tamagawa number of an algebraic group; and conjectures (due to ourselves, due to Tate, and due to others) have proliferated.

The rich tapestry of arithmetic conjectures and theory we enjoy today would not exist without the ground-breaking application of computing by Birch and Swinnerton-Dyer. Computations in the 1980s by Mestre were key in convincing Serre that his conjectures on modularity of odd irreducible Galois representations were worthy of serious consideration (see [24]). These conjectures have inspired much recent work; for example, Ribet's proof of the $\epsilon$-conjecture, which played an essential role in Wiles's proof of Fermat's Last Theorem.

My work on the Birch and Swinnerton-Dyer conjecture for modular abelian varieties and search for new examples of modular icosahedral Galois representations has led me to discover and implement algorithms for explicitly computing with modular forms. My research, which involves finding ways to compute with modular forms and modular abelian varieties, is driven by outstanding conjectures in number theory.