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 with work of J. Cremona, N. Elkies, and J.-F. Mestre on explicit modular forms computations. 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.