computational aspects of elliptic curves and modular forms
The ultimate motivation for much of the study of Number Theory is the solution of Diophantine Equations -- finding integer solutions to systems of equations. Elliptic curves comprise a large, and important class of such equations. Throughout the history of their study Elliptic Curves have always had a strong algorithmic component. In the early 1960's Birch and Swinnerton-Dyer developed systematic algorithms to automate a generalization of a procedure called "descent" which went back to Fermat. The data they obtained was instrumental in formulating their famous conjecture, which is now one of the Clay Mathematical Institute's Millenium prizes.
elliptic curves