Computational aspects of elliptic curves and modular forms

Authors:
Victor S. Miller
Affiliations:
IDA, Center for Communications Research, Princeton, NJ, USA
Venue:
Proceedings of the 36th international symposium on Symbolic and algebraic computation
Year:
2011

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Abstract

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.