University of Georgia
Mathematics Department Colloquium
William Stein
UC Berkeley
February 1, 2000
Modularity and the Birch and Swinnerton-Dyer Conjecture
Abstract:
The subject of this lecture is the arithmetic of elliptic curves over
the rational numbers. Elliptic curves play a central role in
number theory and cryptography; they were a key player in the
recent proof of Fermat's Last Theorem, and are used to make and
sometimes break cryptosystems. The Birch and Swinnerton-Dyer
conjecture ties together the arithmetic invariants of an elliptic
curve. The modularity theorem reveals that every elliptic curve
has a vast amount of structure. In this talk I will describe these
ideas and how they relate, then explain some of my computational
investigations into natural analogues of the Birch and Swinnerton-Dyer
conjecture that are suggested by the modularity theorem.