The Birch and Swinnerton-Dyer Conjecture is the Clay Math Institute's million dollar prize problem in algebraic number theory. The variant selected for the prize is the following CONJECTURE: If E is an elliptic curve over Q, then the rank of E equals the order of vanishing of the L-function L(E,s) of E at s=1. In this course I will precisely state the conjecture then briefly discuss its generalizations over number fields, to abelian varieties, motives, and $p$-adic variants. I will also discuss in some detail work of Kolyvagin on the conjecture, and briefly work of Gross-Zagier, Rubin, and Kato. Finally, I will explain work of myself with several students to computationally verify a refinement of the conjecture for a large number of particular curves.

This would be an excellent course to take if you are taking my modular forms course right now, or are interested in the Birch and Swinnerton-Dyer conjecture seminar I am organizing.

Prerequisites: The only prerequisites are complex analysis, and some familiarity with algebraic number theory and algebraic curves.