The Birch and Swinnerton-Dyer conjecture

The Birch and Swinnerton-Dyer conjecture (BSD) was formulated in the 1960s in an attempt to understand rational solutions of polynomial equations. At its core, it predicts a relation between the algebraic and the analytic information of elliptic curves. The conjecture quickly became one of the central problems in number theory, and in the year 2000, it was chosen by the Clay Mathematics Institute as one of the seven "Millennium Prize Problems" for which a one million dollar prize is offered.

In this talk, we will explain the mathematical objects involved in the BSD conjecture. We are going to discuss the classical origins of the conjecture, and view its modern formulation using the language of L-functions. In the end, we hope to be able to convince an undergraduate audience of the importance of this conjecture to number theory.