Two Million Dollars of Math: An Introduction to the Birch and Swinnerton-Dyer Conjecture

04/28/2009 - 12:15pm
04/28/2009 - 1:10pm
Christopher Towse (Scripps College)

Two of the Clay Institute's seven Millennium ("million dollar") Problems involve the location of the zeros of certain zeta functions. The more famous of the two, the Riemann Hypothesis, certainly motivated the second, the Birch and Swinnerton-Dyer Conjecture (BSD). Roughly speaking, BSD predicts whether an elliptic curve has a finite or an infinite number of rational points. For this talk, we will concentrate on the definition and basic properties of the L-function of an elliptic curve. Beginning with a specific family of elliptic curves, we will show how one uses basic character theory to count points on the curves, modulo $p$. Eventually, we will construct the Hasse-Weil L-function and sketch a proof (using integral transforms) of its analytic continuation and functional equation. These fundamental definitions, constructions, and proofs are analogous (parallel) to similar constructions for the Riemann Zeta Function.

This is probably better presented as a semester long course!

ML 211