Elliptic curves, L-functions and a conjecture of Mazur and Tate

The conjecture of Birch and Swinnerton-Dyer famously relates rational points on elliptic curves to special values of their associated L-functions (certain Dirichlet series formed out of point counting on the elliptic curve over finite fields). In this talk we introduce elliptic curves and L-functions, discuss the history and known results on Birch-Swinnertyon-Dyer, and then turn to recent work (joint with Emerton and Pollack) on a closely related conjecture of Mazur and Tate.