This talk is intended as an introduction to the introductory workshop. It will introduce some of the main players -- special cohomology classes, Euler systems, p-adic L-functions -- largely by focusing on the special case of cyclotomic units and Dirichlet characters.

This talk will describe some recent applications of p-adic L-functions and Iwasawa theory to the arithmetic of elliptic curves, especially to establishing the p-parts of the the Birch-Swinnerton-Dyer formula in cases of rank at most one.