  These are the notes from a course of five lectures at the 2009 Park City Math
Institute. The focus is on elliptic curves over function fields over finite
fields. In the first three lectures, we explain the main classical results
(mainly due to Tate) on the Birch and Swinnerton-Dyer conjecture in this
context and its connection to the Tate conjecture about divisors on surfaces.
This is preceded by a "Lecture 0" on background material. In the remaining two
lectures, we discuss more recent developments on elliptic curves of large rank
and constructions of explicit points in high rank situations.
