See the Park City notes of Ulmer, and maybe other stuff in the same book.
arXiv:1101.1939 Park City lectures on elliptic curves over function fields from arXiv Front: math.AG by Douglas Ulmer 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.
nLab page on Function fields