Park City lectures on elliptic curves over function fields

TL;DR

This paper provides an overview of classical and recent results on elliptic curves over function fields, focusing on the Birch and Swinnerton-Dyer conjecture, Tate conjecture, and high-rank constructions.

Contribution

It synthesizes classical theorems with recent advances on elliptic curves over function fields, highlighting connections and new explicit point constructions.

Findings

Classical results on BSD conjecture over function fields

Connections between BSD and Tate conjecture for surfaces

Recent methods for constructing high-rank elliptic curves

Abstract

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.