Nonarchimedean equidistribution on elliptic curves with global applications

Abstract

Let K be an algebraically closed, complete, nonarchimedean field, let E/K be an elliptic curve, and let E denote the Berkovich analytic space associated to E/K. We study the μ-equidistribution of finite subsets of E(K), where μ is a certain canonical unit Borel measure on E. Our main result is an inequality bounding the error term when testing against a certain class of continuous functions on E. We then give two applications to elliptic curves over global function fields: We prove a function field analogue of the Szpiro-Ullmo-Zhang equidistribution theorem for small points, and a function field analogue of a result of Baker, Ih, and Rumely on the finiteness of S-integral torsion points. Both applications are given in explicit quantitative form.