Non-archimedean equidistribution on elliptic curves with global applications

TL;DR

This paper establishes a non-archimedean equidistribution theorem for elliptic curves over algebraically closed fields, providing explicit bounds and applications to global function fields, including small points and torsion points.

Contribution

It introduces a new inequality for equidistribution error bounds on Berkovich spaces and applies it to function field analogues of key theorems in arithmetic geometry.

Findings

Proved a bound on the error term in equidistribution testing.

Established a function field analogue of the Szpiro-Ullmo-Zhang theorem.

Derived a quantitative result on the finiteness of S-integral torsion points.

Abstract

Let $E$ be an elliptic curve over an algebraically closed, complete, non-archimedean field $K$, and let ${\mathsf E}$ denote the Berkovich analytic space associated to $E/K$. We study the $\mu$-equidistribution of finite subsets of $E(K)$, where $\mu$ is a certain canonical unit Borel measure on ${\mathsf E}$. Our main result is an inequality bounding the error term when testing against a certain class of continuous functions on ${\mathsf E}$. 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-Rumely on the finiteness of $S$-integral torsion points. Both applications are given in explicit quantitative form.