Non-archimedean equidistribution on elliptic curves with global applications
Abstract
Description
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 $μ$-equidistribution of finite subsets of $E(K)$, where $μ$ 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.
Final version. Inserted a full proof of Corollary 3, clarified the definition of the topology on the Berkovich unit disc, revised the proof of Proposition 7, included some new references, and made some other minor changes
Final version. Inserted a full proof of Corollary 3, clarified the definition of the topology on the Berkovich unit disc, revised the proof of Proposition 7, included some new references, and made some other minor changes