Non-archimedean equidistribution on elliptic curves with global applications

Details Non-archimedean equidistribution on elliptic curves with global applications Non-archimedean equidistribution on elliptic curves with global applications

2007-10-21

arxiv.org/abs/0710.3957v3

Mathematics

Number Theory

Final version. Inserted a full proof of Corollary 3, clarified the definition of the topology on the Berkovich unit disc, revi

Scientific paper

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.

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