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.
Keywords
elliptic curves, nonarchimedean fields, function fields,
Berkovich analytic spaces, equidistribution, integral
points
