Equidistribution over function fields

TL;DR

This paper proves equidistribution results for small points and subvarieties over function fields, extending known number field results and providing explicit descriptions in special cases.

Contribution

It generalizes Yuan's equidistribution theorem from number fields to function fields and describes measures explicitly for subvarieties of abelian varieties.

Findings

Proves equidistribution of small points in projective varieties over function fields.

Extends equidistribution to subvarieties in algebraic dynamical systems.

Provides explicit measure descriptions for subvarieties of abelian varieties.

Abstract

We prove equidistribution of a generic net of small points in a projective variety X over a function field K. For an algebraic dynamical system over K, we generalize this equidistribution theorem to a small generic net of subvarieties. For number fields, these results were proved by Yuan and we transfer here his methods to function fields. If X is a closed subvariety of an abelian variety, then we can describe the equidistribution measure explicitly in terms of convex geometry.