Non-archimedean canonical measures on abelian varieties

TL;DR

This paper provides an explicit convex geometric description of canonical measures on subvarieties of abelian varieties over non-archimedean fields, with applications to equidistribution of small points.

Contribution

It introduces a convex geometric framework for understanding canonical measures on abelian varieties using tropicalization and Raynaud extensions.

Findings

Explicit description of measures in convex geometric terms

Connection between tropicalization and canonical measures

Applications to equidistribution of small points

Abstract

For a closed d-dimensional subvariety X of an abelian variety A and a canonically metrized line bundle L on A, Chambert-Loir has introduced measures $c_1(L|_X)^{\wedge d}$ on the Berkovich analytic space associated to A with respect to the discrete valuation of the ground field. In this paper, we give an explicit description of these canonical measures in terms of convex geometry. We use a generalization of the tropicalization related to the Raynaud extension of A and Mumford's construction. The results have applications to the equidistribution of small points.