Points of Small Height on Semiabelian Varieties

TL;DR

This paper proves the Equidistribution Conjecture for all semiabelian varieties over number fields, extending previous results limited to almost split cases, and introduces new techniques to handle negative heights.

Contribution

It extends equidistribution results to general semiabelian varieties by developing an asymptotic approach, also providing new proofs for the Bogomolov and Strong Equidistribution Conjectures.

Findings

Proved the Equidistribution Conjecture for all semiabelian varieties.

Developed an asymptotic method to handle negative heights.

Provided new proofs for the Bogomolov and Strong Equidistribution Conjectures.

Abstract

The Equidistribution Conjecture is proved for general semiabelian varieties over number fields. Previously, this conjecture was only known in the special case of almost split semiabelian varieties through work of Chambert-Loir. The general case has remained intractable so far because the height of a semiabelian variety is negative unless it is almost split. In fact, this places the conjecture outside the scope of Yuan's equidistribution theorem on algebraic dynamical systems. To overcome this, an asymptotic adaption of the equidistribution technique invented by Szpiro, Ullmo, and Zhang is used here. It also allows a new proof of the Bogomolov Conjecture and hence a self-contained proof of the Strong Equidistribution Conjecture in the same general setting.