Trace of abelian varieties over function fields and the geometric Bogomolov conjecture

TL;DR

This paper reduces the geometric Bogomolov conjecture for abelian varieties to a special case involving nowhere degenerate varieties with trivial trace, and confirms it for certain isogenous abelian varieties.

Contribution

It establishes a reduction of the geometric Bogomolov conjecture to specific cases and proves it for abelian varieties with particular structural properties.

Findings

Reduction of the conjecture to nowhere degenerate abelian varieties with trivial trace

Verification of the conjecture for abelian varieties with isogenous constant subvarieties

Investigation of subvarieties of abelian schemes over constant varieties

Abstract

We prove that the geometric Bogomolov conjecture for any abelian varieties is reduced to that for nowhere degenerate abelian varieties with trivial trace. In particular, the geometric Bogomolov conjecture holds for abelian varieties whose maximal nowhere degenerate abelian subvariety is isogenous to a constant abelian variety. To prove the results, we investigate closed subvarieties of abelian schemes over constant varieties, where constant varieties are varieties over a function field which can be defined over the constant field of the function field.