Torsion points and height jumping in higher-dimensional families of abelian varieties

TL;DR

This paper explores the behavior of torsion points and height jumping in higher-dimensional families of abelian varieties, proposing a conjecture that generalizes a classical result and linking it to the Uniform Boundedness Conjecture.

Contribution

It introduces a new conjecture extending Silverman and Tate's finiteness result to higher dimensions and proves some special cases, connecting it to broader conjectures in number theory.

Findings

Proposed a conjecture on non-Zariski density of points with small height in higher dimensions.

Proved special cases of the conjecture.

Linked the conjecture to the Uniform Boundedness Conjecture.

Abstract

In 1983 Silverman and Tate showed that the set of points in a 1-dimensional family of abelian varieties where a section of infinite order has `small height' is finite. We conjecture a generalisation to higher-dimensional families, where we replace `finite' by `not Zariski dense'. We show that this conjecture would imply the Uniform Boundedness Conjecture for torsion points on abelian varieties. We then prove a few special cases of this new conjecture.