Integral torsion points on abelian varieties over function fields

TL;DR

This paper proves a conjecture about the non-density of integral torsion points on abelian varieties over function fields, using a Tate--Voloch type theorem and equidistribution results.

Contribution

It establishes an analogue of Ih's conjecture over function fields and introduces a Tate--Voloch type theorem for abelian varieties in this setting.

Findings

Proves non-Zariski density of integral torsion points over function fields.

Establishes a Tate--Voloch type theorem for abelian varieties over completions.

Provides a logarithmic equidistribution result for Galois orbits of torsion points.

Abstract

We prove an analogue, over global function fields, of a conjecture due to Su-Ion Ih concerning the non-Zariski density of torsion points on abelian varieties that are integral with respect to a given non-special divisor. Along the way, we establish a Tate--Voloch type theorem for abelian varieties over completions of global function fields, which allows us to obtain a logarithmic equidistribution result for Galois orbits of torsion points.