Infinitely $p$-divisible points on abelian varieties defined over function fields of characteristic $p>0$

TL;DR

This paper proves that infinitely p-divisible points on abelian varieties over certain function fields are torsion, and that no such points of p-power order exist when the endomorphism ring is Z, advancing understanding in positive characteristic arithmetic geometry.

Contribution

It establishes new results on the nature of p-divisible points on abelian varieties over function fields in characteristic p, addressing questions posed by Benoist, Bouscaren, and Pillay.

Findings

Infinitely p-divisible points are torsion on abelian varieties over function fields of transcendence degree one.

No infinitely p-divisible points of p-power order exist when the endomorphism ring is Z.

Results contribute to the understanding of p-divisibility in positive characteristic settings.

Abstract

In this article we consider some questions raised by F. Benoist, E. Bouscaren and A. Pillay. We prove that infinitely $p$-divisible points on abelian varieties defined over function fields of transcendence degree one over a finite field are necessarily torsion points. We also prove that when the endomorphism ring of the abelian variety is $\mZ$ then there are no infinitely $p$-divisible points of order a power of $p$.