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

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 Z, then there are no infinitely p-divisible points of order a power of p.