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

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