Let k be a field which is finitely generated over the algebraic closure of a finite field and let $A$ be an abelian variety over k without isotrivial isogeny factors. By the Lang-Néron theorem, the group of k-rational torsion points of A is finite. In this talk we show that the same is true for the group of torsion points defined on a perfect closure of k, giving a positive answer to a question of Esnault. To achieve this, we study the Dieudonné module of A, its overconvergent incarnation and various algebraic groups attached to them. This is a joint work with Marco D'Addezio.