Perfect points of abelian varieties

Abstract: Let $k$ be an algebraic extension of $\mathbb F_p$ and $K/k$ a regular extension of fields (e.g. $\mathbb F_p(T)/\mathbb F_p$). Let $A$ be a $K$-abelian variety such that all the isogeny factors are neither isotrivial nor of $p$-rank zero. We give a necessary and sufficient condition for the finite generation of $A(K^{perf})$ in terms of the action of $End(A)\otimes \mathbb Q_p$ on the $p$-divisible group $A[p^{\infty}]$ of $A$. In particular we prove that if $End(A)\otimes \mathbb Q_p$ is a division algebra then $A(K^{perf})$ is finitely generated. This implies the "full" Mordell-Lang conjecture for these abelian varieties. In addition we prove that all the infinitely $p$-divisible elements in $A(K^{perf})$ are torsion. These reprove and extend previous results to the non ordinary case. One of the main technical intermediate result is an overconvergence theorem for the Dieudonn\'e module of certain semiabelian schemes over smooth varieties.