Perfect points of abelian varieties

Let p be a prime number, k a finite field of characteristic p > 0 and K/k a finitely generated extension of fields. 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(Kperf) in terms of the action of End(A) circle times Q(p) on the p-divisible group A[p(infinity)] of A. In particular, we prove that if End(A) circle times 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.