Perfect points of abelian varieties

Let $p$ be a prime number, $k$ finite field of characteristic $p>0$ and $K/k$ finitely generated extension fields. $A$ $K$ -abelian variety such that all the isogeny factors are neither isotrivial nor -rank zero. We give necessary sufficient condition for generation $A(K^{\mathrm {perf}})$ in terms action $\mathrm {End}(A)\otimes \mathbb {Q}_p$ on -divisible group $A[p^{\infty }]$ . In particular, we prove if is division algebra, then generated. This implies ‘full’ Mordell–Lang conjecture these abelian varieties. addition, infinitely elements torsion. These reprove extend previous results to non-ordinary case.