On the group of purely inseparable points of an abelian variety defined over a function field of positive characteristic

Let $K$ be the function field of a smooth and proper curve $S$ over an algebraically closed field $k$ of characteristic $p>0$. Let $A$ be an ordinary abelian variety over $K$. Suppose that the N'eron model $\text{CA}$ of $A$ over $S$ has a closed fibre ${\text{CA}}_s$, which is an abelian variety of $p$-rank 0. We show that under these assumptions the group $A(K^{\text{perf}})\mathrm{/}{\text{Tr}}_{K\mathrm{\mid }k}(A)(k)$ is finitely generated. Here $K^{\text{perf}}=K^{p^{-\mathrm{\infty }}}$ is the maximal purely inseparable extension of $K$. This result implies that in some circumstances, the “full” Mordell-Lang conjecture, as well as a conjecture of Esnault and Langer, are verified.