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Let $K$ be the function field of a smooth and proper curve $S$ over analgebraically closed field $k$ of characteristic $p>0$. Let $A$ be an ordinaryabelian variety over $K$. Suppose that the N\'eron model $\CA$ of $A$ over $S$has a closed fibre $\CA_s$, which is an abelian variety of $p$-rank 0. We showthat under these assumptions the group $A(K^\perf)/\Tr_{K|k}(A)(k)$ is finitelygenerated. Here $K^\perf=K^{p^{-\infty}}$ is the maximal purely inseparableextension of $K$. This result implies that in some circumstances, the "full"Mordell-Lang conjecture, as well as a conjecture of Esnault and Langer, areverified.

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