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 Dieudonné module of certain semiabelian schemes over smooth varieties.

arXiv commentv3: 16 pages, shortened and final version. To appear in Compositio Mathematica. Some of the results in Part II will appear elsewhere. v2: minor edits. v1:37 pages, comments are very welcome

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