An ordinary abelian variety with an étale self-isogeny of $p$-power degree and no isotrivial factors

We construct, for every prime $p$, a function field $K$ of characteristic $p$ and an ordinary abelian variety $A$ over $K$, with no isotrivial factors, that admits an étale self-isogeny $\phi : A \to A$ of $p$‑power degree. As a consequence, we deduce that there exist ordinary abelian varieties over function fields whose groups of points over the maximal purely inseparable extension is not finitely generated, answering in the negative a question of Thomas Scanlon.

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Received 19 April 2021

Received revised 10 June 2021

Accepted 20 July 2021

Published 29 September 2022