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

Abstract: 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 etale self-isogeny 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.