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

TL;DR

This paper constructs examples of ordinary abelian varieties over function fields of characteristic p that have an etale self-isogeny of p-power degree and no isotrivial factors, demonstrating new phenomena in their arithmetic structure.

Contribution

It provides explicit constructions of such abelian varieties, answering a question about the finiteness of their points over maximal purely inseparable extensions.

Findings

Existence of non-finitely generated groups of points over purely inseparable extensions.

Construction of ordinary abelian varieties with specific self-isogenies.

Negative answer to a question by Thomas Scanlon.

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.