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

TL;DR

This paper proves that for certain abelian varieties over function fields of positive characteristic, the group of purely inseparable points modulo the trace is finitely generated, supporting conjectures like Mordell-Lang in these cases.

Contribution

It establishes the finite generation of purely inseparable points for ordinary abelian varieties with specific reduction properties over function fields.

Findings

The group of purely inseparable points modulo the trace is finitely generated.

Supports the full Mordell-Lang conjecture in certain positive characteristic cases.

Verifies a conjecture of Esnault and Langer under specified conditions.

Abstract

Let $K$ be the function field of a smooth and proper curve $S$ over an algebraically closed field $k$ of characteristic $p>0$. Let $A$ be an ordinary abelian 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 show that under these assumptions the group $A(K^\perf)/\Tr_{K|k}(A)(k)$ is finitely generated. Here $K^\perf=K^{p^{-\infty}}$ is the maximal purely inseparable extension of $K$. This result implies that in some circumstances, the "full" Mordell-Lang conjecture, as well as a conjecture of Esnault and Langer, are verified.