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

## TL;DR

This paper investigates the structure of purely inseparable points on abelian varieties over function fields of positive characteristic, providing conditions for finite generation and finiteness of torsion, and proving a conjecture on Verschiebung divisibility.

## Contribution

It introduces geometric conditions ensuring finite generation of certain groups of points and relates vector bundle filtrations to finite flat group schemes, also proving a conjecture on Verschiebung divisibility.

## Key findings

- Conditions for finite generation of $A(K^{
m perf})$
- Finiteness of the $p$-primary torsion subgroup of $A(K^{
m sep})$
- Complete proof of a conjecture of Esnault and Langer

## Abstract

Let $A$ be an abelian variety over the function field $K$ of a curve over a finite field. We describe several mild geometric conditions ensuring that the group $A(K^{\rm perf})$ is finitely generated and that the $p$-primary torsion subgroup of $A(K^{\rm sep})$ is finite. This gives partial answers to questions of Scanlon, Ghioca and Moosa, and Poonen and Voloch. We also describe a simple theory (used to prove our results) relating the Harder-Narasimhan filtration of vector bundles to the structure of finite flat group schemes of height one over projective curves over perfect fields. Finally, we use our results to give a complete proof of a conjecture of Esnault and Langer on Verschiebung divisibility of points in abelian varieties over function fields.