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

Rössler, D

23 February 2059

## Journal:

Commentarii Mathematici Helvetici

## Last Updated:

2020-09-27T01:32:40.247+01:00

1

90

10.4171/CMH/344

23-32

## abstract:

Let $K$ be the function field of a smooth and proper curve $S$ over an algebraically closed field $k$ of characteristic $p&gt;0$. 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 is verified in the situation described above. The proof relies on the theory of semistability (of vector bundles) in positive characteristic and on the existence of the compactification of the universal abelian scheme constructed by Faltings-Chai.

745035

Submitted

Journal Article