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

### Damian Rössler

Université Paul Sabatier, Toulouse, France

## 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 Néron model $\mathcal A$ of $A$ over $S$ has some closed fibre $\mathcal A_s$, which is an abelian variety of $p$-rank $0$.

We show that in this situation the group $A(K^{perf})$ is finitely generated (thus generalizing a special case of the Lang-Néron theorem). Here $K^{perf}=K^{p^{-\infty}}$ is the maximal purely inseparable extension of $K$. This result implies in particular that 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.

## Cite this article

Damian Rössler, On the group of purely inseparable points of an abelian variety defined over a function field of positive characteristic. Comment. Math. Helv. 90 (2015), no. 1, pp. 23–32

DOI 10.4171/CMH/344