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

Rössler, D

2015

## Journal:

Commentarii Mathematici Helvetici

## Last Updated:

2020-01-21T09:11:33.77+00:00

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 $p>0$. 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.

745035

Not Submitted

Journal Article