# A couple of conjectures in arithmetic dynamics over fields of positive characteristic

Dragos Ghioca

## Affiliation:

University of British Columbia

## Date:

Tue, 01/12/2020 - 12:00pm

## Venue:

RC-4082, The Red Centre, UNSW

## Abstract:

The Dynamical Mordell-Lang Conjecture predicts the structure of the intersection between a subvariety $V$ of a variety $X$ defined over a field $K$ of characteristic $0$ with the orbit of a point in $X(K)$ under an endomorphism $\Phi$ of $X$. The Zariski dense conjecture provides a dichotomy for any rational self-map $\Phi$ of a variety $X$ defined over an algebraically closed field $K$ of characteristic $0$: either there exists a point in $X(K)$ with a well-defined Zariski dense orbit, or $\Phi$ leaves invariant some non-constant rational function $f$. For each one of these two conjectures we formulate an analogue in characteristic $p$; in both cases, the presence of the Frobenius endomorphism in the case $X$ is isotrivial creates significant complications which we will explain in the case of algebraic tori.

This talk is part of the online Number Theory Web Seminar, and will be streamed live on Zoom.