Algebraic flows on Shimura varieties

TL;DR

This paper formulates conjectures about algebraic flows on Shimura varieties and proves key theorems including the logarithmic Ax-Lindemann theorem and results on the closure of totally geodesic subvarieties.

Contribution

It introduces new conjectures on algebraic flows on Shimura varieties and proves significant theorems advancing understanding of their geometric and topological properties.

Findings

Proof of the logarithmic Ax-Lindemann theorem

Results on the topological closure of totally geodesic subvarieties

Progress towards conjectures on algebraic flows

Abstract

In this paper we formulate some conjectures about algebraic flows on Shimura varieties. In the first part of the paper we prove the `logarithmic Ax-Lindemann theorem'. We then prove a result concerning the topological closure of the images of totally geodesic subvarieties of symmetric spaces uniformising Shimura varieties. This is a special case of our conjectures.