Inner Galois Equidistribution in S-Hecke orbits

TL;DR

This paper proves new results on unlikely intersections in Shimura varieties, specifically the Andre-Pink-Zannier conjecture, using ergodic theory and assuming certain conjectures hold for abelian type Shimura varieties.

Contribution

It advances the understanding of the Andre-Pink-Zannier conjecture by applying ergodic methods to obtain stronger results under specific conjectural assumptions.

Findings

Stronger conclusions on unlikely intersections in Shimura varieties.

Application of ergodic theorems to problems in arithmetic geometry.

Conditional results assuming S-Shafarevich and S-semisimplicity conjectures.

Abstract

We obtain results on the so-called Andre-Pink-Zannier conjecture which is a special case of a the Zilber-Pink conjecture on unlikely intersections in Shimura varieties. Our methods rely on an ergodic theorem of Richard-Zamojski and we are able to obtain stronger conclusions that those of the Andre-Pink-Zannier conjecture in the special case we consider. We work under the assumption of the S-Shafarevich conjecture and S-semisimplicity conjecture which hold for Shimura varieties of abelian type.