A characteristic p analogue of the Andr\'e--Pink--Zannier conjecture

TL;DR

This paper proves an analogue of the Andre9--Pink--Zannier conjecture in characteristic p for certain Shimura varieties, establishing results on monodromy, Hecke correspondences, and the distribution of special points.

Contribution

It introduces a characteristic p version of the conjecture, demonstrating its validity for ordinary points with large monodromy and establishing a Hecke-equidistribution analogue.

Findings

Proved the conjecture for ordinary points with big monodromy.

Established an algebraic Hecke-equidistribution in characteristic p.

Showed abundance of Weyl special points in positive characteristic.

Abstract

We investigate the analogue of the Andr\'e--Pink--Zannier conjecture in characteristic $p$. Precisely, we prove it for ordinary function field-valued points with big monodromy, in Shimura varieties of Hodge type. We also prove an algebraic characteristic $p$ analogue of Hecke-equidistribution (as formulated by Mazur) for Shimura varieties of Hodge type. We prove our main results by a global and local analysis of prime-to-$p$ Hecke correspondences, and by showing that Weyl special points are abundant in positive characterstic.