Generalised Andr\'e-Pink-Zannier Conjecture for Shimura varieties of abelian type

TL;DR

This paper proves the generalized Andre9-Pink-Zannier conjecture for Shimura varieties of abelian type, advancing understanding of unlikely intersections in arithmetic geometry.

Contribution

It establishes the conjecture for all Shimura varieties of abelian type under certain broad assumptions, extending previous partial results.

Findings

Proves the generalized Andre9-Pink-Zannier conjecture for abelian type Shimura varieties.

Introduces a p-adic Kempf-Ness theorem relating good reduction and Mumford stability.

Provides a general framework applicable to all Shimura varieties under specific conditions.

Abstract

In this paper, we prove the generalised Andr\'e-Pink-Zannier conjecture (an important case of the Zilber-Pink conjecture) for all Shimura varieties of abelian type. Questions of this type were first asked by Y. Andr\'e in 1989. We actually prove a general statement for all Shimura varieties, subject to certain assumptions that are satisfied for Shimura varieties of abelian type and are expected to hold in general. We also prove another result, a p-adic Kempf-Ness theorem, on the relation between good reduction of homogeneous spaces over p-adic integers with Mumford stability property in p-adic geometric invariant theory.