Height functions on Hecke orbits and the generalised Andr\'e-Pink-Zannier conjecture

TL;DR

This paper introduces a new concept of generalized Hecke orbits in Shimura varieties, defines a height function on them, and uses it to prove a significant conjecture assuming the Mumford-Tate conjecture, advancing understanding in arithmetic geometry.

Contribution

It defines generalized Hecke orbits, establishes a height function on these orbits, and proves the generalized Andre9-Pink-Zannier conjecture under the Mumford-Tate conjecture.

Findings

Established lower bounds for Galois orbits size in terms of height.

Proved the generalized Andre9-Pink-Zannier conjecture assuming Mumford-Tate.

Implemented Pila-Zannier strategy for the proof.

Abstract

We introduce and study the notion of a generalised Hecke orbit in a Shimura variety. We define a height function on such an orbit and study its properties. We obtain a lower bounds for the size of Galois orbits of points in a generalised Hecke orbit in terms of these height, assuming a version of the Mumford-Tate conjecture. We then use it to prove the generalised Andr\'e-Pink-Zannier conjecture under this assumption by implementing the Pila-Zannier strategy.