Height functions on Hecke orbits and the generalised André-Pink-Zannier conjecture

Rodolphe Richard, Andrei Yafaev

Published 2021-09-28, updated 2022-05-27Version 2

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.