Introduction 2. The strategy 3. Some preliminaries . Mumford-Tate groups . Variations of Z-Hodge structure on Shimura varieties . Representations of tori 4. Lower bounds for Galois orbits . Galois orbits and Mumford-Tate groups . Getting rid of G . Proof of Proposition 5. Images under Hecke correspondences 6. Density of Hecke orbits 7. Proof of the main result . The case where i is bounded . The case where i is not bounded 1. Introduction The aim of this article is to prove a special case of the following conjecture of Andr´ and Oort on subvarieties.
