Hecke orbits on Shimura varieties

I will talk about the proof of the Hecke orbit conjecture for Shimura varieties of Hodge type. This is a conjecture proposed by Chai and Oort on the geometry of the reduction modulo p of Shimura varieties. After recalling the statement, I will explain how to linearise the problem using some “generalised Serre-Tate coordinates” on central leaves. Subsequently, I will explain how the monodromy groups of F-isocrystals enter into the picture and I will say some words on how we use the Cartier-Witt stacks, constructed by Drinfeld and Bhatt-Lurie. This is a joint work with Pol van Hoften.