Hecke orbits on Shimura varieties of Hodge type

Abstract

We prove the Hecke orbit conjecture of Chai--Oort for Shimura varieties of Hodge type at odd primes of good reduction. We use a novel result for the local monodromy groups of $F$-isocrystals "coming from geometry", which refines Crew's parabolicity conjecture. In the course of the proof, we also introduce a noncommutative generalisation of Serre--Tate coordinates for formal neighbourhoods of central leaves, built upon the previous work of Caraiani--Scholze and Kim. Using these coordinates, we reinterpret Chai--Oort's notion of strongly Tate-linear subspaces and we establish upper bounds for their monodromy groups. For this step, we employ the notion of Cartier--Witt stacks, as introduced by Drinfeld and Bhatt--Lurie. Another crucial ingredient in the proof is a rigidity result proved by Chai--Oort, which shows that the relevant subspaces are strongly Tate-linear. On the way, we generalise de Jong's full faithfulness theorem for $F$-isocrystals.

arXiv comment59 pages; major revisions and corrections: we now prove the conjecture for every odd prime and for the rigidity result we quote Chai--Oort