On the ordinary Hecke orbit conjecture

Abstract

We prove the ordinary Hecke orbit conjecture for Shimura varieties of Hodge type at primes of good reduction. We make use of the global Serre-Tate coordinates of Chai as well as recent results of D'Addezio about the $p$-adic monodromy of isocrystals. The new ingredients in this paper are a general monodromy theorem for Hecke-stable subvarieties for Shimura varieties of Hodge type, and a rigidity result for the formal completions of ordinary Hecke orbits. Along the way we show that classical Serre--Tate coordinates can be described using unipotent formal groups, generalising results of Howe.

44 pages; v3 is a minorly revised version of v1; main results unchanged

References in corpus (2)

• Picard ranks of K3 surfaces over function fields and the Hecke orbit conjecture
• Parabolicity conjecture of $F$-isocrystals

Cited by in corpus (2)

• Picard rank jumps for K3 surfaces with bad reduction
• Mod $p$ points on Shimura varieties of parahoric level (with an appendix by Rong Zhou)