On the ordinary Hecke orbit conjecture

TL;DR

This paper proves the ordinary Hecke orbit conjecture for Hodge type Shimura varieties at good primes, using advanced monodromy and formal coordinate techniques to establish new rigidity and monodromy results.

Contribution

It introduces a general monodromy theorem and a rigidity result for ordinary Hecke orbits, advancing understanding of their structure in Hodge type Shimura varieties.

Findings

Proved the ordinary Hecke orbit conjecture for Hodge type Shimura varieties.

Developed a monodromy theorem for Hecke-stable subvarieties.

Established a rigidity result for formal completions of ordinary Hecke orbits.

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.