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On the ordinary Hecke orbit conjecture

Pol van Hoften

Vol. 18 (2024), No. 5, 847–898
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 monodromy groups 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 a result of Howe.

Keywords
Shimura varieties, Hecke orbit conjecture, ordinary locus, monodromy theorems, Serre–Tate coordinates, Rigidity theorem, local stabiliser principle
Mathematical Subject Classification
Primary: 11G18
Secondary: 14G35
Milestones
Received: 26 January 2022
Revised: 20 February 2023
Accepted: 20 July 2023
Published: 16 April 2024
Authors
Pol van Hoften
Mathematics Department
Stanford University
Stanford, CA
United States

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