On the ordinary Hecke orbit conjecture

van Hoften, Pol

doi:10.2140/ant.2024.18.847

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On the ordinary Hecke orbit conjecture

Pol van Hoften

Vol. 18 (2024), No. 5, 847–898

DOI: 10.2140/ant.2024.18.847

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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Vol. 18, No. 5, 2024