Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Hecke orbits on Shimura varieties of Hodge type

Marco D'Addezio, Pol van Hoften

TL;DR

This work resolves the Hecke orbit conjecture for Shimura varieties of Hodge type at odd primes of good reduction, showing that prime-to-$p$ Hecke orbits are Zariski dense in central leaves and that isogeny classes are dense in Newton strata. The authors develop a novel framework combining local monodromy of $F$-isocrystals, noncommutative generalized Serre–Tate coordinates, and Cartier–Witt stacks to control formal neighbourhoods of central leaves, enabling a rigidity argument that reduces monodromy to the unipotent radical of parabolic subgroups. A key technical advance is bounding monodromy via a geometric descent analysis and a refined local monodromy theorem, complemented by a general full-faithfulness-type result for $F$-isocrystals. The results have broad implications for the geometry of Shimura varieties of Hodge type, including applications to $ ext{l}$-adic Hecke orbits and to density questions in Newton strata, and they provide a framework potentially extendable to parahoric and ramified settings, with connections to Bragg–Yang-type conjectures.

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.

Hecke orbits on Shimura varieties of Hodge type

TL;DR

This work resolves the Hecke orbit conjecture for Shimura varieties of Hodge type at odd primes of good reduction, showing that prime-to- Hecke orbits are Zariski dense in central leaves and that isogeny classes are dense in Newton strata. The authors develop a novel framework combining local monodromy of -isocrystals, noncommutative generalized Serre–Tate coordinates, and Cartier–Witt stacks to control formal neighbourhoods of central leaves, enabling a rigidity argument that reduces monodromy to the unipotent radical of parabolic subgroups. A key technical advance is bounding monodromy via a geometric descent analysis and a refined local monodromy theorem, complemented by a general full-faithfulness-type result for -isocrystals. The results have broad implications for the geometry of Shimura varieties of Hodge type, including applications to -adic Hecke orbits and to density questions in Newton strata, and they provide a framework potentially extendable to parahoric and ramified settings, with connections to Bragg–Yang-type conjectures.

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 -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 -isocrystals.
Paper Structure (52 sections, 70 theorems, 210 equations)

This paper contains 52 sections, 70 theorems, 210 equations.

Table of Contents

  1. Introduction
  2. The Hecke orbit conjecture
  3. Main result
  4. Local monodromy of F-isocrystals
  5. An overview of the proof of Theorem \ref{['Thm:MainThmHO']}
  6. Structure of the article
  7. Acknowledgements
  8. Preliminaries
  9. Formal schemes
  10. Frobenius-smooth schemes and the Tannakian category of isocrystals
  11. p-divisible groups and Dieudonné theory
  12. Representability
  13. Complete slope divisibility
  14. Local monodromy of F-isocrystals
  15. Two Tannakian criterions
  16. ...and 37 more sections

Key Result

Theorem I

If $Z \subseteq C$ is a non-empty reduced closed subvariety that is stable under the prime-to-$p$ Hecke operators, then $Z=C$.

Theorems & Definitions (172)

  • Conjecture 1
  • Theorem I: Theorem \ref{['Thm:MainTheorem2']}
  • Remark 1.2.1
  • Remark 1.2.2
  • Theorem II: Theorem \ref{['Thm:LocalGlobalMonodromyTheorem']}
  • Theorem 1.3.2: Theorem \ref{['Thm:FullFaithfulness']}
  • Corollary 1.3.3: Corollary \ref{['Cor:ExistenceSlopeFiltration']}
  • Remark 1.4.1
  • Remark 1.4.2
  • Remark 1.4.3
  • ...and 162 more