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Decidability of singularities in the Ekedahl--Oort stratification

Jean-Stefan Koskivirta, Lorenzo La Porta

Abstract

For an abelian type Shimura variety and an odd prime $p$ of good reduction, we characterize the regularity in codimension one of Zariski closures of Ekedahl--Oort strata in terms of the Frobenius action on the root datum. We give an algorithm that detects codimension one singularities for arbitrary Ekedahl--Oort strata. When the Shimura datum is of split type, we relate the singularities of Ekedahl--Oort strata to a stack of $G$-zips over the complex numbers. We study the existence of generalized Hasse invariants on this stack.

Decidability of singularities in the Ekedahl--Oort stratification

Abstract

For an abelian type Shimura variety and an odd prime of good reduction, we characterize the regularity in codimension one of Zariski closures of Ekedahl--Oort strata in terms of the Frobenius action on the root datum. We give an algorithm that detects codimension one singularities for arbitrary Ekedahl--Oort strata. When the Shimura datum is of split type, we relate the singularities of Ekedahl--Oort strata to a stack of -zips over the complex numbers. We study the existence of generalized Hasse invariants on this stack.
Paper Structure (68 sections, 68 theorems, 89 equations, 1 algorithm)

This paper contains 68 sections, 68 theorems, 89 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Algorithm for smoothness
  3. Independence from p
  4. Length 2 strata
  5. The case of a split prime
  6. Hasse invariants in characteristic zero
  7. Acknowledgements
  8. Review of the theory of G-zips
  9. Generalities on stacks of G-zips
  10. Zip data
  11. Frobenius-type zip data
  12. Stack of $G$-zips
  13. Zip strata
  14. Closure relations
  15. Bruhat strata
  16. ...and 53 more sections

Key Result

Proposition 2.3.2

Assume that ${\mathcal{Z}}$ is of Frobenius-type. Let $w\in {}^I W\cup W^J$.

Theorems & Definitions (127)

  • Definition 2.1.1
  • Remark 2.2.1
  • Remark 2.2.2
  • Example 2.3.1
  • Proposition 2.3.2: Goldring-Koskivirta-Strata-Hasse
  • Lemma 2.4.1
  • proof
  • Proposition 2.4.2
  • Lemma 2.4.3: Koskivirta-LaPorta-Reppen-singularities
  • Proposition 2.5.1: Koskivirta-LaPorta-Reppen-singularities
  • ...and 117 more