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Ekedahl-Oort strata in the $\mathsf{GU}(q-2,2)$ Shimura variety

Emerald Andrews, Deewang Bhamidipati, Maria Fox, Heidi Goodson, Steven R. Groen, Sandra Nair

Abstract

This paper concerns the characteristic-$p$ fibers of $\mathsf{GU}(q-2,2)$ Shimura varieties, which classify abelian varieties with additional structure. These Shimura varieties admit two stratifications of interest: the Ekedahl-Oort stratification, based on the isomorphism class of the $p$-torsion subgroup scheme, and the Newton stratification, based on the isogeny class of the $p$-divisible group. In this paper, we present several novel techniques that give a better understanding of the Ekedahl-Oort stratification and of the interaction between the two stratifications for a general signature $(q-2,2)$.

Ekedahl-Oort strata in the $\mathsf{GU}(q-2,2)$ Shimura variety

Abstract

This paper concerns the characteristic- fibers of Shimura varieties, which classify abelian varieties with additional structure. These Shimura varieties admit two stratifications of interest: the Ekedahl-Oort stratification, based on the isomorphism class of the -torsion subgroup scheme, and the Newton stratification, based on the isogeny class of the -divisible group. In this paper, we present several novel techniques that give a better understanding of the Ekedahl-Oort stratification and of the interaction between the two stratifications for a general signature .
Paper Structure (30 sections, 46 theorems, 171 equations, 1 figure)

This paper contains 30 sections, 46 theorems, 171 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Unitary Shimura varieties
  4. Ekedahl-Oort and Newton Stratifications
  5. Weyl Group Cosets
  6. Structure of Ekedahl-Oort Strata for Signature $(q-2,2)$
  7. Index Set & Bruhat Order
  8. Standard Objects
  9. The Closure Order
  10. Definition and properties
  11. Closure relations
  12. Example: $q=11$
  13. Concluding remarks on the closure order
  14. Product Maps: Relations to Other Unitary Shimura Varieties
  15. Background
  16. ...and 15 more sections

Key Result

Theorem 2.2

There is a bijection of sets:

Figures (1)

  • Figure 1: Closure relations for $q=11$.

Theorems & Definitions (97)

  • Definition 2.1
  • Theorem 2.2: Moonen
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • Definition 3.3: The Bruhat order
  • Proposition 3.4
  • proof
  • Remark 3.5
  • Corollary 3.6
  • ...and 87 more