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Intersections of the Ekedahl-Oort and Newton Strata of $\mathcal{A}_{5}$

Steven R. Groen, Elvira Lupoian, Mychelle Parker

Abstract

The moduli space $\mathcal{A}_g$ of principally polarised abelian varieties of dimension $g$, defined over an algebraically closed field of characteristic $p >0$, is studied through various stratifications. The two most prominent ones are the Newton stratification, based on the isogeny class of the $p$-divisible group of an abelian variety, and the Ekedahl-Oort stratification, defined by the isomorphism class of its $p$-torsion group scheme. In general, it is not known how the strata of these two intersect. In this paper we completely determine which of these intersections are non-empty in dimension five. As a consequence, we give an explicit description of the induced Ekedahl-Oort stratification on the supersingular locus $\mathcal{S}_{5}$.

Intersections of the Ekedahl-Oort and Newton Strata of $\mathcal{A}_{5}$

Abstract

The moduli space of principally polarised abelian varieties of dimension , defined over an algebraically closed field of characteristic , is studied through various stratifications. The two most prominent ones are the Newton stratification, based on the isogeny class of the -divisible group of an abelian variety, and the Ekedahl-Oort stratification, defined by the isomorphism class of its -torsion group scheme. In general, it is not known how the strata of these two intersect. In this paper we completely determine which of these intersections are non-empty in dimension five. As a consequence, we give an explicit description of the induced Ekedahl-Oort stratification on the supersingular locus .

Paper Structure

This paper contains 22 sections, 31 theorems, 30 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Main Result
  3. Methodology
  4. Outline
  5. Stratifications of $\mathcal{A}_{g}$
  6. Invariants of abelian varieties
  7. Dieudonné Theory
  8. Newton stratification
  9. Ekedahl-Oort Stratification
  10. Intersection Tools
  11. Containment in $\mathcal{S}_g$, first slopes and $a$-numbers
  12. Inductive Arguments
  13. Minimal $p$-divisible groups
  14. Positive $p$-rank intersections
  15. Decomposing Positive $p$-rank Intersections
  16. ...and 7 more sections

Key Result

Theorem 1

The intersections of Ekedahl-Oort strata and Newton strata of $\mathcal{A}_5$ are described in Table intstable. All intersections are non-empty. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (72)

  • Theorem 1
  • Corollary 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Proposition 7
  • proof
  • Definition 8
  • Definition 9
  • ...and 62 more