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arxiv: 2509.19878 · v2 · submitted 2025-09-24 · 🧮 math.AG · math.NT

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

Steven R. Groen , Elvira Lupoian , Mychelle Parker This is my paper

Pith reviewed 2026-05-18 14:35 UTC · model grok-4.3

classification 🧮 math.AG math.NT
keywords Ekedahl-Oort strataNewton stratamoduli space of abelian varietiessupersingular locusp-divisible groupsp-torsion group schemesprincipally polarized abelian varieties
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The pith

The non-empty intersections of Ekedahl-Oort and Newton strata are completely determined for the moduli space of principally polarized abelian varieties in dimension five.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The moduli space A_5 of principally polarized abelian varieties over an algebraically closed field of positive characteristic carries two main stratifications. Newton strata are indexed by isogeny classes of p-divisible groups while Ekedahl-Oort strata are indexed by isomorphism classes of p-torsion group schemes. The paper identifies precisely which pairs of these strata meet non-emptily when the dimension equals five. This list immediately produces an explicit description of the Ekedahl-Oort strata that appear inside the supersingular locus.

Core claim

We completely determine which of the intersections of Ekedahl-Oort and Newton strata are non-empty in dimension five. As a consequence, we give an explicit description of the induced Ekedahl-Oort stratification on the supersingular locus S_5.

What carries the argument

Combinatorial parametrizations of the strata by final types for Ekedahl-Oort strata and by Newton polygons or Dieudonné modules for Newton strata, used to test compatibility of pairs.

If this is right

  • Every compatible pair of Ekedahl-Oort and Newton types in dimension five is realized by some abelian variety.
  • The supersingular locus S_5 carries a complete and explicit Ekedahl-Oort stratification.
  • Incompatible pairs produce empty intersections with no further geometric reasons for emptiness in this dimension.
  • The determination supplies a full list of which Ekedahl-Oort strata appear inside each Newton stratum when g equals five.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same combinatorial method may extend to dimension six or higher if no new obstructions arise.
  • The refined stratification of the supersingular locus could be used to compute its Chow ring or its cohomology with coefficients in the constant sheaf.
  • Results on non-emptiness may inform questions about the geometry of the superspecial locus inside S_5.

Load-bearing premise

The known combinatorial parametrizations of the two stratifications detect every geometric non-emptiness statement and do not miss obstructions that appear only in dimension five.

What would settle it

Existence of a principally polarized abelian variety of dimension five whose p-divisible group realizes a Newton type paired with a p-torsion type that the combinatorial rules declare incompatible.

Figures

Figures reproduced from arXiv: 2509.19878 by Elvira Lupoian, Mychelle Parker, Steven R. Groen.

Figure 1
Figure 1. Figure 1: Ekedahl-Oort Strata in dimension 5 of p-rank 0. Two strata are con￾nected by a line if the lower one is contained in the Zariski closure of the upper one. 3. Closure Relations in the Ekedahl-Oort Stratification To begin our study of the Ekedahl-Oort stratification of S5 we find the closure relations in the Ekedahl-Oort strata of p-rank 0. More precisely, we want to determine for which pairs of elementary s… view at source ↗
read the original 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 $\mathcal{S}_{5}$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims to completely determine the non-empty intersections of the Ekedahl-Oort and Newton strata in the moduli space A_5 of principally polarized abelian varieties over an algebraically closed field of positive characteristic p. It does so via combinatorial compatibility of Newton polygons with Ekedahl-Oort final types (using Dieudonné modules and covariant Dieudonné modules respecting the principal polarization), and as a consequence gives an explicit description of the induced Ekedahl-Oort stratification on the supersingular locus S_5.

Significance. If the classification is complete, the result supplies a concrete, usable description of how the two stratifications interact in dimension 5 and on the supersingular locus, which is a useful benchmark for the general problem in A_g. The reliance on existing combinatorial parametrizations rather than new ad-hoc constructions is a methodological strength.

major comments (2)
  1. [§4] §4 (Compatibility of final types with Newton polygons): the argument that every combinatorially admissible pair (Newton polygon, EO type) is realized geometrically in A_5 rests on local Dieudonné-module conditions; the manuscript must explicitly rule out the possibility that the symplectic form or global moduli geometry in dimension 5 imposes additional obstructions not visible in the local data, as this is load-bearing for the claim of completeness.
  2. [Theorem 5.3] Theorem 5.3 (induced EO stratification on S_5): the explicit list of non-empty EO strata inside the supersingular locus is derived from the intersection table; a summary count or verification that all supersingular Newton polygons have been cross-checked against the known dimension formulas for EO strata would strengthen the claim that no pairs were missed.
minor comments (2)
  1. [§2] Notation for final types and covariant Dieudonné modules is introduced in §2 but could be accompanied by a small reference table of the g=5 cases for quick lookup.
  2. [Figure 1] Figure 1 (Newton polygons in dimension 5) would benefit from explicit labeling of the supersingular polygon to make the later restriction to S_5 easier to follow.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments, which help clarify the exposition of our results on the intersections of Ekedahl-Oort and Newton strata in A_5. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [§4] §4 (Compatibility of final types with Newton polygons): the argument that every combinatorially admissible pair (Newton polygon, EO type) is realized geometrically in A_5 rests on local Dieudonné-module conditions; the manuscript must explicitly rule out the possibility that the symplectic form or global moduli geometry in dimension 5 imposes additional obstructions not visible in the local data, as this is load-bearing for the claim of completeness.

    Authors: The compatibility conditions in §4 are formulated in terms of polarized Dieudonné modules (both covariant and contravariant), which by definition incorporate the principal polarization and the associated symplectic form on the Dieudonné module. Thus the local data already encode the symplectic constraints. For the passage from local conditions to geometric realization in A_5, we rely on the standard deformation theory of principally polarized p-divisible groups: when a polarized Dieudonné module satisfies the compatibility conditions, it lifts to a principally polarized abelian variety over an algebraically closed field (see, e.g., the results of Oort and others on the existence of deformations in the principally polarized setting). In dimension 5 the moduli space A_5 is smooth and the strata are locally closed, so no additional global obstructions arise beyond the local polarized data. We will add a short clarifying paragraph at the end of §4 making this reasoning explicit. revision: yes

  2. Referee: [Theorem 5.3] Theorem 5.3 (induced EO stratification on S_5): the explicit list of non-empty EO strata inside the supersingular locus is derived from the intersection table; a summary count or verification that all supersingular Newton polygons have been cross-checked against the known dimension formulas for EO strata would strengthen the claim that no pairs were missed.

    Authors: We agree that an explicit cross-check would strengthen the presentation. The list in Theorem 5.3 is obtained by intersecting the complete table of admissible (Newton, EO) pairs with the supersingular Newton polygons; each supersingular polygon was verified to satisfy the dimension formula for the corresponding EO stratum (using the formulas of Ekedahl-Oort and subsequent dimension computations). In the revised manuscript we will insert a brief summary paragraph (or small table) immediately before Theorem 5.3 that records the number of supersingular Newton polygons considered and confirms that each has been checked against the EO dimension formulas, thereby documenting that no admissible pairs were overlooked. revision: yes

Circularity Check

0 steps flagged

No circularity: intersections determined by external combinatorial compatibility checks

full rationale

The paper's central result classifies non-empty Newton-EO intersections in A_5 by verifying compatibility of known Newton polygon data with Ekedahl-Oort final types (via Dieudonné modules and principal polarizations). These parametrizations are treated as established external inputs from prior literature, not derived or fitted within the paper itself. No step equates a claimed prediction to a fitted parameter, renames a known result, or reduces the non-emptiness statements to a self-citation chain. The explicit EO stratification on S_5 follows directly from the enumerated compatible pairs without load-bearing self-reference. The derivation remains self-contained against the combinatorial descriptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on standard facts about p-divisible groups, Dieudonné theory, and the known parametrizations of Newton and Ekedahl-Oort strata; no new free parameters or invented entities are introduced.

axioms (2)
  • domain assumption The Newton and Ekedahl-Oort strata admit combinatorial parametrizations via final types or Dieudonné modules that capture all geometric possibilities.
    Invoked to reduce the intersection question to a combinatorial check.
  • standard math Closure relations and dimension formulas for the strata are known and can be used to detect emptiness.
    Standard background from the theory of moduli spaces in positive characteristic.

pith-pipeline@v0.9.0 · 5669 in / 1390 out tokens · 41292 ms · 2026-05-18T14:35:10.769059+00:00 · methodology

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Reference graph

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