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

1. Problem Statement

The paper addresses the intersection theory of two fundamental stratifications on the moduli space $\mathcal{A}_g$ of principally polarized abelian varieties (PPAVs) of dimension $g$ over an algebraically closed field $k$ of characteristic $p > 0$:

1. The Newton Stratification: Defined by the isogeny class of the $p$-divisible group $X[p^\infty]$, characterized by the Newton polygon $\xi$.
2. The Ekedahl-Oort (EO) Stratification: Defined by the isomorphism class of the $p$-torsion group scheme $X[p]$, characterized by an elementary sequence $\phi$.

The Core Question: For a given dimension $g$ and specific strata $N(\xi)$ (Newton) and $S_\phi$ (EO), is the intersection $S_\phi \cap N(\xi)$ non-empty?
While this is well-understood for dimensions $g \le 3$ and partially for $g=4$, the case of dimension $g=5$ presents significant complexity due to the combinatorial explosion of possible strata and the lack of general theoretical tools to determine all intersections. The authors aim to completely classify these intersections for $g=5$, with a specific focus on the supersingular locus ($S_5$), which corresponds to the Newton polygon with all slopes equal to $1/2$.

2. Methodology

The authors employ a multi-faceted approach combining theoretical reductions, combinatorial computations, and explicit constructions of Dieudonné modules.

A. Reduction to $p$-Rank Zero

The stratification by $p$-rank ($f$) serves as a stepping stone.

• For positive $p$-rank ($f > 0$), the problem reduces to lower-dimensional cases via direct sums. Specifically, an abelian variety with $p$-rank $f$ can be decomposed into a product of a supersingular part and a part with $p$-rank 0.
• The authors prove that intersections for $f > 0$ in dimension 5 reduce to known results in dimensions $g < 5$ and $p$-rank 0 (Proposition 36).
• Consequently, the primary technical challenge lies in the $p$-rank zero case.

B. Theoretical Tools for Non-Empty Intersections

To prove that an intersection is non-empty without explicit construction, the authors utilize:

1. Minimal $p$-divisible Groups: Using Oort's theory, they identify "minimal" groups $G(\xi)$ associated with a Newton polygon $\xi$. The elementary sequence of $G(\xi)[p]$ provides a "minimal" EO stratum $S_{\phi_\xi}$ guaranteed to lie inside $N(\xi)$ (Corollary 32).
2. First Slope Analysis (Harashita): They utilize the "first Newton slope" $\lambda_\phi$ associated with an EO stratum $S_\phi$. If a Newton stratum $N(\xi)$ has a unique first slope equal to $\lambda_\phi$, the intersection is non-empty (Corollary 20).
3. Inductive Arguments (Direct Sums): Using Harashita's theory of final types, they compute the elementary sequence of a direct sum of abelian varieties ($\phi \oplus \phi'$). This allows them to construct intersections in dimension 5 from known intersections in lower dimensions (Proposition 22).

C. Explicit Dieudonné Modules

For cases where theoretical tools are insufficient, the authors construct explicit principally quasi-polarized free Dieudonné modules $M$ over the Witt vectors $W(k)$.

• Existence: They define operators $F$ (Frobenius) and $V$ (Verschiebung) on a free module $M$ and a pairing $\langle \cdot, \cdot \rangle$ satisfying the required axioms.
• Verification: They verify the mod-$p$ reduction $M/pM$ corresponds to the target EO stratum and compute the Newton polygon of $M$ (via the slopes of $F^n$) to match the target Newton stratum.
• Non-Existence: Conversely, to prove an intersection is empty, they prove that no such Dieudonné module can exist. This involves analyzing the $p$-adic valuations of matrix entries of powers of $F$ to show that the required Newton slopes are incompatible with the given EO type (e.g., Lemma 50).

D. Computational Implementation

The authors utilized the Magma computer algebra system to:

• Compute decompositions of elementary sequences into indecomposable final types.
• Calculate first slopes for all EO strata of $p$-rank 0.
• Verify matrix conditions for explicit Dieudonné module constructions.
• The code is publicly available in a GitHub repository.

3. Key Contributions and Results

A. Complete Classification for Dimension 5

The main result (Theorem 1) is a complete table (Table 1) listing all pairs $(\xi, \phi)$ such that $S_\phi \cap N(\xi) \neq \emptyset$ for $g=5$.

• Total Count: The paper identifies exactly which of the 42 possible EO strata intersect with the various Newton strata.
• Supersingular Locus ($S_5$): As a direct consequence, the authors provide an explicit description of the EO stratification restricted to the supersingular locus $S_5$ (Corollary 2). They determine exactly which EO strata have non-empty intersection with $S_5$.

B. Discovery of New Phenomena

The paper highlights behaviors in dimension 5 that do not occur in lower dimensions ($g \le 4$):

1. Non-Monotonicity of Slopes: In dimensions $g \le 4$, the set of Newton strata intersecting a given EO stratum $S_\phi$ is determined by a lower and upper bound on the Newton polygon. In dimension 5, this "interval" property fails.
   • Example: The stratum $S_{(0,1,1,1,2)}$ intersects the Newton strata with first slopes $1/3$ and $1/2$ (supersingular) but does not intersect the Newton stratum with first slope $2/5$, even though $1/3 < 2/5 < 1/2$. This is proven via Lemma 50.
2. Existence of "Higher" Lifts: The authors construct a principally quasi-polarized Dieudonné module $M$ where the Newton polygon of $M$ is strictly "higher" (has larger slopes) than the Newton polygon of its tautological lift of the mod-$p$ reduction. This phenomenon was previously unknown and does not exist for $g \le 4$ (Remark 5.1).

C. Specific Technical Results

• Table 4: Lists the elementary sequences associated with minimal $p$-divisible groups for all Newton polygons of $p$-rank 0 and 1 in dimension 5.
• Table 5: Summarizes intersections for positive $p$-rank cases, derived from lower-dimensional results.
• Lemma 48 & Corollary 49: Provides a general criterion for determining the first slope of a Newton polygon based on the structure of the mod-$p$ reduction, used to prove $S_{(0,0,1,2,2)} \subset N_{1/3}$.

4. Significance

• Completeness: This work provides the first complete classification of EO-Newton intersections for dimension 5, a critical step in understanding the geometry of $\mathcal{A}_g$ in higher dimensions.
• Refinement of Theory: By demonstrating the failure of the "interval property" for Newton strata in dimension 5, the paper challenges the intuition built from lower-dimensional cases and suggests that the geometry of the moduli space becomes significantly more complex as $g$ increases.
• Methodological Advancement: The combination of theoretical reduction (Harashita's slopes) with heavy computational verification and explicit Dieudonné module construction sets a new standard for tackling stratification problems in arithmetic geometry.
• Supersingular Locus: The explicit description of the EO stratification on $S_5$ is a valuable resource for researchers studying supersingular abelian varieties, which are central to the study of Shimura varieties and the Langlands program in positive characteristic.

In summary, the paper resolves a long-standing open problem for dimension 5, revealing unexpected geometric complexities and providing a robust framework (both theoretical and computational) for future investigations into higher-dimensional moduli spaces.