# Ekedahl-Oort strata and the supersingular locus in
the $\boldsymbol{\mathsf{GU}(q-2,2)}$ Shimura variety

Emerald Anne  Department of Mathematics and Computer Science, Washington
College, Chestertown, Maryland 21620, USA<EMAIL_ADDRESS>,  Deewang
Bhamidipati  Mathematics Department, University of California, Santa Cruz, CA
95064, USA<EMAIL_ADDRESS>,  Maria Fox  Department of Mathematics,
Oklahoma State University, Stillwater, OK 74078, USA<EMAIL_ADDRESS>,
Heidi Goodson  Department of Mathematics, Brooklyn College, City University of
New York, Brooklyn, NY 11210 USA<EMAIL_ADDRESS>,  Steven
Groen  Department of Mathematics, Lehigh University, Bethlehem, PA 18015, USA
<EMAIL_ADDRESS>and  Sandra Nair  Department of Mathematics, Colorado State
University, Fort Collins, CO 80523, USA<EMAIL_ADDRESS>
###### 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. It is natural to ask which Ekedahl-Oort strata
intersect the unique closed Newton stratum, called the _supersingular locus_.
In this paper, we present several novel techniques that give information about
the interaction between the two stratifications for general signature
$(q-2,2)$, and as an application, we completely answer this question for the
signature $(3,2)$.

##  1\. Introduction

This paper studies the characteristic-$p$ fibers of Shimura varieties of PEL
type. Shimura varieties of PEL type are moduli spaces of abelian varieties
with additional structure, and this moduli interpretation gives rise to two
stratifications of the characteristic-$p$ fiber: the Newton stratification and
the Ekedahl-Oort stratification. The Ekedahl-Oort stratification is based on
the isomorphism class of the $p$-torsion subgroup scheme of the parameterized
abelian varieties, while the Newton stratification is based on the isogeny
class of their $p$-divisible group. Though each stratification is compelling
in its own right, it is also interesting to study the interactions between the
two stratifications. For instance, the two stratifications coincide on the
modular curve, both distinguishing between _ordinary_ and _supersingular_
elliptic curves. However, when studying abelian varieties of increasing
dimension, the stratifications diverge further away from one another.

The interaction between the Ekedahl-Oort and Newton stratifications on general
Shimura varieties of PEL type has been widely studied (see, for example, the
presentation of results in [16]). More precise statements can be made when one
restricts attention to a smaller class of Shimura varieties. In this paper, we
study the interaction between these stratifications for _unitary Shimura
varieties_ $\mathcal{M}(q-a,a)$. These are moduli spaces of abelian varieties
of dimension $q$ with an action of an imaginary quadratic field $K$ that meets
the “signature $(q-a,a)$” condition. In [18], Wooding studies several aspects
of the Ekedahl-Oort and Newton stratifications of $\mathcal{M}(q-a,a)$.
However, the following question is still open in general.

######  Question 1.1.

Which Ekedahl-Oort strata of $\mathcal{M}(q-a,a)$ intersect the supersingular
locus?

The answer to this question is known for some signatures. For example, in
signature $(q,0)$ the Shimura variety is zero-dimensional, consisting of a
single Ekedahl-Oort stratum. By a classical theorem of Deuring, this stratum
coincides with the supersingular locus if and only if $p$ does not split in
$K$. The signature $(q-1,1)$ case is already much more complex, but it is
established in [15] which of the Ekedahl-Oort strata intersect the
supersingular locus. Following from the results of [7], it is known which
Ekedahl-Oort strata of $\mathcal{M}(2,2)$ intersect the supersingular locus.
It is known ([4, Theorem A]) that these are the only signatures for which the
supersingular locus is a union of Ekedahl-Oort strata, and so answering
Question 1.1 should ostensibly be significantly more complicated for Shimura
varieties of other signatures.

This paper focuses on unitary Shimura varieties of signature $(q-2,2)$ at a
prime inert in the relevant quadratic imaginary field. The geometry of the
supersingular locus of $\mathcal{M}(q-2,2)$ is described in [3], but little is
known about which Ekedahl-Oort strata intersect the supersingular locus. In
the tradition of [15], such information may be useful when studying more
subtle aspects of the supersingular locus. We present several techniques that
reveal information about the interaction between the Ekedahl-Oort
stratification and the Newton stratification. As an application of these
general methods, we give a complete answer to Question 1.1 when the signature
is $(3,2)$.

######  Theorem A (Theorem 6.7).

Four of the Ekedahl-Oort strata of $\mathcal{M}(3,2)$ do not intersect the
supersingular locus. The remaining six Ekedahl-Oort strata of
$\mathcal{M}(3,2)$ do intersect the supersingular locus. Furthermore, of these
six, two strata are completely contained in the supersingular locus and one
intersects but is not contained in the supersingular locus.

The precise statement of Theorem 6.7 identifies each relevant Ekedahl-Oort
stratum in terms of the corresponding $p$-torsion group scheme, using the
notation that we develop in Section 3.

The structure of our paper is as follows. We first recall some relevant
notation and background information in Section 2 that will be used throughout
the paper. In Section 3, we study the basic properties of the Ekedahl-Oort
stratification. Using results of Moonen (see [9, Theorem 6.7]), we index the
Ekedahl-Oort strata of $\mathcal{M}(q-2,2)$ by certain Weyl group cosets and,
via Dieudonné theory, concretely describe the $p$-torsion subgroup of the
corresponding Ekedahl-Oort stratum. In Section 3.4, we study the topological
closure relations among the strata and prove that a number of topological
closure relations hold (see Theorems 3.9 and 3.15).

In Sections 4 and 5, we obtain information about the Ekedahl-Oort
stratification of $\mathcal{M}(q-2,2)$ by leveraging its relation to Shimura
varieties whose Ekedahl-Oort stratifications are better understood. First, in
Section 4, we relate $\mathcal{M}(q-2,2)$ to simpler unitary Shimura varieties
via the natural product maps

$\mathcal{M}(a,b)\times\mathcal{M}{(c,d})\to\mathcal{M}(q-2,2),$

induced by taking products of the parameterized abelian varieties. In Theorems
4.10 and 4.16, we explicitly describe the effect of these product maps on the
Ekedahl-Oort stratifications.

In Section 5, we relate $\mathcal{M}(q-2,2)$ to the Siegel modular variety via
a forgetful map

$\mathcal{M}(q-2,2)\to\mathcal{A}_{q},$

which ”forgets” the unitary structure of $\mathcal{M}(q-2,2)$.

More precisely, in Theorem 5.2, we concretely describe the effect of this map
on the Ekedahl-Oort stratifications.

By taking advantage of prior results on the interaction between the Ekedahl-
Oort and Newton stratifications of these better-understood Shimura varieties,
the techniques in Sections 4 and 5 both yield information regarding the
interaction between the Ekedahl-Oort stratification and the supersingular
locus in signature $(q-2,2)$.

In Section 6, we apply the amassed tools to the unitary Shimura variety of
signature $(3,2)$ to prove Theorem A. For all but two of the Ekedahl-Oort
strata of $\mathcal{M}(3,2)$, the results in Sections 4 and 5 can be used to
detect whether or not these strata intersect the supersingular locus. We prove
that the remaining two strata intersect the supersingular locus by explicitly
constructing a point in the intersection. Our classification result for
signature $(3,2)$ is summarized in Table LABEL:table-classification.

### Acknowledgements

This project started at the Rethinking Number Theory workshop in June 2022. We
want to thank the organizers for the workshop, and are grateful for the
supportive, collaborative research environment this workshop provided. The
workshop was supported by the Number Theory Foundation, the American Institute
of Mathematics, and the University of Wisconsin-Eau Claire. Additionally, we
would like to thank Rachel Pries, Damiano Testa, and Martin Weissman for
useful conversations. We would also like to thank Satyam Patel for programming
assistance and Mohamed Tawfik for his early contributions to the project.

H.G. was supported by NSF grant DMS-2201085 and a PSC-CUNY Award, jointly
funded by The Professional Staff Congress and The City University of New York.
M.F. was supported in part by NSF MSPRF Grant 2103150.

##  2\. Background

###  2.1. Unitary Shimura varieties

Unitary Shimura varieties are moduli spaces of abelian varieties equipped with
extra structure, including an action of a quadratic imaginary field. To define
an integral model of such a Shimura variety, we fix a prime $p>2$, a positive
integer $q$, non-negative integers $a$ and $b$ such that $a+b=q$, and a
quadratic imaginary field $K$. We further assume that the prime $p$ is inert
in $K$, and so we identify $\mathcal{O}_{K}/(p)$ as $\mathbx F_{p^{2}}$
throughout.

######  Definition 2.1.

We use the PEL datum
$(K,\mathscr{O}_{K}\otimes_{\mathbx{Z}}\mathbx{Z}_{(p)},{}^{*},V,(\cdot,\cdot),\Lambda,\mathbf{G},h)$
of Kottwitz [8], defined as follows:

  * •

$K$ is the quadratic imaginary field introduced above, with ∗ being the
nontrivial automorphism of $K$ over $\mathbx{Q}$.

  * •

$V$ is a $K$-vector space of dimension $q$, equipped with a perfect
alternating $\mathbx Q$-bilinear pairing $(\cdot,\cdot):V\times V\to\mathbx Q$
such that $(xv,w)=(v,x^{*}w)$ for all $x\in K$ and $v,w\in V$.

  * •

$\mathbf{G}$ is the algebraic group of $K$-linear symplectic similitudes of
$(V,(\cdot,\cdot))$. We assume that $\mathbf{G}_{\mathbx{R}}$ is isomorphic to
the real algebraic group $\mathsf{GU}(a,b)$.

  * •

$\Lambda$ is an
$\mathscr{O}_{K}\otimes_{\mathbx{Z}}\mathbx{Z}_{(p)}$-invariant lattice of
$V\otimes_{\mathbx{Q}}\mathbx{Q}_{p}$ such that the alternating form induced
by $(\cdot,\cdot)$ is a perfect $\mathbx{Z}_{p}$-form.

  * •

$h:\mathrm{Res}_{\mathbx C/\mathbx R}(\mathbx{G}_{m,\mathbx C})\to\mathbf{G}R$
is the homomorphism of real algebraic groups that maps $z\in\mathbx
C^{\times}$ to $\mathrm{diag}(z^{a},\overline{z}^{b})$.

Let $L$ be the reflex field associated to the PEL-datum
$(K,\mathscr{O}_{K}\otimes_{\mathbx{Z}}\mathbx{Z}_{(p)},*,V,(\cdot,\cdot),\Lambda,\mathbf{G},h)$;
if $a=b$ then $L=\mathbx{Q}$, and $L=K$ otherwise. Let $\mathbx{A}_{f}^{p}$
denote the ring of finite adeles with a trivial component at $p$. Fix a
compact open subgroup $C^{p}\subset\mathbf{G}(\mathbx{A}_{f}^{p})$. For
$C^{p}$ small enough, the construction of Kottwitz [8] attaches to this PEL
datum a smooth, quasi-projective scheme $\mathbf{M}(a,b)_{C^{p}}$ over
$\mathrm{Spec}(\mathscr{O}_{L,(p)})$ with the following moduli interpretation.

Let $S$ be an $\mathscr{O}_{L,(p)}$-scheme. Then the set
$\mathbf{M}(a,b)_{C^{p}}(S)$ parameterizes isomorphism classes of tuples
$(A,\iota,\lambda,\xi)$, where:

  * •

$A$ is an abelian variety over $S$ of dimension $q$.

  * •

$\iota:\mathscr{O}_{K}\otimes_{\mathbx Z}\mathbx
Z_{(p)}\to\mathrm{End}(A)\otimes Z\mathbx Z_{(p)}$ is a nonzero homomorphism
of $\mathbx Z_{(p)}$-algebras such that the Rosati involution on
$\mathrm{End}(A)\otimes Z\mathbx Z_{(p)}$ induces the involution ∗ on
$\mathscr{O}_{K}\otimes_{\mathbx Z}\mathbx Z_{(p)}$.

  * •

$\lambda$ is a one-dimensional $\mathbx Q$-subspace of
$\mathrm{Hom}(A,A^{\vee})\otimes Z\mathbx Q$ that contains a $p$-principal
$\mathscr{O}_{K}\otimes_{\mathbx Z}\mathbx Z_{(p)}$-linear polarization.

  * •

$\xi:\mathrm{H}_{1}(A,\mathbx{A}_{f}^{p})\to V\otimes Q\mathbx{A}_{f}^{p}\
\mathrm{mod}\ C^{p}$ is a $C^{p}$-level structure.

We also require that $(A,\iota)$ meets Kottwitz’s determinant condition of
signature $(a,b)$. Two tuples $(A,\iota,\lambda,\xi)$ and
$(A^{\prime},\iota^{\prime},\lambda^{\prime},\xi^{\prime})$ are isomorphic if
there exists a prime-to-$p$ isogeny from $A$ to $A^{\prime}$, commuting with
the action of $\mathscr{O}_{K}\otimes_{\mathbx Z}\mathbx Z_{(p)}$, mapping
$\xi$ to $\xi^{\prime}$ and $\lambda$ to $\lambda^{\prime}$.

The integral model $\mathbf{M}(a,b)_{C^{p}}$ has relative dimension of $ab$.
The main object of study for this paper is the characteristic $p$ unitary
Shimura variety, denoted by $\mathcal{M}(a,b)$, which is the fiber at $p$ of
$\mathbf{M}(a,b)_{C^{p}}$. In particular, $\mathcal{M}(a,b)$ is defined over
the residue field $\mathbx F$ of $L$ at $p$ and is of dimension $ab$. Since
$\mathbf{M}(a,b)_{C^{p}}\cong\mathbf{M}(b,a)_{C^{p}}$, we assume without loss
of generality that $0\leqslant b\leqslant a$.

Frequently, important properties of $\mathcal{M}(a,b)$ can be understood in
terms of its geometric points. For this reason, we fix an algebraic closure
$\mathbx k$ of $\mathbx F$.

###  2.2. Ekedahl-Oort and Newton Stratifications

In this section, we recall the definitions of the Ekedahl-Oort and Newton
stratifications of $\mathcal{M}(a,b)$. For more details, see [16].

The Ekedahl-Oort stratification is based on the isomorphism class of the
$p$-torsion group scheme of the parameterized abelian varieties. Two field-
valued points $(A,\iota,\lambda,\xi)$ and
$(A^{\prime},\iota^{\prime},\lambda^{\prime},\xi^{\prime})$ of
$\mathcal{M}(a,b)$ are in the same Ekedahl-Oort stratum if and only if the
$p$-torsion group schemes equipped with induced action and polarization,
$(A[p],\iota,\lambda)$ and $(A^{\prime}[p],\iota^{\prime},\lambda^{\prime})$,
are isomorphic over $\mathbx k$. The Ekedahl-Oort strata are locally closed,
and the closure of each Ekedahl-Oort stratum is a union of Ekedahl-Oort
strata.

The Newton stratification is based on the isogeny class of the $p$-divisible
group of the parameterized abelian varieties. Two field-valued points
$(A,\iota,\lambda,\xi)$ and
$(A^{\prime},\iota^{\prime},\lambda^{\prime},\xi^{\prime})$ of
$\mathcal{M}(a,b)$ are in the same Newton stratum if and only if the
$p$-divisible groups equipped with induced action and polarization,
$(A[p^{\infty}],\iota,\lambda)$ and
$(A^{\prime}[p^{\infty}],\iota^{\prime},\lambda^{\prime})$, are isogenous (in
a way that respects the actions and polarizations) over $\mathbx k$.

The Newton strata are locally closed, and the closure of each Newton stratum
is a union of Newton strata. The unique closed Newton stratum of
$\mathcal{M}(a,b)$ is the supersingular locus, which we denote as
$\mathcal{M}(a,b)^{ss}$. In particular, a point $(A,\iota,\lambda,\xi)$ of
$\mathcal{M}(a,b)(\mathbx k)$ is contained in the supersingular locus if and
only if $A$ is a supersingular abelian variety.

###  2.3. Weyl Group Cosets

Results of [9] relate the study of the Ekedahl-Oort strata to cosets in a
certain Weyl group. This section introduces the relevant Weyl group cosets and
their minimal-length coset representatives.

The Weyl group that is relevant for the study of $\mathcal{M}(a,b)$ is
$W=\mathfrak{S}_{q}$, the symmetric group on $q$ elements. We consider $W$ as
a Coxeter group with a set of simple reflections

$S=\\{s_{1},\dots,s_{q-1}\\},\text{where }s_{i}=(i,\ i+1).$

The length of $w\in W$, denoted $\ell(w)$, is the length of a shortest
expression for $w$ as a product of simple reflections. It is proved in [1,
Proposition 1.5.2] that the length of an element $w$ in $\mathfrak{S}_{q}$ can
be computed as the number of inversions, i.e., the cardinality of the set

(2.3.1) $\displaystyle\mathrm{Inv}(w):=\\{(i,\,j)\,|\,i<j\text{ and
}w(i)>w(j)\\}.$

In particular, $W$ has a unique element $w_{0}$ of maximal length, where
$w_{0}(k)=q+1-k$.

For $J\subset S$, let $W_{J}$ denote the subgroup of $W$ generated by $J$.
Note that $W_{J}$ is a parabolic subgroup of $W$, and it follows from
Proposition 2.4.4 of [1] that every coset of $W_{J}\setminus W$ contains a
unique minimal-length coset representative. Let ${}^{J}W$ be the collection of
such minimal-length coset representatives for $W_{J}\setminus W$.

For the subset $J_{(a,b)}=\\{s_{1},...,s_{q-1}\\}\setminus\\{s_{b}\\}$ of $S$,
we let $W_{(a,b)}:=W_{J_{(a,b)}}$, and $\mathbf{W}(a,b):={}^{J_{(a,b)}}W$. The
following theorem is paraphrased from Theorem 6.7 of [9]:

######  Theorem 2.2 (Moonen).

There is a bijection of sets:

$\left\\{\text{Ekedahl-Oort Strata of
$\mathcal{M}(a,b)$}\right\\}\longleftrightarrow\mathbf{W}(a,b).$

We now recall some properties of Moonen’s construction. (See also [18].) Let
$G$ be the group of
$\mathcal{O}_{K}\otimes_{\mathbx{Z}}\mathbx{Z}_{(p)}$-linear symplectic
similitudes of $\Lambda$. Then $G$ is a group scheme over $\mathbx{Z}_{p}$,
and we let $\overline{G}$ be its special fiber. Moonen [9] gives an explicit
identification of the Ekedahl-Oort strata with $W_{X}\backslash
W_{\overline{G}}$, for a certain subgroup $W_{X}$ depending on the signature.
Concretely, $W_{X}\backslash W_{\overline{G}}$ can be described as

$\\{(w_{1},w_{2})\in\mathbf{W}(a,b)\times\mathbf{W}(b,a)\ |\
w_{2}=w_{0}w_{1}w_{0}\\}.$

As the map $\pi\mapsto w_{0}\pi w_{0}$ gives an isomorphism between
$\mathbf{W}(a,b)$ and $\mathbf{W}(b,a)$, one has $W_{X}\backslash
W_{\overline{G}}\cong\mathbf{W}(a,b)$ which agrees with our statement of
Theorem 2.2.

For any $w\in\mathbf{W}(a,b)$, let $\mathcal{M}(a,b)_{w}$ denote the
corresponding Ekedahl-Oort stratum and $(G_{w},\iota_{w},\lambda_{w})$ denote
the corresponding $p$-torsion group scheme. One observes that since $G_{w}$ is
a $p$-torsion group scheme, the action of $\mathcal{O}_{K}$ on $G_{w}$, via
$\iota_{w}$, factors through $\mathcal{O}_{K}/(p)$. Since $p$ is assumed to be
inert, $\mathcal{O}_{K}/(p)=\mathbx F_{p^{2}}$, and we abuse notation and
refer to this induced action of $\mathbx F_{p^{2}}$ as $\iota_{w}$ as well.
Moonen uses (contravariant) Dieudonné theory to describe
$(G_{w},\iota_{w},\lambda_{w})$. For each $w\in\mathbf{W}(a,b)$, he constructs
the standard object $(N_{w},F,V)$. This is the Dieudonné module of $G_{w}$,
consisting of a vector space $N_{w}$ of dimension $2q$ over $\mathbx k$, $F$ a
$\text{Frob}_{\mathbx k}$-semilinear operator on $N_{w}$, and $V$ a
$\text{Frob}^{-1}_{\mathbx k}$-semilinear operator on $N_{w}$, described
explicitly on a basis. The action $\iota_{w}$ is recorded by a splitting
$N_{w}=N_{w,1}\oplus N_{w,2}$. By Theorem 6.7 of [9], $\lambda_{w}$ is
uniquely determined by $(G_{w},\iota_{w})$, so it is unnecessary to record the
corresponding polarization of $N_{w}$.

We now outline the inverse of the assignment $\omega\mapsto N_{\omega}$. Given
a Dieudonné module $(N,F,V)$ with an $\mathbx F_{p^{2}}$-action, one
constructs a _final filtration_

$0\subset W_{1}\subset W_{2}\subset\cdots\subset W_{2q}=N$

that is stable under $F$ and $V^{-1}$ and has the property $\dim k(W_{i})=i$.

The action of $\mathbx F_{p^{2}}$ on $N$ induces a decomposition
$N=N_{1}\oplus N_{2}$. Intersecting the filtration $W_{\bullet}$ of $N$ with
the subspace $N_{i}$ (for $i=1,2$) gives:

$0\subset C_{i,1}\subset C_{i,2}\subset\cdots\subset C_{i,q}=N_{i}.$

From here we define the functions

$\eta_{i}(j)=\dim(C_{i,j}\cap N[F]).$

We focus on $\eta_{1}$, as $\eta_{2}$ is determined by $\eta_{1}$ via the
Rosati involution condition. Since

$\eta_{1}(q)=\dim(N_{1}[F])=b,$

there are $b$ indices where $\eta_{1}$ jumps (meaning
$\eta_{1}(j)=\eta_{1}(j-1)+1$). We denote these integers by $1\leqslant
j_{1}\leqslant\ldots\leqslant j_{b}\leqslant q$ and the remaining integers by
$1\leqslant i_{1}\leqslant\ldots i_{a}\leqslant q$. The permutation
$\omega_{N}\in\mathbf{W}(a,b)$ is

(2.3.2) $\omega_{N}(j_{l})=l\text{ and }\omega_{N}(i_{m})=b+m.$

By construction, $N$ is the standard object corresponding to the permutation
$\omega_{N}$.

In Section 3, we give more explicit descriptions in signature $(q-2,2)$ of the
minimal length coset representatives of $W_{J}\setminus W$ and the standard
objects.

##  3\. Structure of Ekedahl-Oort Strata for Signature $(q-2,2)$

The goal of this section is to index the Ekedahl-Oort strata of
$\mathcal{M}(q-2,2)$ and study the topological closure relations between them.
First, we explicitly describe the elements of the set $\textbf{W}(q-2,2)$.
Following that, we recall the Bruhat partial ordering, and its generalization
due to He [5] that corresponds to topological closure relations between the
Ekedahl-Oort strata. Finally, we present some results on the closure relations
between the elements of $\mathbf{W}(q-2,2)$.

###  3.1. Index Set

In this section, we explicitly describe the set $\mathbf{W}(q-2,2)$.

For $1\leqslant u<v\leqslant q$ we define the following element of $W$

(3.1.1) $\gamma_{u,v}:=(2,3,\ldots,v)(1,2,\ldots,u).$

We note that $\gamma_{u,v}$ fixes every element in the set $\left\\{j\
\middle|\ v<j\leqslant q\right\\}$ and

(3.1.2) $\gamma_{u,v}(j)=\begin{cases}j+2&\text{if $1\leqslant j\leqslant
u-1$,}\\\ 1&\text{if $j=u$,}\\\ j+1&\text{if $u+1\leqslant j\leqslant
v-1$,}\\\ 2&\text{if $j=v$.}\end{cases}$

######  Lemma 3.1.

$\mathbf{W}(q-2,2)=\left\\{\gamma_{u,v}\ \middle|\ 1\leqslant u<v\leqslant
q\right\\}$ with $\ell(\gamma_{u,v})=u+v-3$.

###### Proof.

Recall that the length satisfies
$\ell(\gamma_{u,v})=\\#\mathrm{Inv}(\gamma_{u,v})$ where
$\mathrm{Inv}(\gamma_{u,v})$ is the set of inversions, as defined in Equation
(2.3.1). It is apparent from the description of $\gamma_{u,v}$ as a function
in Equation (3.1.2) that (understood appropriately for $u=1$)

$\mathrm{Inv}(\gamma_{u,v})=\left\\{(j,\,u),\,(j,\,v),\,(u+k,\,v)\ \middle|\
1\leqslant j\leqslant u-1,\,1\leqslant k\leqslant v-u-1\right\\}.$

Thus, $\ell(\gamma_{u,v})=u+v-3$. Further, we note that for $\sigma\in
W_{(q-2,2)}$ we obtain
$\mathrm{Inv}(\gamma_{u,v})\subseteq\mathrm{Inv}(\sigma\gamma_{u,v})$ and thus
$\ell(\gamma_{u,v})$ is minimal in the coset $W_{(q-2,2)}\gamma_{u,v}$.

Lastly, to prove that any two distinct
$\gamma_{u_{1},v_{1}},\gamma_{u_{2},v_{2}}$ lie in distinct cosets one can
verify that $\gamma_{u_{2},v_{2}}\gamma_{u_{1},v_{1}}^{-1}$ is not an element
of $W_{(q-2,2)}$. ∎

######  Remark 3.2.

Each $\gamma_{u,v}\in\mathbf{W}(q-2,2)$ corresponds to an Ekedahl-Oort stratum
labelled $\mathcal{M}(q-2,2)_{\gamma_{u,v}}$. By [9], the dimension of
$\mathcal{M}(q-2,2)_{\gamma_{u,v}}$ is equal to $\ell(\gamma_{u,v})$. Let
$n_{d}$ be the number of Ekedahl-Oort strata there are of dimension
$0\leqslant d\leqslant 2(q-2)$, i.e.,

$n_{d}=\\#\left\\{\gamma_{u,v}\ \middle|\ \ell(\gamma_{u,v})=d\right\\}.$

Since $\ell(\gamma_{u,v})=u+v-3$, we have that

$n_{d}=\begin{cases}{\displaystyle\lfloor d/2\rfloor+1}&\text{if }d\leqslant
q-2,\\\\[5.0pt] {\displaystyle\lfloor d/2\rfloor+1}-(d-(q-2))&\text{if
}d>q-2.\end{cases}$

###  3.2. Standard Objects

In this section, we describe the standard objects of the Ekedahl-Oort strata
of $\mathcal{M}(q-2,2)$. These standard objects are mod-$p$ Dieudonné modules,
and by Dieudonné theory each determines a corresponding $p$-torsion group
scheme.

######  Lemma 3.3.

Let $(N_{\gamma_{u,v}},F,V)$ be the standard object corresponding to
$\gamma_{u,v}$. Then,

$N_{\gamma_{u,v}}=\mathrm{Span}k\\{e_{i,j}\\},$

where $1\leqslant i\leqslant 2$ and $1\leqslant j\leqslant q$. Further, $F$ is
the $\mathrm{Frob}_{\mathbx k}$-semilinear operator on $N_{\gamma_{u,v}}$ and
$V$ is the $\mathrm{Frob}^{-1}_{\mathbx k}$-semilinear operator on
$N_{\gamma_{u,v}}$ extended from the following action on the basis:

$\displaystyle F(e_{1,j})$ $\displaystyle=\begin{cases}0&\text{if
$j=u,\,j=v$}\\\ e_{2,j}&\text{if $1\leqslant j\leqslant u-1$}\\\
e_{2,j-1}&\text{if $u<j\leqslant v-1$}\\\ e_{2,j-2}&\text{if
$j>v$}\end{cases}$ $\displaystyle V(e_{1,j})$
$\displaystyle=\begin{cases}0&\text{if $j=1,\,2$}\\\ e_{2,j-2}&\text{if
$2<j\leqslant q-v+2$}\\\ e_{2,j-1}&\text{if $q-v+2<j\leqslant q-u+1$}\\\
e_{2,j}&\text{if $j>q-u+1$}\end{cases}$ $\displaystyle F(e_{2,j})$
$\displaystyle=\begin{cases}e_{1,1}&\text{if $j=q-v+1$}\\\ e_{1,2}&\text{if
$j=q-u+1$}\\\ 0&\text{otherwise}\end{cases}$ $\displaystyle V(e_{2,j})$
$\displaystyle=\begin{cases}e_{1,u}&\text{if $j=q-1$}\\\ e_{1,v}&\text{if
$j=q$}\\\ 0&\text{otherwise}.\end{cases}$

###### Proof.

We use the explicit description of $\gamma_{u,v}$ to unwind the description of
the corresponding standard object given in [9, 4.9], under the assumption that
the prime $p$ is inert in the quadratic imaginary field $K$. ∎

###  3.3. Bruhat Order

In this section we use the index set defined in Section 3.1 to give an
explicit description of the Bruhat order on $\mathbf{W}(q-2,2)$. We first
recall the definition of the Bruhat order on a Weyl group.

######  Definition 3.4 (The Bruhat order).

Consider the Coxeter system $(W,S)$. For $w$ and $w^{\prime}$ in $W$, we say
$w^{\prime}\leqslant w$, if $\ell(w^{\prime})\leqslant\ell(w)$ and there
exists a sequence $w^{\prime}=v_{0},\,v_{1},\,\ldots,\,v_{m}=w$ such that

  * •

$\ell(v_{i-1})\leqslant\ell(v_{i})$; and

  * •

$v_{i-1}^{-1}v_{i}$ is a reflection.

By [1, Proposition 2.5.1], the Bruhat order on $W$ induces a partial order on
$\mathbf{W}(q-2,2)$, also called the Bruhat order and denoted $\leqslant$.
Using the notation in Section 3.1, we can completely characterize the Bruhat
relations for the Weyl group cosets $\mathbf{W}(q-2,2)$ associated to
$\mathcal{M}(q-2,2)$.

######  Proposition 3.5.

Let $\gamma_{u_{1},v_{1}}$ and $\gamma_{u_{2},v_{2}}$ be elements of
$\mathbf{W}(q-2,2)$. Then $\gamma_{u_{1},v_{1}}\leqslant\gamma_{u_{2},v_{2}}$
with respect to the Bruhat order if and only if $u_{1}\leqslant u_{2}$ and
$v_{1}\leqslant v_{2}$.

###### Proof.

For a permutation $x\in\mathfrak{S}_{n}$, define

$x[i,j]:=\\#\\{a\leqslant i\;|\;x(a)\geqslant j\\}.$

Then $x\leqslant y$ with respect to the Bruhat order if and only if
$x[i,j]\leqslant y[i,j]$ for all $i,j\leqslant n$ (see Theorem 2.1.5 of [1]).
Using the definition of $\gamma_{u,v}$ and its inverse we see that

$\gamma_{u,v}^{-1}[i,j]=\begin{cases}i-j+1&\text{ if }j<i\\\ 1&\text{ if
}j\leqslant u\text{ and }i=1\\\ 2&\text{ if }j\leqslant u\text{ and
}1<i\leqslant j\\\ 0&\text{ if }u<j\leqslant v\text{ and }i=1\\\ 1&\text{ if
}u<j\leqslant v\text{ and }1<i\leqslant j\\\ 0&\text{ if }j>v\text{ and
}i<j\\\ 1&\text{ if }j>v\text{ and }i=j.\\\ \end{cases}$

Comparing the values $\gamma_{u_{1},v_{1}}^{-1}[i,j]$ and
$\gamma_{u_{2},v_{2}}^{-1}[i,j]$, we see that
$\gamma_{u_{1},v_{1}}^{-1}\leqslant\gamma_{u_{2},v_{2}}^{-1}$ if and only if
$u_{1}\leqslant u_{2}$ and $v_{1}\leqslant v_{2}$. Since
$\gamma_{u_{1},v_{1}}^{-1}\leqslant\gamma_{u_{2},v_{2}}^{-1}$ is equivalent to
$\gamma_{u_{1},v_{1}}\leqslant\gamma_{u_{2},v_{2}}$, the statement follows. ∎

###  3.4. The Closure Order

####  3.4.1. Definition and properties

There are two partial orders on $\mathbf{W}(q-2,2)$ that we are concerned with
in this paper: the Bruhat order defined above, and a generalization of this
due to He [5] that we refer to as the Closure order. In this section we define
the Closure order and discuss how both orders relate to Ekedahl-Oort strata.
As in Section 3.3, we focus on $\mathbf{W}(q-2,2)$ but the constructions here
can all be re-framed for more general $\mathbf{W}(a,b)$. We refer to [5, 12,
16, 17, 18] for more information.

Recall that the Frobenius automorphism $F$ of the algebraic group
$\overline{G}$ induces a $\psi\in\mathrm{Aut}(W,S)$ of the Coxeter system: for
any element $w\in W$, $\psi(w)=w_{0}ww_{0}$, where $w_{0}$ is the unique
maximal element in $W$.

As in Section 2.3, $W_{(q-2,2)}$ denotes the subgroup of the Weyl group $W$
generated by $J_{(a,b)}=\\{s_{1},...,s_{q-1}\\}\setminus\\{s_{b}\\}$. For ease
of notation, let $J:=J_{(q-2,2)}$. The unique maximal element in
$W_{(q-2,2)}$, with respect to the Bruhat order, is denoted by $w_{0,J}$. Let
$W^{\psi(J)}$ be the set of elements $w\in W$ that have minimal length in
their coset $wW_{\psi(J)}$, and let $J^{\prime}:={}^{w_{0}}\psi(J)$ denote the
set $w_{0}\psi(J)w_{0}$. In the notation of Section 2.3,
${}^{J^{\prime}}W^{\psi(J)}$ denotes the collection of minimal-length coset
representatives for $W_{J^{\prime}}\big{\backslash}W\big{/}\psi(J)$.

The definition of the Closure order involves the element $x$ of minimal length
in the double coset $W_{J^{\prime}}w_{0}W_{\psi(J)}$. One can show that this
element satisfies

$x=w_{0,J^{\prime}}w_{0}=w_{0}w_{0,\psi(J)}$

and so $x$ is the unique maximal length element amongst the minimal length
coset representatives ${}^{J^{\prime}}W^{\psi(J)}$.

We now define the Closure order corresponding to $\psi$ on
$\mathbf{W}(q-2,2)$.

######  Definition 3.6 (The Closure order $\operatorname{\preccurlyeq}$).

Let $w,w^{\prime}$ be elements of $\mathbf{W}(q-2,2)$. Then
$w\operatorname{\preccurlyeq}w^{\prime}$ if there exists an $h\in W_{(q-2,2)}$
such that

(3.4.1) $hwx\psi(h^{-1})x^{-1}\leqslant w^{\prime},$

where $\leqslant$ denotes the Bruhat order111In the notation of [5],
Inequality (3.4.1) actually implies
$wx\operatorname{\preccurlyeq}w^{\prime}x$. In our work, as in [16], we drop
the $x$ from the notation and simply write
$w\operatorname{\preccurlyeq}w^{\prime}$..

Using the fact that $\psi(h)=w_{0}hw_{0}$, we have

$x\psi(h^{-1})x^{-1}=(w_{0}w_{0,\psi(J)})(w_{0}{h^{-1}}w_{0})(w_{0,\psi(J)}w_{0})=w_{0,J}h^{-1}w_{0,J},$

and so Inequality (3.4.1) can be rewritten as

(3.4.2) $hww_{0,J}h^{-1}w_{0,J}\leqslant w^{\prime}.$

Note that the Bruhat order is the special case of the Closure order where
Inequality (3.4.1) is satisfied for $h$ equal to the identity in
$W_{(q-2,2)}$.

In Section 3.4 we give some partial results on the Closure order on
$\mathbf{W}(q-2,2)$. One difficulty that arises in giving a complete
description of the Closure order on $\mathbf{W}(q-2,2)$ is that the expression
$hww_{0,J}h^{-1}w_{0,J}$ is not a coset representative in $\mathbf{W}(q-2,2)$.
In other words, the Bruhat relation that we are checking in Inequality (3.4.1)
is in the Weyl group $W$ and not in $\mathbf{W}(q-2,2)$, and so we are not
able to apply our results on the Bruhat order in Section 3.3 to the Closure
order.

The Closure order is of interest because it captures topological closure
relations among the Ekedahl-Oort strata. Let
$\overline{\mathcal{M}(q-2,2)}_{\gamma_{u^{\prime},v^{\prime}}}$ denote the
topological closure of the stratum
$\mathcal{M}(q-2,2)_{\gamma_{u^{\prime},v^{\prime}}}$. By [16, Theorem 1.2],

$\overline{\mathcal{M}(q-2,2)}_{\gamma_{u^{\prime},v^{\prime}}}=\\!\\!\\!\\!\\!\\!\bigcup_{\gamma_{u,v}\operatorname{\preccurlyeq}\gamma_{u^{\prime},v^{\prime}}}\\!\\!\\!\\!\\!\\!\mathcal{M}(q-2,2)_{\gamma_{{u,v}}}.$

Thus, to understand closure relations among Ekedahl-Oort strata, it suffices
to analyze the Closure order relations in $\mathbf{W}(q-2,2)$.

####  3.4.2. Closure relations

In this section, we prove several results that give a partial classification
of the Closure order for signature $(q-2,2)$. We begin with a lemma that we
will use in order to prove the two sets of closure relations in Theorems 3.9
and 3.15.

######  Lemma 3.7.

Let $s_{k}$ with $k>2$ denote the simple reflection $(k,k+1)$. For $1\leqslant
u<v\leqslant q$, the following hold:

  * •

$\gamma_{u,v+1}s_{v}=\gamma_{u,v}$ and
$s_{v}\gamma^{-1}_{u,v+1}=\gamma^{-1}_{u,v},$

  * •

$\gamma_{u+1,v}s_{u}=\gamma_{u,v}$ and
$s_{u}\gamma^{-1}_{u+1,v}=\gamma^{-1}_{u,v}$.

Furthermore, $s_{q-(j-1)}s_{q-(j-2)}\cdots
s_{q-(j-i)}\gamma^{-1}_{j+1,q+1-j+i}=\gamma^{-1}_{j+1,q+1-j}\,$ for any
$4\leqslant j\leqslant\frac{q}{2}$ and $1\leqslant i\leqslant j-3$.

###### Proof.

We can prove these results using the non-disjoint cycle definition of
$\gamma_{u,v}$:

$\gamma_{u,v}=(2,3,\ldots,v-1,v)(1,2,\ldots,u-1,u).$

For example, to prove the first equality, note that since $v>u$, the
reflection $s_{v}$ commutes with $(1,2,\ldots,u-1,u)$. Hence,

$\displaystyle\gamma_{u,v+1}s_{v}$
$\displaystyle=(2,3,\ldots,v,v+1)(1,2,\ldots,u-1,u)(v,v+1)$
$\displaystyle=(2,3,\ldots,v,v+1)(v,v+1)(1,2,\ldots,u-1,u)$
$\displaystyle=(2,3,\ldots,v)(1,2,\ldots,u-1,u)$

The other relations in the bullet-points are proved in a similar manner, and
the last result follows from applying these repeatedly. ∎

We define an action on elements $w\in\mathbf{W}(q-2,2)$ by elements $h\in
W_{(q-2,2)}$ by

(3.4.3) $h\bullet w:=hww_{0,J}h^{-1}w_{0,J},$

which is the expression that appears on the left side of Inequality 3.4.2. In
other words,
$\gamma_{u_{1},v_{1}}\operatorname{\preccurlyeq}\gamma_{u_{2},v_{2}}$ if there
is an $h\in W_{(q-2,2)}$ such that the Bruhat relation
$h\bullet\gamma_{u_{1},v_{1}}\leqslant\gamma_{u_{2},v_{2}}$ holds. In our
work, we express $h$ as a product of simple reflections, and the following
lemma will be used throughout.

######  Lemma 3.8.

For any simple reflection $s_{k}=(k,k+1)$ with $k>2$,

$w_{0,J}s_{k}w_{0,J}=s_{q+2-k}.$

###### Proof.

Since $w_{0,J}$, defined as $w_{0,J}(x)=q+3-x$ for all $3\leqslant x\leqslant
q$, is of order two and $k>2$, we have

$w_{0,J}s_{k}w_{0,J}=(w_{0,J}(k),w_{0,J}(k+1))=s_{q+2-k}.\qed$

The following theorem gives the first set of closure relations.

######  Theorem 3.9.

Let $q\geqslant 5$. Then the following relations hold

  1. (1)

$\gamma_{j+1,q+1-j}\operatorname{\preccurlyeq}\gamma_{j,q+3-j}\,$ for
$3\leqslant j<q/2$,

  2. (2)

$\gamma_{q-j,j+1}\operatorname{\preccurlyeq}\gamma_{q-j+2,j}\,$ for
$q/2+1<j\leqslant q-1$.

######  Remark 3.10.

For both relations, the difference in lengths between the coset
representatives being compared is 1. We restrict to $q\geqslant 5$ since the
specific $\gamma_{u,v}$ in the statement of the theorem are not defined for
$q\leqslant 4$. For smaller values of $q$, there are no closure relations
other than those coming from Bruhat relations.

In many computations, it is easier to work with the inverses
$\gamma_{u,v}^{-1}$ of the coset representatives. To prove the above result,
Theorem 3.9, we will demonstrate that

$s_{j}\bullet\gamma^{-1}_{j+1,q+1-j}\leqslant\gamma^{-1}_{j,q+3-j}\quad\text{and}\quad
s_{j}\bullet\gamma^{-1}_{q-j,j+1}\leqslant\gamma^{-1}_{q+2-j,j}$

with respect to the Bruhat order. There are two conditions to check for each
of the relations in Theorem 3.9, and we split these into separate lemmata. We
first compute the action by $s_{j}$ on $\gamma^{-1}_{j+1,q+1-j}$ and
$\gamma^{-1}_{q-j,j+1}$.

######  Lemma 3.11.

Let $3\leqslant j<q/2$, and let $\tau_{j}$ be defined by

(3.4.4) $\tau_{j}(k)=\begin{cases}j&\text{for $k=1$,}\\\ q+1-j&\text{for
$k=2$,}\\\ k-2&\text{for $3\leqslant k\leqslant j+1$,}\\\ k-1&\text{for
$j+2\leqslant k\leqslant q+1-j$,}\\\ q+3-j&\text{for $k=q+2-j$,}\\\
q+2-j&\text{for $k=q+3-j$,}\\\ k&\text{for $k>q+3-j$.}\end{cases}$

Then $s_{j}\bullet\gamma^{-1}_{j+1,q+1-j}=\tau_{j}$.

###### Proof.

Applying Lemma 3.7 and Lemma 3.8 yields

$s_{j}\bullet\gamma^{-1}_{j+1,q+1-j}=s_{j}\gamma^{-1}_{j+1,q+1-j}s_{q+2-j}=\gamma^{-1}_{j,q+1-j}s_{q+2-j}.$

Recall that $\gamma_{u,v}=(2,3,\ldots,v)(1,2,\ldots,u)$. Thus, the expression
above can be written as

$(1,2,\ldots,j)^{-1}(2,3,\ldots,q+1-j)^{-1}s_{q+2-j}.$

We now determine the image of each $k$ under the above permutation, focusing
first on small values of $k$. Moving right to left through the permutations,
we see that 1 is mapped as: $1\mapsto j$. The element $2$ is mapped as:
$2\mapsto q+1-j$. For all other $k\leqslant j+1$, we have $k\mapsto k-1\mapsto
k-2$ via $(1,2,\ldots,j)^{-1}(2,3,\ldots,q+1-j)^{-1}$.

For $j+2\leqslant k\leqslant q+1-j$, we have $k\mapsto k-1$ via
$(2,3,\ldots,q+1-j)^{-1}$. Since $s_{q+2-j}$ is disjoint from the rest of the
permutations, $q+2-j$ and $q+3-j$ are swapped. Finally, all $k>q+3-j$ are
fixed. ∎

######  Lemma 3.12.

Let $q/2+1<j\leqslant q-1$ , and let $\tau_{j}$ be defined by

(3.4.5) $\tau_{j}(k)=\begin{cases}q-j&\text{for $k=1$,}\\\ j&\text{for
$k=2$,}\\\ k-2&\text{for $3\leqslant k\leqslant j+1$,}\\\ q+2-j&\text{for
$k=q+2-j$,}\\\ q+1-j&\text{for $k=q+3-j$,}\\\ k-1&\text{for $q+4-j\leqslant
k\leqslant j$,}\\\ k&\text{for $k>q+3-j$}.\end{cases}$

Then $s_{j}\bullet\gamma^{-1}_{q-j,j+1}=\tau_{j}$.

###### Proof.

Applying Lemma 3.7 and Lemma 3.8 yields

$s_{j}\bullet\gamma^{-1}_{q-j,j+1}=s_{j}\gamma^{-1}_{q-j,j+1}s_{q+2-j}=\gamma^{-1}_{q-j,j}s_{q+2-j},$

which can be written as

$(1,2,\ldots,q-j)^{-1}(2,3,\ldots,j)^{-1}s_{q+2-j}.$

The result then follows using techniques similar to those used in the proof of
Lemma 3.11. ∎

We now prove the following result for the lengths of the $\tau_{j}$’s.

######  Lemma 3.13.

If $3\leqslant j<q/2$, then
$\ell(s_{j}\bullet\gamma^{-1}_{j+1,q+1-j})=\ell(\gamma^{-1}_{j+1,q+1-j})$.
If $q/2+1<j\leqslant q-1$, then
$\ell(s_{j}\bullet\gamma^{-1}_{q-j,j+1})=\ell(\gamma^{-1}_{q-j,j+1})$.

###### Proof.

Recall that one may compute the length of a permutation as the cardinality of
its set of inversions (Equation 2.3.1). We obtain the result by a
straightforward computation of the relevant sets of inversions. ∎

The following is the final ingredient needed in order to prove Theorem 3.9.

######  Lemma 3.14.

If $3\leqslant j<q/2$ then
$(s_{j}\bullet\gamma^{-1}_{j+1,q+1-j})^{-1}\gamma^{-1}_{j,q+3-j}=(2,q+2-j)$.
If $q/2+1<j\leqslant q-1$ then
$(s_{j}\bullet\gamma^{-1}_{q-j,j+1})^{-1}\gamma^{-1}_{q+2-j,j}=(1,q+2-j).$

###### Proof.

The proofs of these equalities use the above lemmata and simple permutation
composition. For the first equality, we use the definition of $\tau_{j}$ in
Equation (3.4.4) to write

$\displaystyle(s_{j}\bullet\gamma^{-1}_{j+1,q+1-j})^{-1}\gamma^{-1}_{j,q+3-j}$
$\displaystyle=\tau_{j}^{-1}(1,2,\ldots,j)^{-1}(2,3,\ldots,q+3-j)^{-1}.$

We verify that this simplifies to $(2,q+2-j)$ by composition of permutations.
For example, $\gamma^{-1}_{j,q+3-j}$ maps $2\mapsto q+3-j$, which then maps to
$q+2-j$ via $\tau_{j}^{-1}$. Similarly, $\gamma^{-1}_{j,q+3-j}$ maps
$q+2-j\mapsto q+1-j$, which is then mapped to $2$ via $\tau_{j}^{-1}$. All
other values are fixed by the composition.

For the second equality, we use the definition of $\tau_{j}$ in Equation
(3.4.5) to write

$\displaystyle(s_{j}\bullet\gamma^{-1}_{q-j,j+1})^{-1}\gamma^{-1}_{q+2-j,j}$
$\displaystyle=\tau_{j}^{-1}(1,2,\ldots,q+2-j)^{-1}(2,3,\ldots,j)^{-1}.$

We can use the same method as above to prove the desired result. For example,
we see that $1\mapsto q+2-j$ via $\gamma^{-1}_{q+2-j,j}$, which is then fixed
by $\tau_{j}^{-1}$. We also have that $\gamma^{-1}_{q+2-j,j}$ maps
$q+2-j\mapsto q+1-j\mapsto q-j$, which is then mapped to $1$ via
$\tau_{j}^{-1}$. All other values are fixed by the composition. ∎

These results combine to give a proof of Theorem 3.9.

###### Proof of Theorem 3.9.

The above lemmata demonstrate that
$s_{j}\bullet\gamma^{-1}_{j+1,q+1-j}\leqslant\gamma^{-1}_{j,q+3-j}$ and
$s_{j}\bullet\gamma^{-1}_{q-j,j+1}\leqslant\gamma^{-1}_{q+2-j,j}$, for the
appropriately chosen values of $j$, with respect to the Bruhat order, which
proves the two closure relation statements. ∎

The following theorem gives the second set of closure relations.

######  Theorem 3.15.

Let $q\geqslant 7$. Then the following relations hold

  1. (1)

Let $4\leqslant j<q/2$ and $1\leqslant i\leqslant j-3$. Then
$\gamma_{j+1,q+1-j+i}\operatorname{\preccurlyeq}\gamma_{j,q+3-j+i}$.

  2. (2)

Let $q/2+2<j\leqslant q-1$ and $1\leqslant i<j-q/2-1$. Then
$\gamma_{q-j,j+1-i}\operatorname{\preccurlyeq}\gamma_{q+2-j,j-i}$.

######  Remark 3.16.

As in Theorem 3.9, the difference in lengths between the coset representatives
being compared is 1. We again restrict to the values of $q$, this time to
$q\geqslant 7$, since the specific $\gamma_{u,v}$ in the statement of the
theorem are not defined for smaller $q$. Note that letting $i=0$ yields the
two relations given in Theorem 3.9, however we need to act by different
permutations in order to realize the relations.

As we did for Theorem 3.9, we will prove the result, Theorem 3.15, by working
with the inverses of the coset representatives and by proving the
corresponding Bruhat relations. There are two conditions to check for each of
the relations in Theorem 3.15, and we split these into separate lemmata. We
first compute the action by certain permutations $h_{i,j}$ (defined below) on
$\gamma^{-1}_{j+1,q+1-j+i}$ and $\gamma^{-1}_{q-j-i,j+1-i}$.

######  Lemma 3.17.

Let $4\leqslant j<q/2$ and $1\leqslant i\leqslant j-3$. Define $h_{i,j}$ to be
the permutation

(3.4.6) $h_{i,j}=s_{j}s_{j-1}\cdots s_{j-i}\cdot s_{q-(j-1)}s_{q-(j-2)}\cdots
s_{q-(j-i)}$

and let $\tau_{i,j}$ be defined by

(3.4.7) $\tau_{i,j}(k)=\begin{cases}j&\text{for $k=1$,}\\\ q+1-j&\text{for
$k=2$,}\\\ k-2&\text{for $3\leqslant k\leqslant j+1$,}\\\ k-1&\text{for
$j+2\leqslant k\leqslant q+1-j$,}\\\ q+3-j+i&\text{for $k=q+2-j$,}\\\
k-1&\text{for $q+3-j\leqslant k\leqslant q+3-j+i$,}\\\ k&\text{for
$k>q+3-j+i$}.\end{cases}$

Then $h_{i,j}\bullet\gamma^{-1}_{j+1,q+1-j+i}=\tau_{i,j}$.

###### Proof.

The proof of this result relies on computing permutation compositions. We
provide some simplifications of the expression to make the computations
easier. First, use Lemma 3.7 to write

$\displaystyle h_{i,j}\bullet\gamma^{-1}_{j+1,q+1-j+i}$
$\displaystyle=s_{j}s_{j-1}\cdots
s_{j-i}\gamma^{-1}_{j+1,q+1-j}(w_{0,J}h_{i,j}w_{0,J})^{-1},$

where $(w_{0,J}h_{i,j}w_{0,J})^{-1}=s_{j+2-i}\cdots
s_{j}s_{j+1}s_{q+2-(j-i)}\cdots s_{q+2-(j-1)}s_{q+2-j}.$

Recall that $\gamma_{u,v}=(2,3,\ldots,v)(1,2,\ldots,u)$. Thus, the expression
above can be written as

$s_{j}s_{j-1}\cdots
s_{j-i}(1,2,\ldots,j+1)^{-1}(2,3,\ldots,q+1-j)^{-1}(w_{0,J}h_{i,j}w_{0,J})^{-1}.$

We now determine the image of each $k$ under the above permutation, focusing
first on small values of $k$. First note that the permutation
$(w_{0,J}h_{i,j}w_{0,J})^{-1}$ fixes all values less than $j+2-i$. Moving
right to left through the permutations, we see that 1 is mapped as: $1\mapsto
j+1\mapsto j$. The element $2$ is mapped as: $2\mapsto q+1-j$. For all other
$k\leqslant j+2-i$, we have $k\mapsto k-1\mapsto k-2$ via
$(1,2,\ldots,j+1)^{-1}(2,3,\ldots,q+1-j)^{-1}$. Since $k-2<j-i$, it is fixed
by $s_{j}s_{j-1}\cdots s_{j-i}$.

Now suppose $j-i+2\leqslant k<j+2$. The permutation
$(w_{0,J}h_{i,j}w_{0,J})^{-1}$ maps $k$ to $k+1$. This is then mapped as
$k+1\mapsto k\mapsto k-1$ by $\gamma^{-1}_{j+1,q+1-j}$. Moving right to left,
we see that $k-1$ is mapped to $k-2$ by $s_{j-1}\cdots s_{j-i}$.

As a special case, note that the permutation $(w_{0,J}h_{i,j}w_{0,J})^{-1}$
maps $j+2$ as follows: $j+2\mapsto j+1\mapsto\cdots\mapsto j+2-i$. This is
then mapped as $j+2-i\mapsto j+1-i\mapsto j-i$ by $\gamma^{-1}_{j+1,q+1-j}$.
The permutation $s_{j-1}\cdots s_{j-i}$ maps $j-i$ as follows: $j-i\mapsto
j-i+1\mapsto\cdots\mapsto j\mapsto j+1$.

We now consider larger values of $k$. For $j+2<k<q+2-j$, we see that $k$ is
again fixed by $(w_{0,J}h_{i,j}w_{0,J})^{-1}$. Moving right to left from here,
we see that we have $k\mapsto k-1$ by $\gamma^{-1}_{j+1,q+1-j}$.

As another special case, we may also see that $(w_{0,J}h_{i,j}w_{0,J})^{-1}$
maps $q+2-j$ as follows: $q+2-j\mapsto q+3-j\mapsto q+4-j\cdots\mapsto
q+2-(j-i)+1=q+3-j+i$. Moving right to left, we see that $q+3-j+i$ is fixed by
the remaining permutations.

For $q+3-j\leqslant k\leqslant q+3-(j-i)$, the permutation
$(w_{0,J}h_{i,j}w_{0,J})^{-1}$ maps $k$ to $k-1$. This is then fixed by
$\gamma^{-1}_{j+1,q+1-j}$. Lastly, it is easy to check that all $k>q+3-(j-i)$
are fixed by the permutation. ∎

######  Lemma 3.18.

Let $q/2+2<j\leqslant q-1$ and $1\leqslant i<j-q/2-1$. Define $h_{i,j}$ to be
the permutation

(3.4.8) $h_{i,j}=s_{j}s_{j-1}\cdots s_{j-i}\cdot
s_{q+1-(j-1)}s_{q+1-(j-2)}\cdots s_{q+1-(j-i)},$

and let $\tau_{i,j}$ be defined by

(3.4.9) $\tau_{i,j}(k)=\begin{cases}q-j&\text{for $k=1$,}\\\ j-i&\text{for
$k=2$,}\\\ k-2&\text{for $3\leqslant k\leqslant q+1-j$,}\\\ q+2-j+i&\text{for
$k=q+2-j$,}\\\ q+1-j+i&\text{for $k=q+3-j$,}\\\ k-1&\text{for $q+4-j\leqslant
k\leqslant j-i$,}\\\ k&\text{for $k>j-i$}.\end{cases}$

Then $h_{i,j}\bullet\gamma^{-1}_{q-j,j+1-i}=\tau_{i,j}$.

###### Proof.

We omit the proof of the result since it is similar to the proof Lemma 3.17. ∎

Lemma 3.17 and Lemma 3.18 lead to the following corollary.

######  Corollary 3.19.

If $4\leqslant j<q/2$ and $1\leqslant i\leqslant j-3$, then

$(h_{i,j}\bullet\gamma^{-1}_{j+1,q+1-j+i})^{-1}\gamma^{-1}_{j,q+3-j+i}=(2,q+2-j)$

for $h_{i,j}$ defined in Equation (3.4.6). If $q/2+2<j\leqslant q-1$ and
$1\leqslant i<j-q/2-1$ then

$(h_{i,j}\bullet\gamma^{-1}_{q-j,j+1-i})^{-1}\gamma^{-1}_{q+2-j,j-i}=(1,q+2-j)$

for $h_{i,j}$ defined in Equation (3.4.8).

We have the following result for the lengths of the elements $\tau_{i,j}$.

######  Lemma 3.20.

If $4\leqslant j<q/2$ and $1\leqslant i\leqslant j-3$, then
$\ell(h_{i,j}\bullet\gamma^{-1}_{j+1,q+1-j+i})=\ell(\gamma^{-1}_{j+1,q+1-j+i})$
for $h_{i,j}$ defined in Equation (3.4.6). If $q/2+2<j\leqslant q-1$ and
$1\leqslant i<j-q/2-1$ then

$\ell(h_{i,j}\bullet\gamma^{-1}_{q-j,j+1-i})=\ell(\gamma^{-1}_{q-j,j+1-i})$

for $h_{i,j}$ defined in Equation (3.4.8).

###### Proof.

Recall that one may compute the length of a permutation as the cardinality of
its set of inversions (see Equation (2.3.1)). We obtain the result by a
straightforward computation of the relevant sets of inversions. ∎

These results combine to give a proof of Theorem 3.15.

###### Proof of Theorem 3.15.

The above lemmata demonstrate that, for the appropriate values of $i$ and $j$
and for the appropriate $h_{i,j}$ in Equations (3.4.6) and (3.4.8),
$h_{i,j}\bullet\gamma^{-1}_{j+1,q+1-j+i}\leqslant\gamma^{-1}_{j,q+3-j+i}$ and
$h_{i,j}\bullet\gamma^{-1}_{q-j,j+1-i}\leqslant\gamma^{-1}_{q+2-j,j-i}$, which
proves the two closure relation statements. ∎

####  3.4.3. Example: $q=11$

We demonstrate the above theorems with the following example, where $q=11$.
Each node $(u,v)$ of the diagram in Figure 1 represents a coset representative
$\gamma_{u,v}$. We draw an arrow from $(u_{2},v_{2})$ to $(u_{1},v_{1})$ if
$\gamma_{u_{1},v_{1}}\operatorname{\preccurlyeq}\gamma_{u_{2},v_{2}}$. The
vertical position of a node is determined by the length of the coset
representative, descending from $\ell=18$ to $\ell=0$. Note that in the middle
of the diagram, there is an “equator” where the organization of nodes shifts
for ease of reading.

${{(10,11)}}$${{(9,11)}}$${{(8,11)}}$${{(9,10)}}$${{(7,11)}}$${{(8,10)}}$${{(6,11)}}$${{(7,10)}}$${{(8,9)}}$${{(5,11)}}$${{(6,10)}}$${{(7,9)}}$${{(4,11)}}$${{(5,10)}}$${{(6,9)}}$${{(7,8)}}$${{(3,11)}}$${{(4,10)}}$${{(5,9)}}$${{(6,8)}}$${{(2,11)}}$${{(3,10)}}$${{(4,9)}}$${{(5,8)}}$${{(6,7)}}$${{(1,11)}}$${{(2,10)}}$${{(3,9)}}$${{(4,8)}}$${{(5,7)}}$${{(1,10)}}$${{(2,9)}}$${{(3,8)}}$${{(4,7)}}$${{(5,6)}}$${{(1,9)}}$${{(2,8)}}$${{(3,7)}}$${{(4,6)}}$${{(1,8)}}$${{(2,7)}}$${{(3,6)}}$${{(4,5)}}$${{(1,7)}}$${{(2,6)}}$${{(3,5)}}$${{(1,6)}}$${{(2,5)}}$${{(3,4)}}$${{(1,5)}}$${{(2,4)}}$${{(1,4)}}$${{(2,3)}}$${{(1,3)}}$${{(1,2)}}$
Figure 1. Closure relations for $q=11$.

Observe that there are three types of relations shown, as follows:

  * •

Bruhat relations are represented by black, straight arrows, and represent
relations of the following form:
$\gamma_{u,v}\operatorname{\preccurlyeq}\gamma_{u+1,v}$ and
$\gamma_{u,v}\operatorname{\preccurlyeq}\gamma_{u,v+1}$. These are the
relations given in Proposition 3.5.

  * •

The orange squiggle arrows indicate closure relations of the form
$\gamma_{u+1,v-2}\operatorname{\preccurlyeq}\gamma_{u,v}$. These are above the
equator, and point down and to the right.

  * •

The purple squiggle arrows indicate closure relations of the form
$\gamma_{u-2,v+1}\operatorname{\preccurlyeq}\gamma_{u,v}$. These are below the
equator, and point down and to the left.

These last two relations are given in Theorems 3.9 and 3.15.

####  3.4.4. Concluding remarks on the closure order

Based on experiments done in Sage[14], we believe that the relations in
Theorems 3.9 and 3.15 are the only non-Bruhat closure relations between coset
representatives whose lengths differ by 1, and that all other closure
relations comes from a chain of such relations combined with Bruhat relations.
We summarize this in the following conjecture.

######  Conjecture 3.21.

Let $\gamma_{u_{1},v_{1}}$ and $\gamma_{u_{2},v_{2}}$ be elements of
$\mathbf{W}(q-2,2)$ satisfying
$\gamma_{u_{1},v_{1}}\operatorname{\preccurlyeq}\gamma_{u_{2},v_{2}}$. Then
either we have $\gamma_{u_{1},v_{1}}\leqslant\gamma_{u_{2},v_{2}}$ with
respect to the Bruhat order, or $\gamma_{u_{1},v_{1}}$ and
$\gamma_{u_{2},v_{2}}$ satisfy the closure relations in Theorem 3.9 or 3.15,
or $\gamma_{u_{1},v_{1}}$ and $\gamma_{u_{2},v_{2}}$ are related by a chain of
such relations.

We have checked these claims in Sage for $q\leqslant 11$, but it was difficult
to fully verify these statements for larger values of $q$. Up to $q=20$ we
confirmed that we do not get further relations from acting by a single simple
transposition $s_{k}$. Checking the action for all possible $h\in
W_{(q-2,2)}$, which has size $2!(q-2)!$, is computationally challenging. For
example, even for $q=13$ there are over 79 million possible $h$. Nevertheless,
our numerical experiments and our extensive work with the action in Equation
(3.4.3) lead us to believe that Conjecture 3.21 is true and that further work
in this area would be fruitful.

######  Remark 3.22.

There are $2\lceil q/2\rceil-5$ numbers $j$ that satisfy the conditions of
Theorem 3.9. Moreover, there are $(\lceil q/2\rceil-3)^{2}$ pairs $(i,j)$ that
satisfy the conditions of Theorem 3.15. Combined, this gives $(\lceil
q/2\rceil-2)^{2}$ closure relations that are not Bruhat relations. Conjecture
3.21 predicts that this is the total number of closure relations that are not
Bruhat relations.

##  4\. Product Maps: Relations to Other Unitary Shimura Varieties

In this section, we begin our study of the interaction between Ekedahl-Oort
strata and the supersingular locus by considering products of abelian
varieties. Note that when an abelian variety $A$ decomposes as a product of
abelian varieties, the $p$-torsion group scheme of $A$ also decomposes as a
product. Because of this, the natural ”product map” from a pair of unitary
Shimura varieties to our Shimura variety of interest, $\mathcal{M}(q-2,2)$,
will also induce a map in terms of the Ekedahl-Oort strata of these Shimura
varieties. In Theorems 4.10 and 4.16, we explicitly describe these induced
product maps on the index sets $\textbf{W}(a,b)$ for the Ekedahl-Oort
stratifications of the relevant Shimura varieties. Then, we enumerate some
Ekedahl-Oort strata that must intersect the supersingular locus of
$\mathcal{M}(q-2,2)$, using the product maps and the observation that when two
abelian varieties are supersingular, their product is also supersingular.

###  4.1. Background

For any $m_{1},m_{2},n_{1},n_{2}\in\mathbx{N}$, there is a natural _product
map_

$\displaystyle\Phi:\mathcal{M}(m_{1},n_{1})\times\mathcal{M}(m_{2},n_{2})$
$\displaystyle\to\mathcal{M}(m_{1}+m_{2},n_{1}+n_{2})$
$\displaystyle((A_{1},\lambda_{1},\iota_{1},\xi_{1}),(A_{2},\lambda_{2},\iota_{2},\xi_{2}))$
$\displaystyle\mapsto(A_{1}\times
A_{2},\lambda_{1}\times\lambda_{2},\iota_{1}\times\iota_{2},\xi_{1}\times\xi_{2}),$

where $\lambda_{1}\times\lambda_{2}$, $\iota_{1}\times\iota_{2}$, and
$\xi_{1}\times\xi_{2}$ are (respectively) the natural product polarization,
action, and level structure on the abelian variety $A_{1}\times A_{2}$.

Consider an abelian variety $A$ which decomposes as a product $A\cong
A_{1}\times A_{2}$, and note that the $p$-torsion group scheme satisfies
$A[p]\cong A_{1}[p]\times A_{2}[p]$. Replacing $A_{1}$ by an abelian variety
$B_{1}$ such that $A_{1}[p]\cong B_{1}[p]$ does not affect the Ekedahl-Oort
stratum of $A$.

As a result of this, given Ekedahl-Oort strata
$\mathcal{M}(m_{1},n_{1})_{\omega_{1}}$ and
$\mathcal{M}(m_{2},n_{2})_{\omega_{2}}$, there is a unique stratum
$\mathcal{M}(m_{1}+m_{2},n_{1}+n_{2})_{\omega}$ such that:

$\Phi(\mathcal{M}(m_{1},n_{1})_{\omega_{1}}\times\mathcal{M}(m_{2},n_{2})_{\omega_{2}})\subseteq\mathcal{M}(m_{1}+m_{2},n_{1}+n_{2})_{\omega}.$

In particular, the product map $\Phi$ induces a map $\varphi$ on the index
sets for the Ekedahl-Oort strata:

$\varphi:\mathbf{W}(m_{1},n_{1})\times\mathbf{W}(m_{2},n_{2})\rightarrow\mathbf{W}(m_{1}+m_{2},n_{1}+n_{2}),$

where $\varphi(\omega_{1},\omega_{2})$ is the index for the unique Ekedahl-
Oort stratum containing
$\Phi(\mathcal{M}(m_{1},n_{1})_{\omega_{1}}\times\mathcal{M}(m_{2},n_{2})_{\omega_{2}})$.

The goal of this section is to explicitly describe the product map on the
level of Weyl group cosets under the condition
$(m_{1}+m_{2},n_{1}+n_{2})=(q-2,2)$. This is done by first constructing
_standard objects_ of the strata $\mathcal{M}(m_{1},n_{1})_{\omega_{1}}$ and
$\mathcal{M}(m_{2},n_{2})_{\omega_{2}}$ and then computing the permutation
corresponding to the sum of these standard objects.

Without loss of generality, we assume $n_{2}\leqslant n_{1}$, so that either
$n_{1}=1$ or $n_{1}=2$.

The former case is treated in Section 4.2 and the latter in Section 4.3.

###  4.2. The $1\times 1$ Multiplication Map

####  4.2.1. General approach

In this section we study the product map

$\Phi:\mathcal{M}(m,1)\times\mathcal{M}(n,1)\rightarrow\mathcal{M}(m+n,2),$

under the condition $m+n=q-2$. We compute the induced map on Ekedahl-Oort
strata

$\varphi:\mathbf{W}(m,1)\times\mathbf{W}(n,1)\rightarrow\mathbf{W}(m+n,2).$

By [9, Theorem 6.7], the Ekedahl-Oort strata of $\mathcal{M}(m,1)$ are in
correspondence with cosets in the quotient
$(\mathfrak{S}_{1}\mathfrak{S}_{m})\setminus\mathfrak{S}_{m+1}.$ Each coset
has minimal-length representative given by a cyclic permutation

$\delta_{a}:=(1,2,\ldots,a+1),$

for $0\leqslant a\leqslant m$. We therefore have

$\mathbf{W}(m,1)=\\{\delta_{a}\;|\;0\leqslant a\leqslant m\\}.$

Note that the length of $\delta_{a}$ is $a$, and so the corresponding stratum
$\mathcal{M}(m,1)_{\delta_{a}}$ has dimension $a$.

In [15, Theorem F], the interaction between the Ekedahl-Oort stratification
and the Newton stratification of $\mathcal{M}(m,1)$ is completely described.
If $a\leqslant m/2$, then we have containment:

$\mathcal{M}(m,1)_{\delta_{a}}\subseteq\mathcal{M}(m,1)^{ss}.$

On the other hand, if $a>m/2$, then we have disjointedness:

$\mathcal{M}(m,1)_{\delta_{a}}\cap\mathcal{M}(m,1)^{ss}=\varnothing.$

In particular, from here on, given a Dieudonné module $M$ arising as the
standard object for an Ekedahl-Oort stratum $\mathcal{M}(m,1)_{\delta_{a}}$,
we say that $M$ is supersingular if $a\leqslant m/2$.

Fix $0\leqslant a\leqslant m$ and $0\leqslant b\leqslant n$. We aim to
describe $\varphi(\delta_{a},\delta_{b})\in\mathbf{W}(m+n,2)$. Recall the
bijection between $\mathbf{W}(a,b)$ and Ekedahl-Oort strata of
$\mathcal{M}(a,b)$ outlined below Theorem 2.2. Let the Dieudonné module $M$ be
the standard object in the Ekedahl-Oort stratum
$\mathcal{M}(m,1)_{\delta_{a}}$. Recall that the $\mathbx{F}_{p^{2}}$-action
induces a decomposition $M=M_{1}\oplus M_{2}$. We fix a final filtration
$W_{\bullet}$ of $M$, which induces filtrations $C_{i,\bullet}=W_{\bullet}\cap
M_{i}$ of $M_{i}$ for $i=1,2$. In our specific case, we have $(a,b)=(m,1)$ and
$\dim(M_{1}[F])=1$, so the function

$\eta^{M}_{1}(j)=\dim(C_{1,j}\cap M[F])$

jumps, i.e., increases by 1, at exactly one index. This index is $j_{1}=a+1$
by the bijection in Theorem 2.2.

Similarly, we let $\delta_{b}\in\mathbf{W}(n,1)$ represent an Ekedahl-Oort
stratum in $\mathcal{M}(n,1)$, with standard object $N$. Let $D_{i,\bullet}$
be filtrations of $N_{i}$ for $i=1,2$ coming from a final filtration of $N$.
Let $L:=M\oplus N$ and let $E_{i,\bullet}$ be the filtrations of $L_{i}$
coming from a final filtration of $L$. Our goal is to determine the Weyl group
coset corresponding to the Dieudonné module $L$ via Theorem 2.2. We do this by
computing the function

$\eta^{L}_{1}(j)=\dim(E_{1,j}\cap L[F]).$

As noted in Section 2.3, it is sufficient to compute $\eta_{1}$ in our work,
and so this determines the permutation
$\gamma_{u,v}=\varphi(\delta_{a},\delta_{b})$ representing the Ekedahl-Oort
stratum of $L$. The signature of $L$ is $(m+n,2)$, and so we have

$\eta^{L}_{1}(q)=\dim(L_{1}[F])=2.$

Since the function $\eta^{L}_{1}$ is clearly non-decreasing, it suffices to
find the two integers where $\eta^{L}_{1}$ jumps. Then this function
$\eta_{1}^{L}$ corresponds to a Weyl group coset $\gamma_{u,v}$. It can be
seen from the description of the standard object in Lemma 3.3 that $u$ and $v$
are precisely the two integers where $\eta_{1}^{L}$ jumps.

Depending on whether $M$ and $N$ are supersingular or not, there are slight
differences in our method of determining the two places where $\eta^{L}_{1}$
jumps. These cases are treated separately in Propositions 4.7, 4.8, and 4.9
(and summarized in Theorem 4.10). In Section 4.2.2, the necessary parts of the
canonical filtration of $L$ are constructed. In Section 4.2.3, this
information is used to determine the jumps of $\eta^{L}_{1}$, which in turn
yields the resulting Weyl group coset representative $\gamma_{u,v}$ of $L$,
such that

$\Phi\left(\mathcal{M}(m,1)_{\delta_{a}}\times\mathcal{M}(n,1)_{\delta_{b}}\right)\subseteq\mathcal{M}(m+n,2)_{\gamma_{u,v}}.$

####  4.2.2. The canonical filtration

Using the standard objects of $M$ and $N$, we form the canonical filtration of
$L$, which is crucial in computing the Ekedahl-Oort stratum. Let $M$ be the
standard object of the Ekedahl-Oort stratum $\mathcal{M}(m,1)_{\delta_{a}}$.
[9, 4.9] provides the action of $F$ and $V$ on a basis
$\\{e_{i,j}\;|\;1\leqslant i\leqslant 2\;,\;1\leqslant j\leqslant q\\}$ of
$M$.

$\displaystyle F(e_{1,j})=\begin{cases}e_{2,j}&\hbox{if $1\leqslant j\leqslant
a$}\\\ 0&\hbox{if $j=a+1$}\\\ e_{2,j-1}&\hbox{if $j>a+1$}\end{cases}$
$\displaystyle V(e_{1,j})$ $\displaystyle=\begin{cases}0&\hbox{if $j=1$}\\\
e_{2,j-1}&\hbox{if $1<j\leqslant m+1-a$}\\\ e_{2,j}&\hbox{if
$j>m+1-a$}\end{cases}$ $\displaystyle F(e_{2,j})=\begin{cases}0&\hbox{if
$j\neq m+1-a$}\\\ e_{1,1}&\hbox{if $j=m+1-a$}\end{cases}$ $\displaystyle
V(e_{2,j})$ $\displaystyle=\begin{cases}0&\hbox{if $j\neq m+1$}\\\
e_{1,a+1}&\hbox{if $j=m+1$}.\end{cases}$

The following lemma describes the action of $F$ and $V^{-1}$ on the subspaces
$\langle e_{i,l}\;|\;1\leqslant l\leqslant j\rangle$. Since applying $F$ or
$V^{-1}$ always gives another such subspace, we pick a final filtration
$W_{\bullet}$ of $M$ such that the filtrations $C_{i,\bullet}=W_{\bullet}\cap
M_{i}$ are given by

$C_{i,j}=\langle e_{i,l}\;|\;1\leqslant l\leqslant j\rangle.$

######  Lemma 4.1.

$F$ and $V^{-1}$ have the following actions on the spaces $C_{i,j}$:

$\displaystyle F(C_{1,j})=\begin{cases}C_{2,j}&\hbox{if $j\leqslant a$}\\\
C_{2,j-1}&\hbox{if $j>a$}\end{cases}$ $\displaystyle V^{-1}(C_{1,j})\cap
M_{2}$ $\displaystyle=\begin{cases}C_{2,m}&\hbox{if $j\leqslant a$}\\\
C_{2,m}&\hbox{if $j>a$}\end{cases}$ $\displaystyle
F(C_{2,j})=\begin{cases}C_{1,0}&\hbox{if $j<m+1-a$}\\\ C_{1,1}&\hbox{if
$j\geqslant m+1-a$}\end{cases}$ $\displaystyle V^{-1}(C_{2,j})\cap M_{1}$
$\displaystyle=\begin{cases}C_{1,j+1}&\hbox{if $j<m+1-a$}\\\ C_{1,j}&\hbox{if
$j\geqslant m+1-a$}.\end{cases}$

###### Proof.

This is a direct application of the description, given above, of the standard
object $M$. ∎

The same result applies to the standard object $N$ of the Ekedahl-Oort stratum
$\mathcal{M}(n,1)_{\delta_{b}}$, upon replacing $a$ by $b$ and $m$ by $n$.

For the remainder of this section, let $s_{1}:=\min\\{a+1,m+1-a\\}$ and
$s_{2}:=\max\\{a,m+1-a\\}$.

######  Lemma 4.2.

For $1\leqslant j\leqslant q$,

$V^{-1}(F(C_{1,j}))\cap M_{1}=\begin{cases}C_{1,j+1}&j<s_{1}\\\\[5.0pt]
C_{1,j}&j=s_{1}\\\\[5.0pt] C_{1,j-1}&j>s_{2}\\\\[5.0pt]
C_{1,j}&j=s_{2}\end{cases}$

###### Proof.

Suppose $j<s_{1}=\min\\{a+1,m+1-a\\}$. Using Lemma 4.1, we obtain
$F(C_{1,j})=C_{2,j}$ and $V^{-1}(C_{2,j})\cap M_{1}=C_{1,j+1}$ as desired.

We check the second assertion for two cases of $s_{1}$. In the case
$s_{1}=a+1\leqslant m+1-a$ we get $F(C_{1,a+1})=C_{2,a}$ and
$V^{-1}(C_{2,a})\cap M_{1}=C_{1,a+1}$. In the case $s_{1}=m+1-a<a+1$, we get
$F(C_{1,m+1-a})=C_{2,m+1-a}$ and $V^{-1}(C_{2,m+1-a})\cap M_{1}=C_{1,m+1-a}$.

We now treat the analogue with decreasing index: assume
$j>s_{2}=\max\\{a,m+1-a\\}$. In this case, we have $F(C_{1,j})=C_{2,j-1}$ and,
subsequently, $V^{-1}(C_{2,j-1})\cap M_{1}=C_{1,j-1}$.

Finally, we prove the last assertion for both values of $s_{2}$. In the case
$s_{2}=a\geqslant m+1-a$, we get $F(C_{1,a})=C_{2,a}$ and $V^{-1}(C_{2,a})\cap
M_{1}=C_{1,a}$. In the case $s_{2}=m+1-a>a$ we obtain
$F(C_{1,m+1-a})=C_{2,m-a}$ and $V^{-1}(C_{2,m-a})\cap M_{1}=C_{1,m+1-a}$, as
desired. ∎

We now prove a corollary that allows us to ‘move up’ from $C_{1,0}$ to
$C_{1,s_{1}}$, step by step.

######  Corollary 4.3.

For $c\in\mathbx{Z}_{\geqslant 0}$, we have

$(V^{-1}F)^{c}(0)\cap M_{1}=C_{1,\min\\{c,s_{1}\\}}.$

###### Proof.

Apply Lemma 4.2 for $0\leqslant j\leqslant s_{1}$ repeatedly, so that the
result follows by induction starting at $j=0$. ∎

Similarly we can ‘move down’ from $C_{1,m+1}$ to $C_{1,s_{2}}$, as the
following corollary records.

######  Corollary 4.4.

For $c\in\mathbx{Z}_{\geqslant 0}$, we have

$(V^{-1}F)^{c}(M)\cap M_{1}=C_{1,\max\\{m+1-c,s_{2}\\}}.$

###### Proof.

Apply Lemma 4.2 for $s_{2}\leqslant l\leqslant m+1$ repeatedly, starting at
$l=m+1$ and proceeding by decreasing induction. ∎

In the rest of this section, we use these corollaries to construct the crucial
parts of the filtration $E_{1,\bullet}$ of $L_{1}=M_{1}\oplus N_{1}$, induced
by intersecting a final filtration of $L$ with $L_{1}$. Let $s_{1}$ and
$s_{2}$ be defined as above and let $u_{1}:=\min\\{b+1,n+1-b\\}$ and
$u_{2}:=\max\\{b,n+1-b\\}$.

######  Lemma 4.5.

For $c\in\mathbx{Z}_{\geqslant 0}$, we have

$\displaystyle(V^{-1}F)^{c}(0)\cap L_{1}$
$\displaystyle=C_{1,\min\\{c,s_{1}\\}}\oplus
D_{1,\min\\{c,u_{1}\\}}=E_{1,\min\\{c,s_{1}\\}+\min\\{c,u_{1}\\}}$
$\displaystyle(V^{-1}F)^{c}(L)\cap L_{1}$
$\displaystyle=C_{1,\max\\{m+1-c,s_{2}\\}}\oplus
D_{1,\max\\{n+1-c,u_{2}\\}}=E_{1,\max\\{m+1-c,s_{2}\\}+\max\\{n+1-c,u_{2}\\}}.$

###### Proof.

This is a straightforward application of Corollary 4.3 and Corollary 4.4. For
the first equality, note

$(V^{-1}F)^{c}(0)\cap L_{1}=\left((V^{-1}F)^{c}(0)\cap
M_{1}\right)\oplus\left((V^{-1}F)^{c}(0)\cap N_{1}\right).$

Both summands are computed in Corollary 4.3. Similarly, for the second
equality observe

$(V^{-1}F)^{c}(L)\cap L_{1}=\left((V^{-1}F)^{c}(M)\cap
M_{1}\right)\oplus\left((V^{-1}F)^{c}(N)\cap N_{1}\right).$

Both summands are computed in Corollary 4.4. ∎

In some cases, Lemma 4.5 determines every value of
$\eta^{L}_{1}(j)=\dim(E_{1,j}\cap L[F])$, providing all the information we
need. In general, we need to construct more subspaces of $L_{1}$. For that, we
turn to the following lemma.

######  Lemma 4.6.

Suppose one of the following conditions holds:

  1. (i)

$M$ and $N$ are both supersingular and $a>b$.

  2. (ii)

$M$ and $N$ are both not supersingular and $m-a<n-b$.

  3. (iii)

$M$ is not supersingular and $N$ is supersingular.

Then we have that

$E_{1,1}=C_{1,1}\oplus D_{1,0}\;\;\;\text{and}\;\;\;E_{1,q-1}=C_{1,m}\oplus
D_{1,n+1}.$

###### Proof.

We claim that, in all three cases, it suffices to construct a subspace of the
form $C_{2,l}\oplus D_{2,n-b}$, for some $l>m-a$, in the filtration
$E_{2,\bullet}$. We then show that such a subspace can be constructed in all
three cases.

Assume that $C_{2,l}\oplus D_{2,n-b}$ has been constructed with $l>m-a$. Then
Lemma 4.1 yields

$F\left(C_{2,l}\oplus D_{2,n-b}\right)=C_{1,1}\oplus D_{1,0}=E_{1,1},$

proving the claim. For the second statement, we first construct $E_{2,q-1}$
using symplectic complements. Note that $M$ and $N$ are each equipped with a
symplectic pairing coming from the polarization of the abelian varieties, and
so $L$ is equipped with the product of these two pairings. By the Rosati
involution condition, we have $M_{1}^{\perp}=M_{1}$. Furthermore, we may
assume that the final filtration $W_{\bullet}$ of $M$ is stable under taking
symplectic complements. Therefore, for any $1\leqslant j\leqslant q$ we have
$C_{1,j}^{\perp}=M_{1}\oplus C_{2,q-j}$ and likewise for $N$. This implies

$E^{\perp}_{1,1}\cap L_{2}=(C_{1,1}^{\perp}\cap
M_{2})\oplus(D_{1,0}^{\perp}\cap N_{2})=C_{2,m}\oplus D_{2,n+1}=E_{2,q-1}.$

All three cases imply $a>0$, and so applying Lemma 4.1 yields

$V^{-1}(E_{2,q-1})=C_{1,m}\oplus D_{1,n+1}=E_{1,q-1},$

which finishes the proof of the second statement.

We now construct $C_{2,l}\oplus D_{2,n-b}$ in each individual case. In case
(i), Lemma 4.5 allows us to move down to

$(V^{-1}F)^{b}(L)\cap L_{1}=C_{1,m+1-b}\oplus D_{1,n+1-b}=E_{1,q-2b}.$

Here we have used $m+1-b>m+1-a$, since $a>b$. Furthermore, $m+1-a\geqslant a$
since $M$ is supersingular, and $n+1-b\geqslant b$ since $N$ is supersingular.
Together these observations imply $m+1-b\geqslant s_{2}$ and $n+1-b\geqslant
u_{2}$. We then construct

$F(C_{1,m+1-b}\oplus D_{1,n+1-b})=C_{2,m-b}\oplus D_{2,n-b}.$

Since $m-b>m-a$, we are done.

In case (ii), Lemma 4.5 allows us to move up to

$(V^{-1}F)^{n-b}(0)\cap L_{1}=C_{1,m+1-a}\oplus D_{1,n-b}=E_{1,m+n+1-a-b}.$

where we use: $s_{1}=m+1-a$ and $m-a<n-b$. Applying $F$, using $m+1-a\leqslant
a$ and $n-b\leqslant b$, yields

$F\left(C_{1,m+1-a}\oplus D_{1,n-b}\right)=C_{2,m+1-a}\oplus D_{2,n-b}.$

This subspace satisfies our requirements, as $m+1-a>m-a$.

Finally, in case (iii), Lemma 4.5 allows us to move down to

$C_{1,s_{2}}\oplus D_{1,u_{2}}=C_{1,a}\oplus D_{1,n+1-b}.$

Applying $F$ yields

$F(C_{1,a}\oplus D_{1,n+1-b})=C_{2,a}\oplus D_{2,n-b},$

which satisfies our requirement since $M$ is not supersingular, implying
$a>m-a$. ∎

After constructing $E_{1,1}$ and $E_{1,q-1}$ if necessary, under the
conditions of Lemma 4.6, we use the proof of Lemma 4.5 to construct

(4.2.1) $\displaystyle(V^{-1}F)^{c}(E_{1,1})\cap L_{1}$
$\displaystyle=C_{1,\min\\{1+c,s_{1}\\}}\oplus
D_{1,\min\\{c,u_{1}\\}}=E_{1,\min\\{1+c,s_{1}\\}+\min\\{c,u_{1}\\}},$ (4.2.2)
$\displaystyle(V^{-1}F)^{c}(E_{1,q-1})\cap L_{1}$
$\displaystyle=C_{1,\max\\{m-c,s_{2}\\}}\oplus
D_{1,\max\\{n+1-c,u_{2}\\}}=E_{1,\max\\{m-c,s_{2}\\}+\max\\{n+1-c,u_{2}\\}}.$

The information that these lemmata provide about the canonical filtration of
$L$ is sufficient to compute its Ekedahl-Oort stratum.

####  4.2.3. Ekedahl-Oort strata

We now have the tools to completely describe the product map on the level of
Weyl group cosets

$\varphi:\mathbf{W}(m,1)\times\mathbf{W}(n,1)\rightarrow\mathbf{W}(m+n,2)$

In particular, we compute $\varphi(\delta_{a},\delta_{b})$, where
$\mathcal{M}(m,1)_{\delta_{a}}$ is the unique Ekedahl-Oort stratum of
dimension $a$ and likewise for $\delta_{b}$. Let $M$ be a standard object for
the stratum $\delta_{a}$ and let $N$ be a standard object for the stratum
$\delta_{b}$. All possibilities for $M$ and $N$ (up to exchanging $M$ and $N$)
are treated in this section. Recall that $M$ is supersingular if and only if
$a\leqslant m/2$ or, equivalently, $s_{1}=a+1$. $N$ is supersingular if and
only if $b\leqslant n/2$ or, equivalently, $u_{1}=b+1$.

We first treat the case when $M$ and $N$ are both supersingular in full
detail. Exchanging $M$ and $N$ if necessary, we assume $a\geqslant b$.

######  Proposition 4.7.

Assume $M$ and $N$ are both supersingular, with $a\geqslant b$. Then the
Ekedahl-Oort stratum of $L$ is represented by the Weyl group coset of
$\gamma_{2b+1,2b+2}$ if $a=b$ and $\gamma_{2b+2,a+b+2}$ if $a>b$.

###### Proof.

We compute the two places $j_{1}$ and $j_{2}$ where $\eta^{L}_{1}$ jumps.
Using Lemma 4.5 we ‘move up’ to construct the following parts of the canonical
filtration:

$\displaystyle(V^{-1}F)^{b}(0)\cap L_{1}$ $\displaystyle=C_{1,b}\oplus
D_{1,b}=E_{1,2b},$ $\displaystyle(V^{-1}F)^{b+1}(0)\cap L_{1}$
$\displaystyle=C_{1,b+1}\oplus D_{1,b+1}=E_{1,2b+2},$
$\displaystyle(V^{-1}F)^{a}(0)\cap L_{1}$ $\displaystyle=C_{1,a}\oplus
D_{1,b+1}=E_{1,a+b+1}\hskip 28.45274pt\hbox{in the case $a>b$,}$
$\displaystyle(V^{-1}F)^{a+1}(0)\cap L_{1}$ $\displaystyle=C_{1,a+1}\oplus
D_{1,b+1}=E_{1,a+b+2}.$

We use the explicit description $M_{1}[F]=\langle e_{1,a+1}\rangle$ (see Lemma
4.1), and similarly for $N_{1}[F]$. Thus, we compute the following values of
$\eta^{L}_{1}$:

$\displaystyle\eta^{L}_{1}(2b)$ $\displaystyle=0,$
$\displaystyle\eta^{L}_{1}(2b+2)$ $\displaystyle=\begin{cases}1&\hbox{if
$a>b$,}\\\ 2&\hbox{if $a=b$,}\end{cases}$ $\displaystyle\eta^{L}_{1}(a+b+1)$
$\displaystyle=1,$ $\displaystyle\eta^{L}_{1}(a+b+2)$ $\displaystyle=2.$

Hence $\eta^{L}_{1}$ jumps at $j_{2}=a+b+2$. In the case $a=b$, it follows
that $\eta^{L}_{1}$ must also jump at $j_{1}=2b+1$, since
$\eta^{L}_{1}(2b)=0$.

When $a>b$, it is not yet determined whether the first jump occurs at $2b+1$
or $2b+2$ and more work is required. In this case, Lemma 4.6 and Equation
(4.2.1) imply $E_{1,1}=C_{1,1}\oplus D_{1,0},$ and so

$\displaystyle(V^{-1}F)^{b}(E_{1,1})$
$\displaystyle=C_{1,\min\\{b+1,a+1,m+1-a\\}}\oplus
D_{1,\min\\{b,a+1,m+1-a\\}}$ $\displaystyle=C_{1,b+1}\oplus
D_{1,b}=E_{1,2b+1}.$

Finally, using the explicit description of $L_{1}[F]$, we infer
$\eta^{L}_{1}(2b+1)=0$ and hence $\eta^{L}_{1}$ jumps at $j_{1}=2b+2$.

Finally, we prove the statement about the Weyl group coset representative
$\gamma_{u,v}$. Recall that $\omega=\gamma_{u,v}$ is determined by the jumps
of $\eta_{L}^{1}$ via Equation (2.3.2). Comparing this with Equation (3.1.1)
defining $\gamma_{u,v}$ yields $u=j_{1}$ and $v=j_{2}$, completing the proof.
∎

We now treat the case when $M$ and $N$ are both not supersingular, which is
essentially the mirror image of the preceding case. Exchanging $M$ and $N$ if
necessary, assume that $m-a\leqslant n-b$.

######  Proposition 4.8.

Assume $M$ and $N$ are both not supersingular, with $m-a\leqslant n-b$. Then
the Ekedahl-Oort stratum of $L$ is represented by the Weyl group coset of
$\gamma_{a+b+1,2a+n-m+2}$.

###### Proof.

Using the ‘moving down’ of Lemma 4.5, we compute

$\displaystyle(V^{-1}F)^{m-a}(L)\cap L_{1}$ $\displaystyle=C_{1,a+1}\oplus
D_{1,n+1-m+a}=E_{1,2a+n-m+2},$ $\displaystyle(V^{-1}F)^{m-a+1}(L)\cap L_{1}$
$\displaystyle=C_{1,a}\oplus D_{1,n-m+a}=E_{1,2a+n-m},$
$\displaystyle(V^{-1}F)^{n-b}(L)\cap L_{1}$ $\displaystyle=C_{1,a}\oplus
D_{1,b+1}=E_{1,a+b+1}\hskip 28.45274pt\hbox{in the case $m-a<n-b$,}$
$\displaystyle(V^{-1}F)^{n-b+1}(L)\cap L_{1}$ $\displaystyle=C_{1,a}\oplus
D_{1,b}=E_{1,a+b}.$

This yields the values

$\displaystyle\eta^{L}_{1}(a+b)$ $\displaystyle=0,$
$\displaystyle\eta^{L}_{1}(a+b+1)$ $\displaystyle=1,$
$\displaystyle\eta^{L}_{1}(2a+n-m)$ $\displaystyle=1,$
$\displaystyle\eta^{L}_{1}(2a+n-m+2)$ $\displaystyle=2.$

This implies that $\eta^{L}_{1}$ jumps at $j_{1}=a+b+1$. In the case
$m-a=n-b$, the second jump is at $j_{2}=2a+n-m+2=a+b+2.$

When $m-a<n-b$, Lemma 4.6 and Equation (4.2.2) imply $E_{1,q-1}=C_{1,m}\oplus
D_{1,n+1}$, and so

$\displaystyle(V^{-1}F)^{m-a}(E_{1,q-1})$ $\displaystyle=C_{1,a}\oplus
D_{1,n+1-m+a}=E_{1,2a+n-m+1}.$

Using the explicit description of $L_{1}[F]$, it follows that
$\eta^{L}_{1}(2a+n-m+1)=1$. Hence in that case we also have $j_{2}=2a+n-m+2$.

The statement about Weyl group coset representatives follows from the formulae
$u=j_{1}$ and $v=j_{2}$. ∎

Finally, we treat the case when exactly one of the summands $M$ and $N$ is
supersingular. Exchanging $M$ and $N$ if necessary, we assume that $M$ is not
supersingular and $N$ is supersingular.

######  Proposition 4.9.

Assume $M$ is not supersingular and $N$ is supersingular. Then the Ekedahl-
Oort stratum of $L$ is represented by the Weyl group coset of
$\gamma_{b+2+\min\\{b,m-a\\},a+2+\max\\{n-m+a,n-b\\}}$.

###### Proof.

Using Lemma 4.5, Lemma 4.6, Equation (4.2.1) and Equqtion (4.2.2), we compute

$\displaystyle(V^{-1}F)^{b}(E_{1,1})\cap L_{1}$
$\displaystyle=C_{1,\min\\{b+1,m+1-a\\}}\oplus
D_{1,b}=E_{1,b+1+\min\\{b,m-a\\}},$ $\displaystyle(V^{-1}F)^{b+1}(0)\cap
L_{1}$ $\displaystyle=C_{1,\min\\{b+1,m+1-a\\}}\oplus
D_{1,b+1}=E_{1,b+2+\min\\{b,m-a\\}},$
$\displaystyle(V^{-1}F)^{m-a}(E_{1,q-1})\cap L_{1}$
$\displaystyle=C_{1,a}\oplus
D_{1,a+1+\max\\{n-m+a,n-b\\}}=E_{1,a+1+\max\\{n-m+a,n-b\\}},$
$\displaystyle(V^{-1}F)^{m-a}(L)\cap L_{1}$ $\displaystyle=C_{1,a+1}\oplus
D_{1,\max\\{n+1-m+a,n+1-b\\}}=E_{1,a+2+\max\\{n-m+a,n-b\\}}.$

Using the explicit description of $L_{1}[F]$, we infer that the jumps of
$\eta^{L}_{1}$ are

$\displaystyle u$ $\displaystyle=j_{1}=b+2+\min\\{b,m-a\\},\text{ and}$
$\displaystyle v$ $\displaystyle=j_{2}=a+2+\max\\{n-m+a,n-b\\}.\qed$

Proposition 4.7, Proposition 4.8 and Proposition 4.9 are summarized in Theorem
4.10.

######  Theorem 4.10.

The Ekedahl-Oort stratum of $L=M\oplus N$ is given as follows.

  1. (A)

If $M$ and $N$ are both supersingular, with $a\geqslant b$, then the Ekedahl-
Oort stratum of $L$ is represented by the Weyl group coset of
$\gamma_{2b+1,2b+2}$ if $a=b$ and $\gamma_{2b+2,a+b+2}$ if $a>b$.

  2. (B)

If $M$ and $N$ are both not supersingular, with $m-a\leqslant n-b$, then the
Ekedahl-Oort stratum of $L$ is represented by the Weyl group coset of
$\gamma_{a+b+1,2a+n-m+2}$.

  3. (C)

If $M$ is not supersingular and $N$ is supersingular. Then the Ekedahl-Oort
stratum of $L$ is represented by the Weyl group coset of
$\gamma_{b+2+\min\\{b,m-a\\},a+2+\max\\{n-m+a,n-b\\}}$.

Theorem 4.10 has the following implication on the supersingular locus
$\mathcal{M}(q-2,2)^{ss}$.

######  Corollary 4.11.

Let $m,n,a,b$ be integers satisfying:

$m+n=q-2,\quad a\leqslant m/2,\quad b\leqslant n/2,\quad\text{ and }\quad
a\geqslant b.$

Then the Ekedahl-Oort stratum

$\gamma_{u,v}=\begin{cases}\gamma_{2b+1,2b+2}&\hbox{if $a=b$}\\\
\gamma_{2b+2,a+b+2}&\hbox{if $a>b$}\end{cases}$

intersects $\mathcal{M}(q-2,2)^{ss}$.

###### Proof.

A point $(A,\lambda,\iota,\xi)$ in the intersection
$\mathcal{M}(q-2,2)_{\gamma_{u,v}}\cap\mathcal{M}(q-2,2)^{ss}$ is constructed
as the product

$(A,\lambda,\iota,\xi)=(A_{1}\times
A_{2},\lambda_{1}\times\lambda_{2},\iota_{1}\times\iota_{2},\xi_{1}\times\xi_{2}),$

where$(A_{1},\lambda_{1},\iota_{1},\xi_{1})\in\mathcal{M}(m,1)_{\delta_{a}}\text{
and }(A_{2},\lambda_{2},\iota_{2},\xi_{2})\in\mathcal{M}(n,1)_{\delta_{b}}.$
It follows from Theorem 4.10 (A) that $(A,\lambda,\iota,\xi)$ lies in the
Ekedahl-Oort stratum $\mathcal{M}(q-2,2)_{\gamma_{u,v}}$. Moreover,
$A=A_{1}\times A_{2}$ is supersingular, as $A_{1}$ and $A_{2}$ are
supersingular by the conditions $a\leqslant m/2$ and $b\leqslant n/2$
respectively. ∎

###  4.3. The $2\times 0$ Multiplication Map

####  4.3.1. General approach

We now study the product map

$\Phi:\mathcal{M}(m,2)\times\mathcal{M}(n,0)\to\mathcal{M}(q-2,2),$

where $m+n=q-2$. Let $\gamma_{u,v}$ represent an Ekedahl-Oort stratum of
$\mathcal{M}(m,2)$, with standard object $M$. It is well known that the
Shimura variety $\mathcal{M}(n,0)$ consists only of the superspecial Ekedahl-
Oort stratum $\mathcal{M}(n,0)_{\text{id}}=\mathcal{M}(n,0)$ which is
characterized by the property $F^{2}=0$. The standard object of that stratum
is $N^{n}=\bigoplus_{l=1}^{n}N$, where $N$ is the mod-$p$ Dieudonné module
arising from a supersingular elliptic curve that has an action of
$\mathcal{O}_{K}$ with signature $(1,0)$. Hence,
$\mathcal{M}(n,0)=\mathcal{M}(n,0)^{ss}$. Through a study of the sum
$L:=M\oplus N^{n}$, we compute the permutation
$\gamma_{y,s}:=\varphi(\gamma_{u,v},\text{id})$, such that

$\Phi(\mathcal{M}(m,2)_{\gamma_{u,v}}\;,\mathcal{M}(n,0)_{\text{id}})\subseteq\mathcal{M}(q-2,2)_{\gamma_{y,s}}.$

As a result, we obtain information about whether
$\mathcal{M}(q-2,2)_{\gamma_{y,s}}$ intersects the supersingular locus. If
$\mathcal{M}(m,2)_{\gamma_{u,v}}$ intersects the supersingular locus, then so
does $\mathcal{M}(q-2,2)_{\gamma_{y,s}}$. Conversely, if
$\mathcal{M}(m,2)_{\gamma_{u,v}}$ is not contained in the supersingular locus,
then neither is $\mathcal{M}(q-2,2)_{\gamma_{u,v}}$. While we do not always
know how $\mathcal{M}(m,2)_{\gamma_{u,v}}$ interacts with the supersingular
locus, our methods allow us to draw conclusions in some cases (see, for
example, Table LABEL:table-classification).

Our method for computing $\varphi(\gamma_{u,v},\text{id})$ is essentially the
same as the method used in Section 4.2. We describe the standard object $M$
corresponding to $\mathcal{M}(m,2)_{\gamma_{u,v}}$ and then compute the
permutation corresponding to $L=M\oplus N^{n}$. In this context, set
$q_{M}:=m+2$ to be the dimension of $M$. As before, let $C_{i,\bullet}$ be
filtrations of $M_{i}$ coming from a final filtration of $M$. In the same way,
define the filtrations $D_{i,\bullet}$ of $N_{i}$ and $E_{i,\bullet}$ of
$L_{i}$.

As before, define $\eta^{L}_{1}(j):=\dim(E_{1,j}\cap L[F])$, and in this
setting $L_{1}[F]=M_{1}[F]=\langle e_{1,u},e_{1,v}\rangle$. It suffices to
pinpoint the two places where $\eta^{L}_{1}$ increases, as this determines the
permutation $\gamma_{y,s}$ representing the Ekedahl-Oort stratum of $L$ by
Equation (2.3.2).

####  4.3.2. The canonical filtration

We construct parts of the canonical filtration of $L$ by analysing the action
of $F$ and $V^{-1}$ on both $M$ and $N$. First, $F$ and $V$ act on $N$ as
follows:

(4.3.1) $F(N_{1})=V(N_{1})=N_{2}\text{ and }F(N_{2})=V(N_{2})=0.$

The following lemma gives an analogous description for the action of $F$ and
$V^{-1}$ on $M$.

######  Lemma 4.12.

$F$ and $V^{-1}$ have the following actions on the spaces $C_{i,j}$:

$\displaystyle F(C_{1,j})=\begin{cases}C_{2,j}&\hbox{if $j<u$}\\\
C_{2,j-1}&\hbox{if $u\leqslant j<v$}\\\ C_{2,j-2}&\hbox{if $j\geqslant
v$}\end{cases}$ $\displaystyle V^{-1}(C_{1,j})\cap M_{2}$
$\displaystyle=\begin{cases}C_{2,q_{M}-2}&\hbox{if $j<u$}\\\
C_{2,q_{M}-1}&\hbox{if $u\leqslant j<v$}\\\ C_{2,q_{M}}&\hbox{if $j\geqslant
v$}\end{cases}$ $\displaystyle F(C_{2,j})=\begin{cases}C_{1,0}&\hbox{if
$j\leqslant q_{M}-v$}\\\ C_{1,1}&\hbox{if $q_{M}-v<j\leqslant q_{M}-u$}\\\
C_{1,2}&\hbox{if $j>q_{M}-u$}\end{cases}$ $\displaystyle V^{-1}(C_{2,j})\cap
M_{1}$ $\displaystyle=\begin{cases}C_{1,j+2}&\hbox{if $j\leqslant q_{M}-v$}\\\
C_{1,j+1}&\hbox{if $q_{M}-v<j\leqslant q_{M}-u$}\\\ C_{1,j}&\hbox{if
$j>q_{M}-u$}.\end{cases}$

###### Proof.

The result follows from applying Lemma 3.3. ∎

A variant of ‘moving up’ can be used to construct parts of the canonical
filtration of $L$. To record this, define the multiset
$S_{\textup{up}}=\\{u,v,q_{M}-v+2,q_{M}-u+1\\}$ and let
$r_{2}:=\min(S_{\textup{up}}\setminus\left\\{\min S_{\textup{up}}\right\\})$,
i.e., the second element from the multiset when the elements are listed in
increasing order.

######  Lemma 4.13.

For a sufficiently large integer $c$, we have that $(V^{-1}F)^{c}(0)\cap
L_{1}=C_{1,r_{2}}\oplus D_{1,0}^{n}.$

###### Proof.

We compute both summands of

(4.3.2) $(V^{-1}F)^{c}(0)\cap L_{1}=((V^{-1}F)^{c}(C_{1,0})\cap
M_{1})\oplus((V^{-1}F)^{c}(D_{1,0})\cap N_{1})^{n}.$

First note that $F(D_{1,0})=D_{2,0}$ and $V^{-1}(D_{2,0})\cap N_{1}=D_{1,0}$
by Equation (4.3.1). Hence

$V^{-1}(F(D_{1,0}))\cap N_{1}=D_{1,0}$

which implies the second summand of Equation (4.3.2). We use Lemma 4.12 to
compute the first summand.

First, we assume $j<r_{2}$ and show

(4.3.3) $V^{-1}(F(C_{1,j}))\cap M_{1}\supseteq C_{1,j+1}.$

We begin by computing

$F(C_{1,j})=\begin{cases}C_{2,j}&\hbox{if $j<u$}\\\ C_{2,j-1}&\hbox{if
$u\leqslant j<v$}.\end{cases}$

In the case $j<u$, one verifies

$V^{-1}(F(C_{1,j}))\cap M_{1}=V^{-1}(C_{2,j})\cap
M_{1}=\begin{cases}C_{1,j+2}&\hbox{if $j\leqslant q_{M}-v$},\\\
C_{1,j+1}&\hbox{if $j>q_{M}-v$},\end{cases}$

so that Equation (4.3.3) is satisfied. In the case $j\geqslant u$, it follows
from the definition of $r_{2}$ that $j\leqslant q_{M}-v+2$, and therefore

$V^{-1}(F(C_{1,j}))\cap M_{1}=V^{-1}(C_{2,j-1})\cap M_{1}=C_{1,j+1},$

by Lemma 4.12 and, again, Equation (4.3.3) is satisfied. Thus applying
$V^{-1}F$ increases the index in $C_{1,\bullet}$ for $j<r_{2}$. To prove it
does not increase further, we show

(4.3.4) $V^{-1}(F(C_{1,r_{2}}))\cap M_{1}=C_{1,r_{2}}.$

First, Lemma 4.12 yields

$F(C_{1,r_{2}})=\begin{cases}C_{2,r_{2}}&\hbox{if $r_{2}<u$},\\\
C_{2,r_{2}-1}&\hbox{if $u\leqslant r_{2}<v$},\\\ C_{2,r_{2}-2}&\hbox{if
$r_{2}=v$}.\end{cases}$

In the first case we have $r_{2}<u<v$ and it follows that $r_{2}=q_{M}-u+1$.
Hence

$V^{-1}(F(C_{1,r_{2}}))\cap M_{1}=V^{-1}(C_{2,r_{2}})\cap M_{1}=C_{1,r_{2}}.$

In the second case we have $u\leqslant r_{2}<v$ and it follows that
$q_{M}-v+2\leqslant r_{2}\leqslant q_{M}-u+1$. Therefore

$V^{-1}(F(C_{1,r_{2}}))\cap M_{1}=V^{-1}(C_{2,r_{2}-1})\cap
M_{1}=C_{1,r_{2}}.$

Finally, in the third case we have $r_{2}=v$ and it follows that
$r_{2}\leqslant q_{M}-v+2$. We obtain

$V^{-1}(F(C_{1,r_{2}}))\cap M_{1}=V^{-1}(C_{2,r_{2}-2})\cap
M_{1}=C_{1,r_{2}}.$

We conclude that applying $V^{-1}F$ allows us to ’move up’ along
$C_{1,\bullet}$, precisely until we reach $C_{1,r_{2}}$. This proves the
lemma. ∎

Analogously, we record the effect of ‘moving down’. Define the multiset
$S_{\text{down}}:=\\{u-1,v-1,q_{M}-v+2,q_{M}-u+1\\}$ and let
$r_{3}:=\max(S_{\textup{down}}\setminus\left\\{\max
S_{\textup{down}}\right\\})$, i.e., the second element from the multiset when
the elements are listed in increasing order.

######  Lemma 4.14.

For a sufficiently large integer $c$, we have $(V^{-1}F)^{c}(L)\cap
L_{1}=C_{1,r_{3}}\oplus D_{1,1}^{n}$.

###### Proof.

The proof is similar to the proof of Lemma 4.13. We compute both summands of

(4.3.5) $(V^{-1}F)^{c}(L)\cap L_{1}=((V^{-1}F)^{c}(C_{1,q_{M}})\cap
M_{1})\oplus((V^{-1}F)^{c}(D_{1,1})\cap N_{1})^{n}.$

First, using Equation (4.3.1) we obtain $V^{-1}(F(D_{1,1}))\cap N_{1}=D_{1,1}$
for the second summand of Equation (4.3.5). We now compute the first summand
using Lemma 4.12. Assume $l>r_{3}$, so that $l$ exceeds at least three of the
elements of $S_{\text{down}}$. We show

(4.3.6) $V^{-1}(F(C_{1,l}))\cap M_{1}\subseteq C_{1,l-1}.$

First, applying $F$ yields

$F(C_{1,l})=\begin{cases}C_{2,l-1}&\hbox{if $l<v$}\\\ C_{2,l-2}&\hbox{if
$l\geqslant v$}.\end{cases}$

In the first case, it follows that $l>q_{M}-u+1$ and hence

$V^{-1}(F(C_{1,l}))\cap M_{1}=V^{-1}(C_{2,l-1})\cap M_{1}=C_{1,l-1}.$

In the second case, we use $l>q_{M}-v+2$ to obtain

$V^{-1}(F(C_{1,l}))\cap M_{1}=V^{-1}(C_{2,l-2})\cap M_{1}\subseteq C_{1,l-1}.$

In both cases Equation (4.3.6) is satisfied. Thus applying $V^{-1}F$ decreases
the index in $C_{1,\bullet}$ for $l>r_{3}$. We show it cannot decrease
further, meaning

(4.3.7) $V^{-1}(F(C_{1,r_{3}}))\cap M_{1}=C_{1,r_{3}}.$

We begin by applying $F$:

$F(C_{1,r_{3}})=\begin{cases}C_{2,r_{3}}&\hbox{if $r_{3}=u-1$}\\\
C_{2,r_{3}-1}&\hbox{if $u\leqslant r_{3}<v$}\\\ C_{2,r_{3}-2}&\hbox{if
$r_{3}\geqslant v$}.\end{cases}$

In the case $r_{3}=u-1$, it follows that $r_{3}\geqslant q_{M}-u+1$ and
therefore

$V^{-1}(F(C_{1,r_{3}}))\cap M_{1}=V^{-1}(C_{2,r_{3}})\cap M_{1}=C_{1,r_{3}},$

as desired. In the second case we have $u\leqslant r_{3}<v$ and it follows
that $q_{M}-v+2\leqslant r_{3}\leqslant q_{M}-u+1$. Hence

$V^{-1}(F(C_{1,r_{3}}))\cap M_{1}=V^{-1}(C_{2,r_{3}-1})\cap
M_{1}=C_{1,r_{3}}.$

Finally, in the third case we have $r_{3}>v$. It follows that
$r_{3}=q_{M}-v+2$ and thus

$V^{-1}(F(C_{1,r_{3}}))\cap M_{1}=V^{-1}(C_{2,r_{3}-2})\cap
M_{1}=C_{1,r_{3}}.$

In each case, Equation (4.3.7) holds, showing that we cannot ‘move down’
further down than $C_{1,r_{3}}$. This finishes the proof of the lemma. ∎

In order to describe the Ekedahl-Oort stratum of $L$, one more lemma is
needed.

######  Lemma 4.15.

We have $C_{1,r_{2}}\cap M[F]=C_{1,r_{3}}\cap M[F]$.

###### Proof.

Since $r_{2}\leqslant r_{3}$ and, hence, $C_{1,r_{2}}\subseteq C_{1,r_{3}}$
the inclusion “$\subseteq$” is immediate, so we focus on the inclusion
“$\supseteq$”. Recall that $M_{1}[F]=\langle e_{1,u},e_{1,v}\rangle$. Given
the definitions of $S_{\textup{up}}$ and $S_{\textup{down}}$ one can check
that it is not possible to have $r_{2}<u<r_{3}$ or $r_{2}<v<r_{3}$. Therefore
there is no element of $M[F]$ gained when moving from $C_{1,r_{2}}$ to
$C_{1,r_{3}}$. In other words, we have $C_{1,r_{3}}\cap M[F]\subseteq
C_{1,r_{2}}$, which proves the lemma. ∎

####  4.3.3. Ekedahl-Oort strata

Recall the setting. $M$ is the standard object of
$\mathcal{M}(m,2)_{\gamma_{u,v}}$, and $C_{i,\bullet}$ are filtrations of
$M_{i}$ coming from a final filtration on $M$. $N$ is the standard object of
the unique Ekedahl-Oort stratum of $\mathcal{M}(1,0)$, with the filtration
$D_{i,\bullet}$ of $N_{i}$. We form $L:=M\oplus N^{n}$, with the filtrations
$E_{1,\bullet}$ of $L_{i}$. We define

$\displaystyle\eta^{M}_{1}(j)$ $\displaystyle=\dim(C_{1,j}\cap M[F])$
$\displaystyle\eta^{L}_{1}(j)$ $\displaystyle=\dim(E_{1,j}\cap L[F]).$

Recall that $r_{2}=\min(S_{\text{up}\setminus\min S_{\text{up}}}$, the second
smallest element of the multiset
$S_{\textup{up}}:=\\{u,v,q_{M}-v+2,q_{M}-u+1\\}$. We now completely describe
the Ekedahl-Oort stratum of $L$.

######  Theorem 4.16.

The Ekedahl-Oort stratum of $L:=M\oplus N^{n}$ in $\mathcal{M}(m+n,2)$ is
represented by the permutation $\gamma_{y,s}$, with

$y=\begin{cases}u&\hbox{if $u\leqslant r_{2}$}\\\ u+n&\hbox{if
$u>r_{2}$}\end{cases}\qquad\qquad s=\begin{cases}v&\hbox{if $v\leqslant
r_{2}$}\\\ v+n&\hbox{if $v>r_{2}$}.\end{cases}$

###### Proof.

To prove the result, we compute the function $\eta^{L}_{1}$. For $j\leqslant
r_{2}$, Lemma 4.13 implies

$E_{1,j}=C_{1,j}\oplus D_{1,0}^{n}.$

Hence, $\eta^{L}_{1}(j)=\eta^{M}_{1}(j)$, since $D_{1,0}\cap N[F]=0$ by
Equation (4.3.1). For $j\geqslant r_{3}+n$, Lemma 4.14 implies

$E_{1,j}=C_{1,j-n}\oplus D_{1,1}^{n}.$

Hence, $\eta^{L}_{1}(j)=\eta^{M}_{1}(j-n)$, since $D_{1,1}\cap N[F]=0$ by
Equation (4.3.1). Finally, Lemma 4.15 implies that $\eta_{1}^{L}$ stays
constant between $r_{2}$ and $r_{3}$. In summary:

(4.3.8) $\eta^{L}_{1}(j)=\begin{cases}\eta^{M}_{1}(j)&\hbox{if $j\leqslant
r_{2}$},\\\ \eta^{M}_{1}(r_{2})&\hbox{if $r_{2}<j<r_{3}+n$},\\\
\eta^{M}_{1}(j-n)&\hbox{if $j\geqslant r_{3}+n$}.\end{cases}$

Note that $\eta^{M}_{1}$ jumps at $u$ and $v$. If $u\leqslant r_{2}$, then we
are in case 1 of Equation (4.3.8) and $\eta^{L}_{1}$ jumps at $j=u$. If
$u>r_{2}$ then, as noted in the proof of Lemma 4.15, $u$ must also be greater
than $r_{3}$. Hence, we are in case 3 of Equation (4.3.8) and $\eta^{L}_{1}$
jumps at $j=u+n$. Similarly, $\eta^{L}_{1}$ jumps at either $v$ or $v+n$ if
$v\leqslant r_{2}$ or $v>r_{2}$, respectively. This results in the statement
of the theorem. ∎

This theorem has the following implication for supersingular loci.

######  Corollary 4.17.

Let $m+n=q-2$ and
$\mathcal{M}(m,2)_{\gamma_{u,v}}\cap\mathcal{M}(m,2)^{ss}\neq\varnothing$.
Then we have

$\mathcal{M}(q-2,2)_{\gamma_{y,s}}\cap\mathcal{M}(q-2,2)^{ss}\neq\varnothing,$

where $y$ and $s$ are computed from $u$ and $v$ as in Theorem 4.16.

###### Proof.

This follows immediately from Theorem 4.16, the fact
$\mathcal{M}(n,0)=\mathcal{M}(n,0)^{ss}$, and the observation that the product
of two supersingular abelian varieties is supersingular. ∎

Note that Corollary 4.17 allows one to gain information about some Ekedahl-
Oort strata that intersect $\mathcal{M}(q-2,2)$, using information about which
Ekedahl-Oort strata intersect $\mathcal{M}(m,2)^{ss}$, for $m$ smaller than
$q-2$. Fortunately, there is a complete classification of Ekedahl-Oort strata
that intersect the supersingular locus of $\mathcal{M}(m,2)$ for very small
$m$. We first recall results for $m=0$ and 1.

The Shimura variety $\mathcal{M}(0,2)$ is isomorphic to $\mathcal{M}(2,0)$. As
mentioned in Subsection 4.3.1, $\mathcal{M}(0,2)$ has a unique Ekedahl-Oort
stratum. This must coincide with the supersingular locus, and so
$\mathcal{M}(0,2)=\mathcal{M}(0,2)^{ss}$.

The Shimura variety $\mathcal{M}(2,1)$ is isomorphic to $\mathcal{M}(1,2)$,
and the interaction between the Ekedahl-Oort strata and the supersingular
locus in this case is summarized in Subsection 4.2.1.

######  Proposition 4.18.

The supersingular locus of $\mathcal{M}(2,2)$ is exactly the union of those
Ekedahl-Oort strata of dimension two or less:

$\mathcal{M}(2,2)^{ss}=\mathcal{M}(2,2)_{\gamma_{1,2}}\sqcup\mathcal{M}(2,2)_{\gamma_{1,3}}\sqcup\mathcal{M}(2,2)_{\gamma_{1,4}}\sqcup\mathcal{M}(2,2)_{\gamma_{2,3}}$

###### Proof.

By [4], the Shimura variety $\mathcal{M}(2,2)$ is of Coxeter type. In
particular, the supersingular locus is a union of Ekedahl-Oort strata since
every Ekedahl-Oort stratum intersecting the supersingular locus is entirely
contained in the supersingular locus.

There are six Ekedahl-Oort strata of $\mathcal{M}(2,2)$, including exactly two
strata of dimension two: those indexed by $\gamma_{2,3}$ and $\gamma_{1,4}$.
It is known by results of [7] that the supersingular locus
$\mathcal{M}(2,2)^{ss}$ is two-dimensional, and so at least one of
$\mathcal{M}(2,2)_{\gamma_{1,4}}$ or $\mathcal{M}(2,2)_{\gamma_{2,3}}$ must be
contained in the supersingular locus. In what follows, we observe that, in
fact, both are.

There is a natural automorphism of $\mathcal{M}(2,2)(\mathbx k)$, taking
$(A,\iota,\lambda,\xi)$ to $(A,\overline{\iota},\lambda,\xi)$, where
$\overline{\iota}$ is the action $\iota$ composed with the nontrivial
automorphism of $K$. This action stabilizes the supersingular locus, but has
the effect of conjugating the action on the $p$-torsion group schemes defining
the Ekedahl-Oort strata. On the level of the standard objects
$N_{\gamma_{u,v}}$ corresponding to $\gamma_{u,v}$, this action interchanges
the roles of the basis vectors $e_{1,j}$ and $e_{2,j}$ for each $1\leqslant
j\leqslant q$.

By applying Lemma 3.3 to compute the standard objects $N_{1,4}$ and $N_{2,3}$,
one can see immediately that interchanging the roles of $e_{1,j}$ and
$e_{2,j}$ defines an isomorphsim between $N_{1,4}$ and $N_{2,3}$. Therefore,
both $\mathcal{M}(2,2)_{\gamma_{1,4}}$ and $\mathcal{M}(2,2)_{\gamma_{2,3}}$
are contained in the supersingular locus.

Since $\mathcal{M}(2,2)_{\gamma_{1,4}}$ and $\mathcal{M}(2,2)_{\gamma_{2,3}}$
are contained in the supersingular locus and the supersingular locus is
closed, the closure of these strata are also contained in the supersingular
locus. By Proposition 3.5,

$\overline{\mathcal{M}(2,2)_{\gamma_{1,4}}\bigsqcup\mathcal{M}(2,2)_{\gamma_{2,3}}}=\mathcal{M}(2,2)_{\gamma_{1,2}}\sqcup\mathcal{M}(2,2)_{\gamma_{1,3}}\sqcup\mathcal{M}(2,2)_{\gamma_{1,4}}\sqcup\mathcal{M}(2,2)_{\gamma_{2,3}},$

i.e., exactly those strata of dimension less than or equal to two.

Finally, the strata indexed by $\gamma_{2,4}$ and $\gamma_{3,4}$ have
dimension greater than two, and so they cannot be contained in the
supersingular locus. ∎

##  5\. The Forgetful Map: Relation to Siegel Modular Variety

In the previous section we obtained information about the Shimura variety
$\mathcal{M}(q-2,2)$ via product maps that we developed. The results of that
section, particularly Corollary 4.11 and Corollary 4.17, allow us to draw
conclusions about the interaction between Ekedahl-Oort strata and the
supersingular locus in some cases. In this section, we further our study of
this interaction by relating our Shimura variety $\mathcal{M}(q-2,2)$ to the
Siegel modular variety $\mathcal{A}_{q}$.

###  5.1. Background

By “forgetting” the unitary structure of the abelian varieties we are
considering, we obtain a point on the Siegel modular variety
$\mathcal{A}_{q}$. In other words, there is a forgetful map

$\displaystyle\Psi:\mathcal{M}(q-2,2)$ $\displaystyle\to\mathcal{A}_{q}$
$\displaystyle(A,\lambda,\iota,\xi)$ $\displaystyle\mapsto(A,\lambda,\xi).$

The map maintains the polarisation $\lambda$ and the level structure $\xi$ of
the abelian variety $A$, but drops the unitary structure
$\iota:\mathcal{O}_{K}\to\text{End}(A)$ of signature $(q-2,2)$. This forgetful
map induces a map on Ekedahl-Oort strata

$\psi:\mathbf{W}(q-2,2)\to\mathbf{W}_{q},$

where $\mathbf{W}_{q}$ is the set of minimal length Weyl group coset
representatives indexing the Ekedahl-Oort strata of $\mathcal{A}_{q}$. By [9,
3.6], the set $\mathbf{W}_{q}$ consists of the permutations
$\omega\in\mathfrak{S}_{2q}$ satisfying
$\omega^{-1}(1)<\omega^{-1}(2)<\cdots<\omega^{-1}(q)$ and
$\omega(i)+\omega(2q+1-i)=2q+1$. This section is devoted to computing
$\psi(\gamma_{u,v})$ for $\gamma_{u,v}\in\mathbf{W}(q-2,2)$ with the goal of
obtaining information about $\mathcal{M}(q-2,2)$ from various results about
$\mathcal{A}_{q}$.

We now outline how, given $\gamma_{u,v}\in\mathbf{W}(q-2,2)$, the permutation
$\psi(\gamma_{u,v})\in\mathbf{W}_{q}$ is computed. Using the bijection in
Theorem 2.2 and Lemma 3.3, we construct the standard object $M$ corresponding
to the Ekedahl-Oort stratum $\mathcal{M}(q-2,2)_{\gamma_{u,v}}$. Upon
forgetting the $\mathbb{F}_{p^{2}}$-action of $M$, we observe that $M$ is the
Dieudonné module of an Ekedahl-Oort stratum of $\mathcal{A}_{q}$. By [9, 3.6],
this Ekedahl-Oort stratum corresponds to an element of $\mathbf{W}_{q}$. This
correspondence works as follows. Let $W_{\bullet}$ be a final filtration of
$M$ and define the non-decreasing function

$\eta(j):=\dim(W_{j}\cap M[F])$

for $0\leqslant j\leqslant 2q$. Note that $\eta(2q)=\dim(M[F])=q$. Let
$1\leqslant j_{1}<\cdots<j_{q}\leqslant 2q$ be the indices where $\eta$ jumps,
meaning $\eta(j_{l})=\eta(j_{l}-1)+1$, and let $1\leqslant
i_{1}<\cdots<i_{q}\leqslant 2q$ be the remaining indices. We define
$\omega_{u,v}\in\mathfrak{S}_{2q}$ by $\omega_{u,v}(j_{l})=l$ and
$\omega_{u,v}(i_{m})=m+q$. By construction, $\omega_{u,v}$ is an element of
$\mathbf{W}_{q}$ and $\psi(\gamma_{u,v})=\omega_{u,v}$.

The duality coming from the symplectic pairing on $M$ implies that
$\eta(j)=\eta(j-1)$ holds if and only if $\eta(2q+1-j)=\eta(2q-j)+1$ holds. By
induction, it follows that $\eta(j)+q=\eta(2q-j)+j.$ The identity
$i_{m}=2q+1-j_{q-m}$ then follows from induction, implying that

(5.1.1) $\omega_{u,v}(i)+\omega_{u,v}(2q+1-i)=2q+1.$

The preceding paragraphs show that the function $\eta(j)=\dim(W_{j}\cap M[F])$
determines the permutation $\omega_{u,v}=\psi(\gamma_{u,v})$, and so we focus
on determining all the values of $\eta$. Lemma 3.3 yields the description

$M[F]=\langle e_{1,u},e_{1,v}\rangle\oplus\langle
e_{2,j}\;|\;j\notin\\{q+1-u,q+1-v\\}\rangle.$

Recall the filtrations $C_{i,\bullet}=W_{\bullet}\cap M_{i}$ and note that any
$W_{j}$ can be written as $W_{j}=C_{1,l_{1}}\oplus C_{2,l_{2}}$ for some
$l_{1}$ and $l_{2}$ satisfying $l_{1}+l_{2}=j$. In Proposition 5.1, this
decomposition is made explicit, and as a result a final filtration of $M$ is
given in terms of $C_{1,\bullet}$ and $C_{2,\bullet}$. The shape of this final
filtration depends on $u$ and $v$. In Theorem 5.2, this final filtration is
used to determine the permutation $\omega_{u,v}\in\mathbf{W}_{q}$.

There are three possibilities for the Ekedahl-Oort stratum of
$(\mathcal{A}_{q})_{\omega_{u,v}}$. The first possibility is that
$(\mathcal{A}_{q})_{\omega_{u,v}}$ is contained in $\mathcal{A}_{q}^{ss}.$ In
that case, it follows that $\mathcal{M}(q-2,2)_{\gamma_{u,v}}$ is contained in
$\mathcal{M}(q-2,2)^{ss}.$ Corollary 5.4 records the Ekedahl-Oort strata of
$\mathcal{M}(q-2,2)$ that occur in in this way. Another possibility is that
$(\mathcal{A}_{q})_{\omega_{u,v}}$ is disjoint from $\mathcal{A}_{q}^{ss}.$ In
that case, it follows that $\mathcal{M}(q-2,2)_{\gamma_{u,v}}$ is disjoint
from $\mathcal{M}(q-2,2)^{ss}$. Proposition 5.6 and Corollary 5.12 record the
Ekedahl-Oort strata of $\mathcal{M}(q-2,2)$ with this property. Finally, it is
possible that the Ekedahl-Oort stratum $(\mathcal{A}_{q})_{\omega_{u,v}}$
intersects $\mathcal{A}_{q}^{ss}$, but is not contained in it. In this case,
we do not obtain information about the Ekedahl-Oort stratum
$\mathcal{M}(a,b)_{\gamma_{u,v}}$, due to the highly non-surjective nature of
the forgetful map. It could, for instance, be that there exist supersingular
abelian varieties in $(\mathcal{A}_{q})_{\omega_{u,v}}$, but that none of them
admit a compatible unitary structure of signature $(q-2,2)$. For this reason,
our work in this section focuses on the cases when the Ekedahl-Oort
$(\mathcal{A}_{q})_{\omega_{u,v}}$ is either completely contained in
$\mathcal{A}_{q}^{ss}$ or completely disjoint from it.

###  5.2. A Final Filtration of the Standard Object

In order to compute $\eta(j)=\dim(W_{j}\cap M[F])$, we construct a final
filtration $W_{\bullet}$ of $M$ in terms of $C_{1,\bullet}$ and
$C_{2,\bullet}$. Essentially this is done by applying Lemma 3.3 (or its
consequence Lemma 4.12) repeatedly. We could give a canonical filtration
instead, but giving a final filtration makes the exposition shorter and later
computations easier. It is sufficient to construct $W_{j}$, where $1\leqslant
j\leqslant q$, as the remaining parts of the filtration can be constructed by
taking symplectic complements. As seen in the proof of Lemma 4.6, we have

$W_{j}^{\perp}=(C_{1,l_{1}}\oplus C_{2,l_{2}})^{\perp}=C_{1,l_{1}}^{\perp}\cap
C_{2,l_{2}}^{\perp}=(M_{1}\oplus C_{2,q-l_{1}})\cap(C_{1,q-l_{2}}\oplus
M_{2})=C_{1,q-l_{2}}\oplus C_{1,q-l_{1}}=W_{2q-j}.$

The following proposition gives a final filtration $W_{\bullet}$ of $M$ that
is dependent on $u$ and $v$. When there is only one way to fill up a gap
between two subspaces, for instance between $C_{1,1}\oplus C_{2,0}$ and
$C_{1,1}\oplus C_{2,q-2}$, then the subspaces in between (of the form
$C_{1,1}\oplus C_{2,l}$ for $1<l<q-2$) are omitted from the notation. In each
case, Lemma 3.3 yields

$\displaystyle W_{q}$ $\displaystyle=F(M)=\langle
e_{1,1},e_{1,2}\rangle\oplus\langle e_{2,j}\;|\;j\leqslant
q-2\rangle=C_{1,2}\oplus C_{2,q-2},$ (5.2.1) $\displaystyle M[F]$
$\displaystyle=M[F]_{1}\oplus M[F]_{2}=\langle
e_{1,u},e_{1,v}\rangle\oplus\langle
e_{2,j}\;|\;j\notin\\{q+1-v,q+1-u\\}\rangle.$

######  Proposition 5.1.

Let $M$ be the standard object of $\mathcal{M}(q-2,2)_{\gamma_{u,v}}$. In the
case $u=1$, the first half of a final filtration of $M$ is

$0\subset C_{1,1}\oplus C_{2,0}\subset\cdots\subset C_{1,1}\oplus
C_{2,q-2}\subset W_{q}.$

In the case $u=2$, the first half of a final filtration is

$0\subset C_{1,0}\oplus C_{2,1}\subset C_{1,1}\oplus
C_{2,1}\subset\cdots\subset C_{1,1}\oplus C_{2,s_{1}}\subset C_{1,2}\oplus
C_{2,s_{1}}\subset\cdots\subset W_{q}.$

In the case $u>2$ and $v<q-1$, the first half of a final filtration is

$0\subset\cdots\subset C_{1,0}\oplus C_{2,2}\subset\cdots\subset C_{1,2}\oplus
C_{2,2}\subset\cdots\subset W_{q}.$

In the case $u>2$ and $v\geqslant q-1$, the first half of a final filtration
is

$0\subset C_{1,0}\oplus C_{2,1}\subset C_{1,1}\oplus C_{2,1}\subset
C_{1,1}\oplus C_{2,2}\subset C_{1,2}\oplus C_{2,2}\subset\cdots\subset W_{q}.$

###### Proof.

We prove the proposition by constructing the necessary parts of the canonical
filtration. The standard object from Lemma 3.3 is used for determining the
effect on $F$ or $V^{-1}$ on a subspace. We split the proof up into three
cases: $u=1$, $u=2$ and $u>2$.

First, assume $u=1$. In that case, we construct

$F(W_{q})=\begin{cases}C_{1,0}\oplus C_{2,0}=W_{0}&\hbox{if $v=2$},\\\
C_{1,1}\oplus C_{2,1}=W_{2}&\hbox{if $v>2$}.\end{cases}$

In the case $(u,v)=(1,2)$, we have $W_{q}=M[F]$ and therefore $\eta(q)=q$,
implying that $\eta$ has to increase by 1 at each index between $0$ and $q$.
It is not possible to construct a subspace between $0$ and $W_{q}$ using $F$
and $V^{-1}$, so we can extend the canonical filtration to a final filtration
in any way we like; it will not influence the function $\eta$. In the case
$v>2$, we construct

$F(W_{2})=\begin{cases}C_{1,0}\oplus C_{2,0}=W_{0}&\hbox{if $v<q$},\\\
C_{1,1}\oplus C_{2,0}=W_{1}&\hbox{if $v=q$}.\end{cases}$

Here we have used that $F(C_{1,1})=0$, since $u=1$, and $F(C_{2,1})=0$ if and
only if $v<q$. In the first case, the canonical filtration cannot produce
$W_{1}$. This is not required, as $\eta(2)=2$ implies $\eta(1)=1$. In both
cases, we can form a final filtration containing $W_{1}$.

Next, we fill in the canonical filtration between $W_{2}$ and $W_{q}$. This
amounts to discovering when $C_{1,1}$ changes to $C_{1,2}$ in the filtration.
We construct

$V^{-1}(W_{q})=C_{1,q-1}\oplus C_{2,q-1}=W_{2q-2}=W_{2}^{\perp}$

We then apply $F$ to obtain

$F(W_{2q-2})=\begin{cases}C_{1,1}\oplus C_{2,q-2}=W_{q-1}&\hbox{if $v=q$},\\\
C_{1,1}\oplus C_{2,q-3}=W_{q-2}&\hbox{if $v<q$}.\end{cases}$

Thus far, we have constructed $W_{0}$, $W_{1}$, $W_{2}$, $W_{q-1}$, and
$W_{q}$. This forces all the intermediate $W_{i}=C_{1,1}\oplus C_{2,i-1}$ due
to dimension reasons, as we showed that all have the $C_{1,1}$ term in the
first half. In the case $v=q$, we are done: the first half of the canonical
filtration is

(5.2.2) $0\subset C_{1,1}\oplus C_{2,0}\subset C_{1,1}\oplus
C_{2,1}\subset\cdots\subset C_{1,1}\oplus C_{2,q-2}\subset C_{1,2}\oplus
C_{2,q-2}=W_{q}.$

In the case $v=3$, we compute

$\eta(q-2)=\dim\left(W_{q-2}\cap
M[F]\right)=\eta_{1}(1)+\eta_{2}(q-3)=1+(q-3)=q-2.$

Together with $\eta(q)=q-2$, this implies $\eta(q-1)=q-2$, meaning that
$W_{q-1}$ does not need to be constructed. The final filtration in Equation
(5.2.2) is still valid in this case.

In the case $3<v<q$, one more step is needed. We construct

$\displaystyle V^{-1}(W_{2})$ $\displaystyle=V^{-1}(C_{1,1}\oplus
C_{2,1})=C_{1,3}\oplus C_{2,q-1}=W_{q+2},$ $\displaystyle V^{-1}(W_{q+2})$
$\displaystyle=V^{-1}(C_{1,3}\oplus C_{2,q-1})=C_{1,q}\oplus
C_{2,q-1}=W_{2q-1},$ $\displaystyle F(W_{2q-1})$
$\displaystyle=F(C_{1,q}\oplus C_{2,q-1})=C_{1,1}\oplus C_{2,q-2}=W_{q-1}.$

We conclude that the final filtration given in Equation (5.2.2) always works
in the case $u=1$.

We now treat the case $u=2$, which is the most challenging. We again construct

$F(W_{q})=C_{1,1}\oplus C_{2,1}=W_{2}.$

In the case $v=q$, we compute $\eta(2)=0$. Combining this with $\eta(q)=q-2$
fixes the behavior of $\eta$ everywhere; it has to jump everywhere between
$\eta(2)=0$ and $\eta(q)=q-2$. Therefore we are finished with this case and
assume $2<v<q$ from here on. Applying $F$ to $W_{2}$ yields

$F(W_{2})=C_{1,0}\oplus C_{2,1}=W_{1}.$

This implies that in the case $u=2$ the final filtration must be of the form

$\displaystyle 0\subset C_{1,0}\oplus C_{2,1}\subset C_{1,1}\oplus C_{2,1}$
$\displaystyle\subset C_{1,0}\oplus C_{2,2}\subset C_{1,1}\oplus
C_{2,2}\subset\cdots$ $\displaystyle\cdots\subset C_{1,1}\oplus C_{2,m}\subset
C_{1,2}\oplus C_{2,m}\oplus\cdots\subset C_{1,2}\oplus C_{2,q-2}=W_{q}$

for some (not necessarily unique) integer $m$. The remaining goal is to find
such an integer $m$. We do this by constructing $C_{1,1}\oplus C_{2,j}$ for
increasing $j$ (by ‘moving up’) and constructing $C_{1,2}\oplus C_{2,j}$ for
decreasing $j$ (by ‘moving down’).

We first outline the moving down procedure. Define $s_{2}:=\max\\{v-2,q-v\\}$.
Starting from $W_{q}=C_{1,2}\oplus C_{2,q-2}$, we construct
$W_{j+2}=C_{1,2}\oplus C_{2,j}$ for decreasing $j$, by applying $FV^{-1}$,
until we reach $C_{1,2}\oplus C_{2,s_{2}}$. The induction step goes as
follows. Assume $j>s_{2}$. Since $j>q-v$, we have

$V^{-1}(W_{j+2})=V^{-1}(C_{1,2}\oplus C_{2,j})=C_{1,j+1}\oplus
C_{2,q-1}=W_{q+j}.$

As additionally $j+1\geqslant v$ holds, we have

$F(C_{1,j+1}\oplus C_{2,q-1})=C_{1,2}\oplus C_{2,j-1}=W_{j+1}.$

In this way $C_{1,2}\oplus C_{2,j}$ is constructed for decreasing $j$. We now
show that this process halts when $j=s_{2}$ is reached. If $j=q-v$, then we
have

$V^{-1}(C_{1,2}\oplus C_{2,j})=C_{1,j+2}\oplus C_{2,q-1}=W_{q+j+1}.$

and we cannot move down further. Similarly, if $j=v-2$, then we obtain

$F(V^{-1}(C_{1,2}\oplus C_{2,j}))=F(C_{1,j+1}\oplus C_{2,q-1})=C_{1,2}\oplus
C_{2,j}.$

We conclude that we can move down to $W_{2+s_{2}}=C_{1,2}\oplus C_{2,s_{2}}$
and no further.

The moving up procedure works analogously. Define
$s_{1}:=\min\\{v-2,q+1-v\\}$. We begin with $W_{2}=C_{1,1}\oplus C_{2,1}$ and
construct $W_{j+1}=C_{1,1}\oplus C_{2,j}$ for increasing $j$, by applying
$FV^{-1}$, until we reach $C_{1,1}\oplus C_{2,s_{1}}$. Now, for the induction
step we assume $j<s_{1}$. Since $j\leqslant q-v$, we have

$V^{-1}(W_{j+1})=V^{-1}(C_{1,1}\oplus C_{2,j})=C_{1,j+2}\oplus
C_{2,q-2}=W_{q+j}.$

Then, as $j+2<v$ holds, we obtain

$F(W_{q+j})=F(C_{1,j+2}\oplus C_{2,q-2})=C_{1,1}\oplus C_{2,j+1}=W_{j+2}.$

Thus $C_{1,1}\oplus C_{2,j}$ is constructed for increasing $j$. We now show
that this procedure stops when $j=s_{1}$ is reached. In the case $j=q+1-v$, we
have

$V^{-1}(W_{j+1})=V^{-1}(C_{1,1}\oplus C_{2,j})=C_{1,j+1}\oplus
C_{2,q-2}=W_{q+j-1}.$

On the other hand, in the case $j=v-2$, we have

$F(W_{q+j})=F(C_{1,j+2}\oplus C_{2,q-2})=C_{1,1}\oplus C_{2,j}.$

Finally, it is left to prove that this construction suffices, in the sense
that it is not necessary to construct more parts of the filtration in order to
compute the function $\eta$. In general there may be subspaces between
$W_{1+s_{1}}$ and $W_{2+s_{2}}$. We now show that this does not influence the
behavior of $\eta$.

In the case $v-2\geqslant q+1-v$, we have $W_{1+s_{1}}=W_{q+2-v}=C_{1,1}\oplus
C_{2,q+1-v}$. Since $e_{2,q+1-v}\notin M[F]$, we obtain that
$\eta(q+2-v)=q-v$. Then $\eta$ must keep increasing till $\eta(q)=q-2$, so the
canonical filtration does not develop further.

On the other hand, in the case $v-2<q+1-v$ we have
$W_{2+s_{2}}=W_{q+2-v}=C_{1,2}\oplus C_{2,q-v}$. We compute and obtain
$\eta(q+2-v)=q+1-v$. This implies that $\eta$ must keep increasing between
$\eta(2)=1$ and $\eta(q+2-v)=q+1-v$. Hence the canonical filtration does not
develop further. In both cases the first half of a final filtration of $M$ is
given by

$0\subset C_{1,0}\oplus C_{2,1}\subset C_{1,1}\oplus
C_{2,1}\subset\cdots\subset C_{1,1}\oplus C_{2,s_{1}}\subset C_{1,2}\oplus
C_{2,s_{1}}\subset\cdots\subset W_{q}.$

The last case to treat is $u>2$. Here we obtain

$F(W_{q})=C_{1,2}\oplus C_{2,2}=W_{4},$

which already fixes the final filtration between $W_{4}$ and $W_{q}$:
everything must be of the form $C_{1,2}\oplus C_{2,j}$ for $2\leqslant
j\leqslant q-2$. The only task at hand is to compute $\eta(1)$, $\eta(2)$ and
$\eta(3)$. In the case $u=q-1$, we have $F(W_{4})=W_{4}$ and therefore
$\eta(4)=0$, which implies that $\eta(1)=\eta(2)=\eta(3)=0$. We now assume
$u<q-1$ from here on and compute

$F(W_{4})=\begin{cases}C_{1,0}\oplus C_{2,2}=W_{2}&\hbox{if $2\leqslant
q-v$},\\\ C_{1,1}\oplus C_{2,2}=W_{3}&\hbox{if $2>q-v.$}\\\ \end{cases}$

First, we treat the case $2\leqslant q-v$. From the expression for $W_{2}$ and
the fact that a final filtration is a filtration, it follows what $W_{1}$ and
$W_{3}$ must be. This gives the final filtration

$0\subset\cdots\subset C_{1,0}\oplus C_{2,2}\subset\cdots\subset C_{1,2}\oplus
C_{2,2}\subset\cdots\subset W_{q}.$

We finally treat the case $2>q-v$ or equivalently $v\geqslant q-1$. Applying
$F$ again gives

$\displaystyle F(W_{3})$ $\displaystyle=C_{1,1}\oplus C_{2,1}=W_{2},\text{ and
}$ $\displaystyle F(W_{2})$ $\displaystyle=\begin{cases}C_{1,0}\oplus
C_{2,1}=W_{1}&\hbox{if $v=q-1$},\\\ C_{1,1}\oplus C_{2,1}=W_{2}&\hbox{if
$v=q.$}\\\ \end{cases}$

When $v=q$, we have $\eta(2)=0$ and therefore $\eta(1)=0$. On the other hand,
in the case $v=q-1$ we have constructed $W_{1}$, $W_{2}$ and $W_{3}$, which
clearly suffices. In both cases, there is a final filtration

$0\subset C_{1,0}\oplus C_{2,1}\subset C_{1,1}\oplus C_{2,1}\subset
C_{1,1}\oplus C_{2,2}\subset C_{1,2}\oplus C_{2,2}\subset\cdots\subset
W_{q}.\qed$

###  5.3. Weyl group cosets

We now compute the permutation
$\omega_{u,v}=\psi(\gamma_{u,v})\in\mathbf{W}_{q}\subset\mathfrak{S}_{2q}$
using the results of Proposition 5.1. The resulting permutation represents the
Ekedahl-Oort stratum of $M$ in $\mathcal{A}_{q}$.

In the following theorem, only the action of $\omega_{u,v}$ on the integers
$1\leqslant i\leqslant q$ is given, as the remaining information can be
retrieved using Equation (5.1.1).

######  Theorem 5.2.

In the case $(u,v)=(1,2)$, we have $\omega_{u,v}=\text{id}$. For various other
choices of $u,v$, the following tables describe the corresponding
$\omega_{u,v}$:

|  $u=1$ and $v>2$
---|---
$\omega_{u,v}(i)$ | $\begin{cases}i&\hbox{if $i<q+2-v$}\\\ q+1&\hbox{if $i=q+2-v$}\\\ i-1&\hbox{if $q+2-v<i<q$}\\\ q+2&\hbox{if $i=q$}.\end{cases}$

|  $u=2$ and $v=q$ |  $u=2$ and $1<q+1-v\leqslant v-2$ |  $u=2$ and $q+1-v>v-2$
---|---|---|---
$\omega_{u,v}(i)$ | $\begin{cases}q+i&\hbox{if $i\leqslant 2$}\\\ i-2&\hbox{if $2<i\leqslant q$}.\end{cases}$ | $\begin{cases}1&\hbox{if $i=1$}\\\ q+1&\hbox{if $i=2$}\\\ i-1&\hbox{if $2<i<q+2-v$}\\\ q+2&\hbox{if $i=q+2-v$}\\\ i-2&\hbox{if $q+2-v<i\leqslant q$}\end{cases}$ | $\begin{cases}1&\hbox{if $i=1$}\\\ q+1&\hbox{if $i=2$}\\\ i-1&\hbox{if $2<i<q+3-v$}\\\ q+2&\hbox{if $i=q+3-v$}\\\ i-2&\hbox{if $q+3-v<i\leqslant q$}\end{cases}$

$2<u<q-1$ and $v=q-1$ $2<u<q-1$ and $v=q$ $2<u<q-1$ and $v<q-1$
$\omega_{u,v}(i)$ $\begin{cases}1&\hbox{if $i=1$}\\\ q+i-1&\hbox{if
$1<i\leqslant 4$}\\\ i-3&\hbox{if $4<i<q+3-u$}\\\ q+4&\hbox{if $i=q+3-u$}\\\
i-4&\hbox{if $q+3-u<i\leqslant q$}.\end{cases}$ $\begin{cases}q+i&\hbox{if
$1\leqslant i\leqslant 2$}\\\ i-2&\hbox{if $i=3$}\\\ q+3&\hbox{if $i=4$}\\\
i-3&\hbox{if $4<i\leqslant q+3-u$}\\\ q+4&\hbox{if $i=q+3-u$}\\\ i-4&\hbox{if
$q+3-u<i\leqslant q$}.\end{cases}$ $\begin{cases}i&\hbox{if $1\leqslant
i\leqslant 2$}\\\ q+i-2&\hbox{if $3\leqslant i\leqslant 4$}\\\ i-2&\hbox{if
$4<i\leqslant q+3-v$}\\\ q+3&\hbox{if $i=q+3-v$}\\\ i-3&\hbox{if
$q+3-v<i<q+3-u$}\\\ q+4&\hbox{if $i=q+3-u$}\\\ i-4&\hbox{if $q+3-u<i\leqslant
q$}.\end{cases}$

| $u=q-1$
---|---
$\omega_{u,v}(i)$ | $\begin{cases}q+i&\hbox{if $1\leqslant i\leqslant 4$}\\\ i-4&\hbox{if $4<i\leqslant q$}.\end{cases}$

###### Proof.

In the case $(u,v)=(1,2)$, we have $\eta(q)=q$. Therefore the function $\eta$
jumps at every integer $1\leqslant j\leqslant q$. Hence we obtain $j_{l}=l$
and $i_{m}=m+q$. We conclude $\omega_{u,v}=\text{id}$.

In the case $u=1$ and $v>2$, we follow the final filtration given in
Proposition 5.1. We compute

$\eta(j)=\begin{cases}1&\hbox{if $j=1$},\\\ 1+\eta_{2}(j-1)&\hbox{if
$1<j<q$},\\\ 1+\eta_{2}(j-2)&\hbox{if $j=q$}\end{cases}$

using the information we have on $\eta_{1}$ and $\eta_{2}$. We see that $\eta$
jumps everywhere except at the indices where $e_{1,2}$ and $e_{2,q+1-v}$ are
added. The computation above shows where this happens: $i_{1}=q+2-v$ and
$i_{2}=q$. This is the information needed to compute $\omega_{u,v}$:

$\omega_{u,v}(i)=\begin{cases}i&\hbox{if $i<q+2-v$},\\\ q+1&\hbox{if
$i=q+2-v$},\\\ i-1&\hbox{if $q+2-v<i<q$},\\\ q+2&\hbox{if $i=q$}.\end{cases}$

as asserted.

In the case $u=2$, set $s_{1}=\min\\{v-2,q+1-v\\}$. Then by Proposition 5.1 it
follows that

$\eta(j)=\begin{cases}\eta_{2}(j)&\hbox{if $j=1$},\\\ \eta_{2}(j-1)&\hbox{if
$1<j\leqslant s_{1}+1$},\\\ 1+\eta_{2}(j-2)&\hbox{if $s_{1}+1<j\leqslant
q$}.\end{cases}$

The integers $i_{1}$ and $i_{2}$ correspond to the indices where $e_{1,1}$ and
$e_{2,q+1-v}$ are added. In the case $v=q$, this implies $i_{1}=1$ and
$i_{2}=2$. Otherwise, we have $i_{1}=2$. In the case $1<q+1-v\leqslant v-2$,
we have $i_{2}=q+2-v$. In the case $q+1-v>v-2$, we have $i_{2}=q+3-v$.

In the case $2<u<q-1$ and $v<q-1$, the final filtration in Proposition 5.1
implies

$\eta(j)=\begin{cases}\eta_{2}(j)&\hbox{if $j\leqslant 2$},\\\
\eta_{2}(j-1)&\hbox{if $j=3$},\\\ \eta_{2}(j-2)&\hbox{if $3<j\leqslant
q$}.\end{cases}$

Thus $\eta$ jumps everywhere except at the indices $\\{3,4,q+3-v,q+3-u\\}$,
resulting in the permutation $\omega_{u,v}$ from the theorem.

Finally, in the case $u>2$ and $v\geqslant q-1$, Proposition 5.1 provides

$\eta(j)=\begin{cases}\eta_{2}(j)&\hbox{if $j=1$},\\\ \eta_{2}(j-1)&\hbox{if
$j\in\\{2,3\\}$},\\\ \eta_{2}(j-2)&\hbox{if $3<j\leqslant q$}.\end{cases}$

Now there are four indices where $\eta$ does not jump: these are the indices
when $e_{1,1}$, $e_{1,2}$, $e_{2,q+1-u}$ and $e_{2,q+1-v}$ are added. In each
case, $2$ and $4$ are among these indices. In the case $u=q-1$, the remaining
indices are $1$ and $3$. In the case $u<q-1$ and $v=q$, the remaining indices
are $1$ and $q+3-u$. Finally, in the case $u<q-1$ and $v=q-1$, the remaining
indices are $3$ and $q+3-u$. ∎

###  5.4. Ekedahl-Oort strata contained in the supersingular locus

Using the main result of [6], we pinpoint exactly which $\omega_{u,v}$
represent an Ekedahl-Oort stratum of $\mathcal{A}_{q}$ that is contained in
$\mathcal{A}_{q}^{ss}$.

######  Proposition 5.3.

The Ekedahl-Oort stratum $(\mathcal{A}_{q})_{\omega_{u,v}}$ is contained in
$\mathcal{A}_{q}^{ss}$ if and only if $u=1$ and $v<\lfloor q/2\rfloor+2$.

###### Proof.

By [6, Theorem 1.2], the containment holds if and only if $\omega_{u,v}(i)=i$
for each $1\leqslant i\leqslant\lceil q/2\rceil$. For $u=1$, by Theorem 5.2
this holds exactly when $\lceil q/2\rceil<q+2-v$, i.e., when $v<\lfloor
q/2\rfloor+2$.

We now show that containment is not possible for $u>1$. In the case $u=2$, we
have $\omega_{u,v}(2)>q$, so the containment can only hold if $q\leqslant 2$.
That, however, contradicts the assumption $u=2$.

In the case $u>2$, we also have $\omega_{u,v}(2)>q$, except when $2<u<q-1$ and
$v<q-1$. In this case we have $\omega_{u,v}(3)>q$ and hence $\lceil
q/2\rceil<3$. This implies $q\leqslant 4$, which contradicts
$2<u<v<q-1\leqslant 3$. ∎

######  Corollary 5.4.

Assume $u=1$ and $v<\lfloor q/2\rfloor+2$. Then the Ekedahl-Oort stratum
$\mathcal{M}(q-2,2)_{\gamma_{u,v}}$ is contained in $\mathcal{M}(q-2,2)^{ss}$.

###### Proof.

Proposition 5.3 shows that $\omega_{u,v}$ is completely contained in
$\mathcal{A}_{q}^{ss}$ under these assumptions. Therefore any $4$-tuple
$(A,\lambda,\iota,\xi)$ in $\mathcal{M}(q-2,2)_{\gamma_{u,v}}$ has the
property that $(A,\lambda,\iota)$ is supersingular. This implies that the
$4$-tuple is supersingular. ∎

###  5.5. Ekedahl-Oort strata on which $\mathbold{F}$ and $\mathbold{V}$ are
not nilpotent

If $A[p]$ has a non-trivial subgroup scheme on which $F$ or $V$ act
bijectively, then the slopes $0$ and $1$ occur in the Newton polygon of $A$.
Thus $F$ and $V$ act nilpotently on supersingular abelian varieties
(equivalently, their so-called _$p$ -rank_ is zero). By duality, $F$ is
nilpotent if and only if $V$ is nilpotent. The following lemma shows how this
is measured by the permutation $\omega_{u,v}$.

######  Lemma 5.5.

The action of $F$ is nilpotent on $M$ if and only if $\omega_{u,v}(1)=1$.

###### Proof.

Assume the action of $F$ is nilpotent on $M$. Then we must have $F(W_{1})=0$,
because otherwise $F$ acts bijectively on $W_{1}$. Hence

$\eta(1)=\dim(W_{1}\cap M[F])=1$

and therefore $j_{1}=1$ and $\omega_{u,v}(1)=1$.

On the other hand, assume that $F$ is not nilpotent on $M$. This implies that
$F^{n}(M)\neq 0$ for every $n$. Since applying $F$ gives a subspace in the
canonical filtration, we infer that there is an $l>0$ such that
$F(W_{l})=W_{l}$. Thus $\eta(l)=0$ and, therefore, $i_{1}=1$ and
$\omega_{u,v}(1)=q+1$. ∎

This allows us show that several Ekedahl-Oort strata cannot intersect the
supersingular locus.

######  Proposition 5.6.

Assume $u>1$. Then the Ekedahl-Oort stratum
$\mathcal{M}(q-2,2)_{\gamma_{u,q}}$ does not intersect
$\mathcal{M}(q-2,2)^{ss}$.

###### Proof.

By Lemma 5.5, it suffices to check that $\omega_{u,q}(1)>1$. Appealing to
Theorem 5.2, we conclude that this happens exactly when $u>1$ and $v=q$. Since
$(\mathcal{A}_{q})_{\omega_{u,q}}$ does not intersect $\mathcal{A}_{q}^{ss}$,
it follows that $\mathcal{M}(q-2,2)_{\gamma_{u,q}}$ does not intersect
$\mathcal{M}(q-2,2)^{ss}$. ∎

###  5.6. Minimal Ekedahl-Oort strata of non-supersingular Newton strata

We now describe another method that allows us to show Ekedahl-Oort strata are
disjoint from the supersingular locus. This method is based on _minimal_
Ekedahl-Oort strata, which are completely contained in a non-supersingular
Newton stratum.

######  Definition 5.7.

An Ekedahl-Oort stratum $S$ of a Shimura variety of PEL type $\mathcal{M}$ is
_minimal_ if $A[p^{\infty}]\cong B[p^{\infty}]$ holds for any $A,B\in
S(\mathbx k)$.

It follows immediately from this definition that the Ekedahl-Oort stratum $S$
is completely contained in one Newton stratum. Unfortunately, it is not known
what the minimal Ekedahl-Oort strata of $\mathcal{M}(q-2,2)$ look like, or
whether each Newton stratum of $\mathcal{M}(q-2,2)$ contains a (unique)
minimal Ekedahl-Oort stratum, since the group $\mathsf{GU}(q-2,2)$ is not
split for $q>4$ (see [16]).

Luckily things are different for $\mathcal{A}_{q}$. Oort proved in [10, 11]
that each Newton stratum of $\mathcal{A}_{q}$ contains a unique minimal
Ekedahl-Oort stratum. Given a Newton polygon, results of [2, 5.3] give an
explicit description of the $p$-divisible group and the $p$-torsion group
scheme of the minimal Ekedahl-Oort stratum contained in that Newton stratum.
Hence, the minimal Ekedahl-Oort stratum of a non-supersingular Newton stratum
does not intersect $\mathcal{A}_{q}^{ss}$, and we can use Theorem 5.2 to
determine Ekedahl-Oort strata of $\mathcal{M}(q-2,2)$ that do not intersect
$\mathcal{M}(q-2,2)^{ss}$.

######  Definition 5.8.

Let $m$ and $n$ be non-negative integers. Define the mod-$p$ Dieudonné module
$M_{m,n}:=\operatorname{span}k\\{e_{0},\ldots,e_{m+n-1}\\}$ with following
action of $F$ and $V$:

(5.6.1) $\displaystyle F(e_{i})$ $\displaystyle=\begin{cases}e_{i+n}&\hbox{if
$i\leqslant m-1$},\\\ 0&\hbox{else},\end{cases}$ (5.6.2) $\displaystyle
V(e_{i})$ $\displaystyle=\begin{cases}e_{i+m}&\hbox{if $i\leqslant n-1$},\\\
0&\hbox{else.}\end{cases}$

Let $\alpha$ be a Newton stratum of $\mathcal{A}_{q}$ with slopes
$\frac{n_{1}}{m_{1}+n_{1}},\ldots,\frac{n_{r}}{m_{r}+n_{r}}$, where each slope
may occur with multiplicity greater than or equal to 1. Oort shows in [10]
that the unique minimal Ekedahl-Oort stratum in $\alpha$ has the Dieudonné
module

(5.6.3) $M_{\alpha}:=\bigoplus_{l=1}^{r}M_{m_{l},n_{l}}.$

The Dieudonné module $M_{\alpha}$ corresponds to a permutation
$\omega_{\alpha}\in\mathbf{W}_{q}$, which we determine in the results below.

######  Lemma 5.9.

$M_{m,n}$ has a final filtration given by

$W_{j}^{m,n}:=\operatorname{span}k\\{e_{m+n-j},\ldots,e_{m+n-1}\\}$

(5.6.4) $\displaystyle F(W_{j}^{m,n})$
$\displaystyle=W^{m,n}_{\max\\{0,j-n\\}},$ (5.6.5) $\displaystyle
V^{-1}(W_{j}^{m,n})$ $\displaystyle=W^{m,n}_{\min\\{m+n,j+m\\}},$ (5.6.6)
$\displaystyle M_{m,n}[F]$ $\displaystyle=W^{m,n}_{n}.$

###### Proof.

Equation (5.6.4) follows from Equation (5.6.1); all indices are shifted by $n$
when $F$ is applied, except if the resulting index exceeds $m+n-1$, in which
case $F$ acts as $0$. Similarly, Equation (5.6.5) is proved using Equation
(5.6.2). Since the filtration $W^{m,n}_{\bullet}$ is stable under $F$ and
$V^{-1}$ and $\dim k(W_{j})=j$, it is a final filtration of $M_{m,n}$.
Finally, Equation (5.6.6) also follows directly from Equation (5.6.1). ∎

Given integers $m,n$ and a word $w$ in the letters $F$ and $V^{-1}$, define
the integer $w(m,n)$ recursively as follows:

  * •

If $w$ is the empty word, then $w(m,n)=m+n$,

  * •

$(Fw)(m,n)=\max\\{0,w(m,n)-n\\}$,

  * •

$(V^{-1}w)(m,n)=\min\\{m+n,w(m,n)+m\\}$.

######  Corollary 5.10.

We have $w(M_{m,n})=W^{m,n}_{w(m,n)}$.

###### Proof.

Apply Lemma 5.9 repeatedly. ∎

Recall the definition of the mod-$p$ Dieudonné module $M_{\alpha}$ from
Equation (5.6.3). Combining this with Corollary 5.10 gives, for any word $w$
in the alphabet $\\{F,V^{-1}\\}$, the formula

$w(M_{\alpha})=w\left(\bigoplus_{l=1}^{r}M_{m_{l},n_{l}}\right)=\bigoplus_{l=1}^{r}w(M_{m_{l},n_{l}})=\bigoplus_{l=1}^{r}W^{m_{l},n_{l}}_{w(m_{l},n_{l})}.$

We now introduce the function $\eta_{\alpha}(j)=\dim\left(M_{\alpha}\cap
M[F]\right).$ We have the following restrictions on $\eta_{\alpha}$:

$\displaystyle\eta_{\alpha}\left(\sum_{l=1}^{r}w(m_{l},n_{l})\right)$
$\displaystyle=\dim
k\left(\left(\bigoplus_{l=1}^{r}W^{m_{l},n_{l}}_{w(m_{l},n_{l})}\right)\cap
M_{\alpha}[F]\right)$ $\displaystyle=\dim
k\left(\bigoplus_{l=1}^{r}W^{m_{l},n_{l}}_{w(m_{l},n_{l})}\cap
W^{m_{l},n_{l}}_{n_{l}}\right)$
$\displaystyle=\sum_{l=1}^{r}\min\\{n_{l},w(m_{l},n_{l})\\}.$

By letting $w$ range over sufficiently many words, we obtain enough
restrictions to determine the non-decreasing function
$\eta_{\alpha}:\\{1,\ldots,2q\\}\to\\{1,\ldots,q\\}$ uniquely. This is because
$\eta_{\alpha}$ is determined by the canonical filtration of $M_{\alpha}$.
Finally, this function $\eta_{\alpha}$ gives rise to a permutation
$\omega_{\alpha}\in\mathbf{W}_{q}$ via the steps given in [9, 3.6].

######  Proposition 5.11.

Let $n_{1},\ldots,n_{r}$ be non-negative integers satisfying the following
conditions:

  1. (i)

$\sum_{l=1}^{r}n_{l}=q$,

  2. (ii)

$\operatorname{GCD}(n_{l},n_{r+1-l})=1$ for every $l\in\\{1,\ldots,r\\}$,

  3. (iii)

there exists $l\in\\{1,\ldots,r\\}$ such that $n_{l}+n_{r+1-l}\neq 2$.

Define the mod-$p$ Dieudonné module $M_{\alpha}:=\bigoplus
M_{n_{r+1-l},n_{l}}$ and let $\omega_{\alpha}\in\mathfrak{S}_{2q}$ be as
above. Then the Ekedahl-Oort stratum $\mathcal{A}_{\omega_{\alpha}}$ does not
intersect $\mathcal{A}_{q}^{ss}$.

###### Proof.

By conditions (i) and (ii), there exists a symmetric Newton polygon from
$(0,0)$ to $(2q,q)$ with slopes $\frac{n_{l}}{n_{l}+n_{r+1-l}}$ (possibly with
multiplicity greater than 1). By condition (iii), this Newton polygon has a
slope that is not $1/2$. By construction, $M_{\alpha}$ and $\omega_{\alpha}$
correspond to a minimal Ekedahl-Oort stratum contained in a Newton stratum
that is not supersingular. Thus it does not intersect $\mathcal{A}_{q}^{ss}$.
∎

######  Corollary 5.12.

Suppose $\omega_{u,v}=\omega_{\alpha}$ under the conditions of Proposition
5.11. Then the Ekedahl-Oort stratum of $\mathcal{M}(q-2,2)_{\gamma_{u,v}}$
does not intersect $\mathcal{M}(q-2,2)^{ss}.$

###### Proof.

This follows directly from Proposition 5.11, as (forgetting) the unitary
structure does not affect whether an abelian variety is supersingular or not.
∎

##  6\. Ekedahl-Oort Strata of $\mathcal{M}(3,2)$ and the Supersingular Locus

Recall our motivating question.

######  Question 1.1.

Which Ekedahl-Oort strata of $\mathcal{M}(q-a,a)$ intersect the supersingular
locus?

As discussed in the introduction, the lowest-dimensional unitary Shimura
variety without a complete answer to the above question is $\mathcal{M}(3,2)$.
This Shimura variety is also the lowest-dimensional unitary Shimura variety
that is not of “Coxeter type” (see [4]). Because $\mathcal{M}(3,2)$ is not of
Coxeter type, by [4] it is known that there must exist some Ekedahl-Oort
stratum that intersects the supersingular locus, but is not entirely contained
in the supersingular locus. In this sense, the interaction between the
Ekedahl-Oort strata and the supersingular locus is significantly more
complicated for $\mathcal{M}(3,2)$ than for previously studied unitary Shimura
varieties.

While the techniques explored in the previous sections (including closure
properties of the Ekedahl-Oort strata, the product maps, and the forgetful
map) can be used to study the intersection of Ekedahl-Oort strata with the
supersingular locus for unitary Shimura varieties of signature $(q-2,2)$ for
any $q\geqslant 2$, in this section we consider $\mathcal{M}(3,2)$, i.e.
$q=5$, as a concrete example.

Of the ten Ekedahl-Oort strata of $\mathcal{M}(3,2)$, all but two can be
studied with the methods developed so far; the remaining strata are those
indexed by $\gamma_{1,4}$ and $\gamma_{3,4}$. We will first show that these
strata do in fact intersect the supersingular locus, by directly constructing
a point in the intersection. Then, we will combine this approach with the
previous techniques to give a complete answer to Question 1.1 for the Shimura
variety $\mathcal{M}(3,2)$, in Theorem 6.7.

###  6.1. Explicit Construction of Points

In this section we explicitly construct $\mathbx k$-points in the
intersections $\mathcal{M}(3,2)_{\gamma_{1,4}}\cap\mathcal{M}(3,2)^{ss}$ and
$\mathcal{M}(3,2)_{\gamma_{3,4}}\cap\mathcal{M}(3,2)^{ss}$. To do this, we
will first use $p$-adic Dieudonné theory to construct the $p$-divisible groups
of these points; then we will use the $p$-divisible groups and Rapoport-Zink
uniformization to construct points of $\mathcal{M}(3,2)$.

Let $\breve{\mathbx{Z}}_{p}=W(\mathbx k)$ be the ring of Witt vectors of
$\mathbx k$, denote by $\mathrm{Frob}=W(\mathrm{Frob}_{\mathbx k})$ the lift
of $\mathrm{Frob}_{\mathbx k}$ to $\breve{\mathbx{Z}}_{p}$, and let
$\breve{\mathbx{Q}}_{p}=\breve{\mathbx{Z}}_{p}[\frac{1}{p}]$. Note that
$\breve{\mathbx{Q}}_{p}$ is isomorphic to the completion of the maximal
unramified extension of $\mathbx{Q}_{p}$, with ring of integers isomorphic to
$\breve{\mathbx{Z}}_{p}$, and that the residue field of
$\breve{\mathbx{Z}}_{p}$ is $\mathbx k$. Let $\varphi_{1}$, $\varphi_{2}$ be
the two embeddings of $\mathcal{O}_{K}$ into $\breve{\mathbx{Q}}_{p}$.

######  Definition 6.1.

For any scheme $S$ over $\mathbx k$, a _unitary $p$-divisible group of
signature $(3,2)$_ over $S$ is a triple $(X,\iota_{X},\lambda_{X})$, where:

  * •

$X$ is a $p$-divisible group over $S$.

  * •

$\iota_{X}:\mathcal{O}_{K}\otimes_{\mathbx{Z}}\mathbx{Z}_{p}\rightarrow\mathrm{End}(X)$
is an action satisfying the signature (3,2) condition, for all
$a\in\mathcal{O}_{K}$:

$\mathrm{charpol}(\iota(a)\mid\mathrm{Lie}(X))=(T-\varphi_{1}(a))^{3}(T-\varphi_{2}(a))^{2}\in\breve{\mathbx{Z}_{p}}[T].$

  * •

$\lambda_{X}:X\rightarrow X^{\vee}$ is a $p$-principal polarization, meeting
the following $\mathcal{O}_{K}$-linearity condition, for all
$a\in\mathcal{O}_{K}$:

$\lambda_{X}\circ\iota_{X}(a)=\iota_{X}(\overline{a})^{\vee}\circ\lambda_{X}.$

Over an algebraically closed field, we may study unitary $p$-divisible groups
linear-algebraically:

######  Definition 6.2.

A _unitary $p$-adic Dieudonné module of signature $(3,2)$_ over $\mathbx k$ is
a tuple $(M,M=M_{1}\oplus M_{2},F,V,\langle\cdot,\cdot\rangle)$, where:

  * •

$M$ is a free $\breve{\mathbx{Z}}_{p}$-module of rank 10,

  * •

$M=M_{1}\oplus M_{2}$ is a decomposition into rank-5 summands,

  * •

$F:M\rightarrow M$ is a $\mathrm{Frob}$-semilinear operator, $V:M\rightarrow
M$ is a $\mathrm{Frob}^{-1}$-semilinear operator, with $F\circ V=V\circ F=p$,

  * •

$\langle\cdot,\cdot\rangle$ is a perfect alternating
$\breve{\mathbx{Z}}_{p}$-bilinear pairing on M such that $\langle
Fx,y\rangle=\langle x,Vy\rangle^{\mathrm{Frob}},$ for all $x,y\in M$,

  * •

$\mathrm{dim}_{\mathbx k}(M_{1}/FM_{2})=3$ and $\mathrm{dim}_{\mathbx
k}(M_{2}/FM_{1})=2$,

and further $F$ and $V$ are homogeneous of degree 1 with respect to the
decomposition $M=M_{1}\oplus M_{2}$, and $M_{1}$ and $M_{2}$ are each totally
isotropic with respect to $\langle\cdot,\cdot\rangle$.

Using contravariant $p$-adic Dieudonné theory, there is an equivalence of
categories between unitary $p$-divisible groups of signature $(3,2)$ over
$\mathbx k$ and unitary $p$-adic Dieudonné module of signature $(3,2)$ over
$\mathbx k$. When no confusion is possible, we will drop the extra structure
and abbreviate unitary $p$-divisible groups simply as $X$ and unitary $p$-adic
Dieudonné modules simply as $M$.

######  Lemma 6.3.

Let $G_{\gamma_{1,4}}$be the $p$-torsion group scheme occurring as $p$-torsion
subgroup for points in the Ekedahl-Oort stratum
$\mathcal{M}(3,2)_{\gamma_{1,4}}$. There exists a supersingular unitary
$p$-divisible group $X_{1,4}$ of signature $(3,2)$ over $\mathbx k$ such that
$X_{1,4}[p]\cong G_{\gamma_{1,4}}$, respecting action and polarization.

###### Proof.

We aim to construct a supersingular unitary $p$-divisible group $X_{1,4}$ of
signature $(3,2)$ such that $X_{1,4}[p]\cong G_{\gamma_{1,4}}$. Using
contravariant $p$-adic Dieudonné theory, it suffices to construct a unitary
$p$-adic Dieudonné module $M$ of signature $(3,2)$ such that all of the slopes