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.

When the modular curve is replaced by a general Shimura variety of PEL type, the interaction between the Ekedahl-Oort and Newton stratifications is very subtle; for example, see the results in [1].

More precise statements can be made when one restricts attention to a smaller class of Shimura varieties. We study the interaction between these two stratifications for unitary Shimura varieties \(\mathcal{M}(q-b, b)\). These are moduli spaces of abelian varieties of dimension \(q\) with an action of an order in an imaginary quadratic field \(K\) that meets the “signature \((q-b, b)\)" condition. In addition to the main focus of her thesis, especially under the assumption of the relevant prime \(p\) being split in \(K\), Wooding [2] serves as an excellent source of background information on the Newton and Ekedahl-Oort stratifications of \(\mathcal{M}(q-b, b)\).

The unitary Shimura varieties of signature \((q,0)\) are zero-dimensional and those of signature \((q-1,1)\) have been extensively studied [3], [4], we therefore focus on unitary Shimura varieties of signature \((q-2,2)\). Unlike those of smaller signatures, the Shimura varieties \(\mathcal{M}(q-2,2)\) are in general not of Coxeter type (see [5] for details). For this reason, the interaction between Ekedahl-Oort and Newton strata is much more complicated than in previously studied cases, and is also likely a better reflection of the situation for general signature \((q-b,b)\).

In this paper we study three aspects of the Ekedahl-Oort stratification of \(\mathcal{M}(q-2,2)\), under the assumption that the relevant prime \(p\) is inert in \(K\), in order to develop tools for studying the interaction between the Ekedahl-Oort stratification and Newton stratification.

First, in 3, we study the topological closure relations among the Ekedahl-Oort strata. In 5 7, we prove that a number of topological closure relations hold. While topological closure relations between Ekedahl-Oort strata of Shimura varieties of PEL type have been previously studied [6], [7], we are able to express closure relations for \(\mathcal{M}(q-2,2)\) much more explicitly than is possible in the general setting.

Second, in 4, we relate \(\mathcal{M}(q-2,2)\) to simpler unitary Shimura varieties via the 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 14 15, we explicitly describe the effect of these product maps on the Ekedahl-Oort stratifications. These results give a solution to the analogue of a long-standing question of Moonen and Oort [8], when the Siegel modular variety is replaced by \(\mathcal{M}(q-2,2)\).

Third, in 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 action of the field \(K\) on the abelian varieties parameterized by \(\mathcal{M}(q-2,2)\).

More precisely, in 18, we concretely describe the effect of this map on the Ekedahl-Oort stratifications. As much more is known about the Ekedahl-Oort and Newton stratifications of \(\mathcal{A}_q\) than other Shimura varieties, this is an especially powerful tool.

These three aspects of the Ekedahl-Oort stratification can each be leveraged to illuminate the relationship between the Ekedahl-Oort and Newton stratifications of \(\mathcal{M}(q-2,2)\). As a running example, we apply these techniques to make some observations about the interaction between the Ekedahl-Oort stratification and the supersingular locus (the unique closed Newton stratum) of \(\mathcal{M}(q-2,2)\); with more care, our techniques are equally well-suited to the study of any Newton stratum. In the tradition of [4], we hope a better understanding of the interaction between the Ekedahl-Oort and Newton strata may be useful when studying more subtle aspects of the geometry of \(\mathcal{M}(q-2,2)\).

We apply the techniques developed in this article to the case of \(\mathcal{M}(3,2)\) in [9].

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.

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

2 Background↩︎

2.1 Unitary Shimura varieties↩︎

Unitary Shimura varieties are moduli spaces of abelian varieties equipped with extra structure, including an action of an order in an imaginary quadratic 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 an imaginary quadratic 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, where \(\mathcal{O}_K\) is the ring of integers of \(K\).

Let \(L\) be the reflex field associated to the PEL-datum \((K, \mathscr O_K \otimes_{\mathbx{Z} } \mathbx{Z}_{(p)}, *, V, (\cdot,\cdot),\Lambda, \boldsymbol{\mathsf{G}}, h)\); 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 \(C^p \subset \boldsymbol{\mathsf{G}}(\mathbx{A}_f^p)\). For \(C^p\) small enough, the construction of Kottwitz [10] 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 \(q\);

• \(\iota: \mathscr O_K \otimes_{\mathbx Z} \mathbx Z_{(p)}\to \mathrm{End}(A)\otimes_\mathbx Z\mathbx Z_{(p)}\) is a nonzero homomorphism of \(\mathbx Z_{(p)}\)-algebras such that the Rosati involution on \(\mathrm{End}(A)\otimes_\mathbx Z\mathbx Z_{(p)}\) induces the involution \({}^*\) on \(\mathscr O_K \otimes_{\mathbx Z} \mathbx Z_{(p)}\);

• \(\lambda\) is a one-dimensional \(\mathbx Q\)-subspace of \(\mathrm{Hom}(A, A^\vee)\otimes_\mathbx Z\mathbx Q\) that contains a \(p\)-principal \(\mathscr O_K \otimes_{\mathbx Z} \mathbx Z_{(p)}\)-linear polarization;

• \(\xi: \mathrm{H}_1(A, \mathbx{A}_f^p) \to V \otimes_\mathbx Q\mathbx{A}_f^p \;\mathrm{mod } \;C^p\) 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 \((A', \iota', \lambda', \xi')\) are isomorphic if there exists a prime-to-\(p\) isogeny from \(A\) to \(A'\), commuting with the action of \(\mathscr O_K \otimes_{\mathbx Z} \mathbx Z_{(p)}\), mapping \(\xi\) to \(\xi'\) and \(\lambda\) to \(\lambda'\).

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 [1].

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 \((A', \iota', \lambda', \xi')\) 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 \((A'[p], \iota', \lambda')\), 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 \((A', \iota', \lambda', \xi')\) 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 \((A'[p^\infty], \iota', \lambda')\), 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 [11] relate 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 \(q = a+b\) 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 [12] 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 \[\begin{align} \label{eq:SnInversions} \mathrm{Inv}(w):=\{(i,\, j)\,|\, i<j \text{ and } w(i)>w(j)\}. \end{align}\tag{1}\]

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 [12] that every right coset of \(W_J\setminus W\) contains a unique minimal-length coset representative. Let \({}^JW\) be the collection of such minimal-length coset representatives.

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 [11]:

We now recall some properties of Moonen’s construction. (See also [2].) 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 [11] 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 1.

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 [11], \(\lambda_w\) 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 \(\dim_\mathbx k(W_i)=i\).

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 \[\label{eqn:omegafromDD} \omega_N(j_l)=l \text{ and } \omega_N(i_m)=b+m.\tag{2}\]

By construction, \(N\) is the standard object corresponding to the permutation \(\omega_N\).

In 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 \(\boldsymbol{W}(q-2,2)\) and recall the Bruhat partial ordering of these elements. Following that, we discuss a generalization of the Bruhat order due to He [7] 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 & Bruhat Order↩︎

In this section, we explicitly describe the set \(\mathbf{W}(a,b)\) and the Bruhat order between its elements for a general signature \((a,b)\). We end the section by summarizing our results for the case \((a,b) = (q-2,2)\).

The result in [12] shows that \[\mathbf{W}(a,b) = \left\{\gamma \in W\ \middle|\ \gamma^{-1}(1) < \cdots < \gamma^{-1}(b) \text{ and } \gamma^{-1}(b + 1) < \cdots < \gamma^{-1}(a + b)\right\}.\] Observe that any \(\gamma \in \mathbf{W}(a,b)\) is uniquely determined by \(\underline{u} := (u_1,\ldots,u_b)=(\gamma^{-1}(1),\ldots,\gamma^{-1}(b))\). Therefore, by \(\gamma_{\underline{u}}\), we mean the unique permutation in \(W\) such that \(\gamma_{\underline{u}}(u_i) = i\) and \(\gamma_{\underline{u}}(z_j) = b + j\), where \(z_1 < \cdots < z_a\) is the complement of the set \(\left\{u_1,\ldots,u_b\right\}\) in \(\left\{1,\ldots,a+b\right\}\). Necessarily, we have \[\begin{align} \label{eq:coset-rep} \mathbf{W}(a,b) = \left\{\gamma_{\underline{u}}\ \middle|\ \underline{u} = (u_1,\ldots,u_b) \text{ such that }1\leqslant u_1 < \cdots < u_b \leqslant a + b\right\}. \end{align}\tag{3}\] In particular, \(\gamma_{\underline{u}}\) is an order-preserving bijection on both \(\left\{u_1,\ldots,u_b\right\}\) and its complement.

We can make the permutation \(\gamma_{\underline{u}}\) explicit.

By [12], the Bruhat order on \(W\) induces a partial order on \(\mathbf{W}(a,b)\), also called the Bruhat order and denoted \(\leqslant\). This can described explicitly, and we record this as the following proposition.

We specialize our descriptions to the case \((a,b) = (q-2,2)\) below.

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.

3.3 The Closure Order↩︎

3.3.1 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 [7] 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. 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 [1], [2], [6], [7], [13] 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 \(w\in W\), \(\psi(w)=w_0ww_0\), where \(w_0\) is the unique maximal element in \(W\).

As in 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 \(J':={}^{w_0}\psi(J)\) denote the set \(w_0 \psi(J) w_0\). In the notation of 2.3, \({}^{J'}W^{\psi(J)}\) denotes the collection of minimal-length coset representatives for \(W_{J'}\big\backslash W\big\slash\psi(J)\).

The definition of the Closure order involves the element \(x\) of minimal length in the double coset \(W_{J'} w_0 W_{\psi(J)}\). One can show that this element satisfies \[x=w_{0,J'} w_0 = w_0 w_{0,\psi(J)},\] and so \(x\) is the unique maximal length element amongst the minimal length coset representatives \({}^{J'} W^{\psi(J)}\).

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

Using the fact that \(\psi(h)=w_0hw_0\), we have \[x\psi(h^{-1})x^{-1}=(w_0w_{0,\psi(J)})(w_0{ h^{-1}}w_0)(w_{0,\psi(J)}w_0)= w_{0,J}h^{-1}w_{0,J},\] and so Inequality 4 can be rewritten as \[\label{eq:HeOrderDef2} hww_{0,J}h^{-1}w_{0,J} \leqslant w'.\tag{5}\] Note that the Bruhat order is the special case of the Closure order where Inequality 4 is satisfied for \(h\) equal to the identity in \(W_{(q-2,2)}\).

In 3.3 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 4 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 3 to the Closure order.

The Closure order is of interest because it captures topological closure relations among the Ekedahl-Oort strata. Let \(\overline{\mathcal{M}(q-2,2)}_{\gamma_{u',v'}}\) denote the topological closure of the stratum \(\mathcal{M}(q-2,2)_{\gamma_{u',v'}}\). By [1], \[\overline{\mathcal{M}(q-2,2)}_{\gamma_{u',v'}} = \!\!\!\!\!\!\bigcup_{\gamma_{u,v} \operatorname{\preccurlyeq}\gamma_{u',v'} } \!\!\!\!\!\!\mathcal{M}(q-2,2)_{\gamma_{{u,v}}}.\]

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

3.3.2 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 5 and 7.

We define an action on elements \(w\in\mathbf{W}(q-2,2)\) by elements \(h\in W_{(q-2,2)}\) by \[\label{eqn:He95action} h\bullet w:= hw w_{0,J}h^{-1}w_{0,J},\tag{6}\] which is the expression that appears on the left side of Inequality 5 . 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.

The following theorem gives the first set of closure 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, 5, 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 5, and we split these into separate lemmata. We first compute the action by \(s_j\) on \(\gamma^{-1}_{j+1,q+1-j}\) and \(\gamma^{-1}_{q-j,j+1}\).

We now prove the following result for the lengths of the \(\tau_j\)’s.

The following is the final ingredient needed in order to prove 5.

These results combine to give a proof of 5.

The following theorem gives the second set of closure relations.

As we did for 5, we will prove the result, 7, 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 7, 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 \(\gamma^{-1}_{q-j-i,j+1-i}\).

9 and 10 lead to the following corollary.

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

These results combine to give a proof of 7.

3.3.3 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.

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 3.

• 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 5 and 7.

3.3.4 Concluding remarks on the closure order↩︎

Based on experiments done in Sage [14], we believe that the relations in Theorems 5 and 7 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.

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 6 lead us to believe that Conjecture 9 is true and that further work in this area would be fruitful.

4 Product Maps: Relations to Other Unitary Shimura Varieties↩︎

In this section, we begin our study of the interaction between Ekedahl-Oort strata and Newton strata 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 14 15, we explicitly describe these induced product maps on the index sets \(\boldsymbol{W}(a,b)\) 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 \[\begin{align} \Phi: \mathcal{M}(m_1,n_1) \times \mathcal{M}(m_2,n_2) &\to \mathcal{M}(m_1+m_2, n_1 + n_2) \\ ((A_1, \lambda_1, \iota_1, \xi_1), (A_2, \lambda_2, \iota_2, \xi_2)) &\mapsto (A_1 \times A_2, \lambda_1 \times \lambda_2, \iota_1 \times \iota_2, \xi_1 \times \xi_2), \end{align}\] 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 4.2 and the latter in 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 [11], 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 \[\gamma_{a+1} := (1,2,\ldots,a+1),\] for \(0 \leqslant a \leqslant m\). We therefore have \[\mathbf{W}(m,1) = \{ \gamma_{a+1} \; | \; 0 \leqslant a \leqslant m\}.\] Note that the length of \(\gamma_{a+1}\) is \(a\), and so the corresponding stratum \(\mathcal{M}(m,1)_{\gamma_{a+1}}\) has dimension \(a\).

In [4], 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: \[\mathcal{M}(m,1)_{\gamma_{a+1}} \subseteq \mathcal{M}(m,1)^{ss}.\] On the other hand, if \(a > m/2\), then we have disjointedness: \[\mathcal{M}(m,1)_{\gamma_{a+1}} \cap \mathcal{M}(m,1)^{ss} = \varnothing.\] In particular, from here on, given a Dieudonné module \(M\) arising as the standard object for an Ekedahl-Oort stratum \(\mathcal{M}(m,1)_{\gamma_{a+1}}\), 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 \(\varphi(\gamma_{a+1}, \gamma_{b+1})\in \mathbf{W}(m+n, 2)\). Recall the bijection between \(\mathbf{W}(a,b)\) and Ekedahl-Oort strata of \(\mathcal{M}(a,b)\) outlined below 1. Let the Dieudonné module \(M\) be the standard object in the Ekedahl-Oort stratum \(\mathcal{M}(m,1)_{\gamma_{a+1}}\). 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 1.

Similarly, we let \(\gamma_{b+1} \in \mathbf{W}(n,1)\) 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 1. We do this by computing the function \[\eta^L_1(j)= \dim (E_{1,j} \cap L[F] ).\] As noted in 2.3, it is sufficient to compute \(\eta_1\) in our work, and so this determines the permutation \(\gamma_{u,v} = \varphi(\gamma_{a+1},\gamma_{b+1})\) 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 2 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 11 12 13 (and summarized in 14). In 4.2.2, the necessary parts of the canonical filtration of \(L\) are constructed. In 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 \[\Phi \left( \mathcal{M}(m,1)_{\gamma_{a+1}} \times \mathcal{M}(n,1)_{\gamma_{b+1}} \right) \subseteq \mathcal{M}(m+n,2)_{\gamma_{u,v}}.\]

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 \(\mathcal{M}(m,1)_{\gamma_{a+1}}\). [11] provides the action of \(F\) and \(V\) on a basis \(\{ e_{i,j} \; | \; 1\leqslant i \leqslant 2 \; , \; 1 \leqslant j \leqslant q \}\) of \(M\). \[\begin{align} &F(e_{1,j}) = \begin{cases} e_{2,j} &if 1 \leqslant j \leqslant a, \\ 0 &if j=a+1, \\ e_{2,j-1} &if j>a+1, \end{cases} &V(e_{1,j}) &= \begin{cases} 0 &if j=1, \\ e_{2,j-1} &if 1 < j \leqslant m+1-a, \\ e_{2,j} &if j> m+1-a , \end{cases} \\ &F(e_{2,j}) = \begin{cases} 0 &if j\neq m+1-a, \\ e_{1,1} &if j=m+1-a,\end{cases} &V(e_{2,j}) &= \begin{cases} 0 &if j\neq m+1, \\ e_{1,a+1} &if j=m+1. \end{cases} \end{align}\]

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

The same result applies to the standard object \(N\) of the Ekedahl-Oort stratum \(\mathcal{M}(n,1)_{\gamma_{b+1}}\), 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 \(s_2:=\max\{a,m+1-a\}\).

We now prove a corollary that allows us to ‘move up’ from \(C_{1,0}\) to \(C_{1,s_1}\), step by step.

Similarly we can ‘move down’ from \(C_{1,m+1}\) to \(C_{1,s_2}\), as the following corollary records.

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 \(u_2:=\max\{b,n+1-b\}\).

In some cases, 14 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.

After constructing \(E_{1,1}\) and \(E_{1,q-1}\) if necessary, under the conditions of 15, we use the proof of 14 to construct \[\begin{align} (V^{-1}F)^c(E_{1,1}) \cap L_1 &= C_{1,\min\{1+c,s_1\}} \oplus D_{1,\min \{c,u_1\}} = E_{1,\min\{1+c,s_1\} + \min\{c,u_1\}}, \text{ and } \tag{13} \\ (V^{-1}F)^c(E_{1,q-1}) \cap L_1 &= C_{1,\max\{m-c,s_2\}} \oplus D_{1,\max\{n+1-c,u_2\}} = E_{1,\max\{m-c,s_2\} + \max\{n+1-c,u_2\}}. \tag{14} \end{align}\]

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 \(\varphi(\gamma_{a+1}, \gamma_{b+1})\), where \(\mathcal{M}(m,1)_{\gamma_{a+1}}\) is the unique Ekedahl-Oort stratum of dimension \(a\) and likewise for \(\gamma_{b+1}\). Let \(M\) be a standard object for the stratum \(\gamma_{a+1}\) and let \(N\) be a standard object for the stratum \(\gamma_{b+1}\). 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 \(a \geqslant b\).

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 \(m-a \leqslant n-b\).

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.

11, 12 and 13 are summarized in 14.

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

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 \(\mathcal{M}(n,0)_{\mathrm{id}}= \mathcal{M}(n,0)\) 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 \(\gamma_{y,s} := \varphi(\gamma_{u,v} , \mathrm{id})\), such that \[\Phi(\mathcal{M}(m,2)_{\gamma_{u,v}} \; , \mathcal{M}(n,0)_{\mathrm{id}}) \subseteq \mathcal{M}(q-2,2)_{\gamma_{y,s}}.\] 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.

Our method for computing \(\varphi(\gamma_{u,v}, \mathrm{id})\) is essentially the same as the method used in 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 .

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: \[\label{eqssKraft} F(N_1)=V(N_1)=N_2 \text{ and } F(N_2)=V(N_2)=0.\tag{15}\]

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

A variant of ‘moving up’ can be used to construct parts of the canonical filtration of \(L\). To record this, define the multiset \(S_{\mathrm{up}}=\{u, v, q_M-v+2, q_M-u+1\}\) and let \(r_2:= \min (S_{\mathrm{up}}\setminus \left\{\min S_{\mathrm{up}}\right\})\), i.e., the second element from the multiset when the elements are listed in increasing order.

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 \(r_3 : = \max (S_{\mathrm{down}}\setminus \left\{\max S_{\mathrm{down}}\right\})\), i.e., the second element from the multiset when the elements are listed in decreasing order.

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

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 \[\begin{align} \eta^M_1(j)&=\dim (C_{1,j} \cap M[F]), \text{ and}\\ \eta^L_1(j) &= \dim (E_{1,j} \cap L[F]). \end{align}\] Recall that \(r_2 = \min(S_{\text{up}}) \setminus \left\{\min S_{\text{up}}\right\}\), the second smallest element of the multiset \(S_{\mathrm{up}} := \{u,v,q_M-v+2,q_M-u+1\}\). We now completely describe the Ekedahl-Oort stratum of \(L\).

This theorem has the following implication for supersingular loci.

Note that 6 allows one to gain information about some Ekedahl-Oort strata that intersect \(\mathcal{M}(q-2,2)^{ss}\), 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 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 4.2.1.

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. The results of that section, particularly 5 6, allow us to draw conclusions about the interaction between Ekedahl-Oort strata and Newton strata 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 \[\begin{align} \Psi: \mathcal{M}(q-2,2) &\to \mathcal{A}_q \\ (A, \lambda, \iota, \xi) &\mapsto (A,\lambda,\xi). \end{align}\] 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 [11], 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 1 and 2, we construct the standard object \(M\) corresponding to the Ekedahl-Oort stratum \(\mathcal{M}(q-2,2)_{\gamma_{u,v}}\). Upon forgetting the \(\mathbx{F}_{p^2}\)-action of \(M\), we observe that \(M\) is the Dieudonné module of an Ekedahl-Oort stratum of \(\mathcal{A}_q\). By [11], 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 \[\label{eqomegaduality} \omega_{u,v}(i) + \omega_{u,v}(2q+1-i) = 2q+1.\tag{24}\]

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 \(\eta\). 2 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 17, 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 18, 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 \(\mathcal{M}(q-2,2)^{ss}.\) 7 records the Ekedahl-Oort strata of \(\mathcal{M}(q-2,2)\) that occur 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 \(\mathcal{M}(q-2,2)^{ss}\). 20 and 9 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 2 (or its consequence 16) 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 15, 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, 2 yields \[\begin{align} W_q &= F(M)= \langle e_{1,1}, e_{1,2} \rangle \oplus \langle e_{2,j} \; | \; j \leqslant q-2 \rangle =C_{1,2} \oplus C_{2,q-2}, \nonumber \\ M[F] &= M[F]_1 \oplus M[F]_2 = \langle e_{1,u}, e_{1,v} \rangle \oplus \langle e_{2,j} \; | \; j \notin \{q+1-v,q+1-u\} \rangle. \label{eq:M[F]} \end{align}\tag{25}\]

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 17. 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 24 .

5.4 Ekedahl-Oort strata contained in the supersingular locus↩︎

Using the main result of [16], we pinpoint exactly which \(\omega_{u,v}\) represent an Ekedahl-Oort stratum of \(\mathcal{A}_q\) that is contained in \(\mathcal{A}_q^{ss}\).

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 \(\omega_{u,v}\).

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

5.6 Minimal Ekedahl-Oort strata of non-supersingular Newton strata↩︎

There is another method that shows when Ekedahl-Oort strata are disjoint from a given Newton stratum. We focus on interactions with the supersingular locus, though the techniques can be applied more generally. This method is based on minimal Ekedahl-Oort strata, which are completely contained in a non-supersingular Newton stratum.

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 [1]).

Luckily things are different for \(\mathcal{A}_q\). Oort proved in [17], [18] that each Newton stratum of \(\mathcal{A}_q\) contains a unique minimal Ekedahl-Oort stratum. Given a Newton polygon, results of [19] 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 18 to determine Ekedahl-Oort strata of \(\mathcal{M}(q-2,2)\) that do not intersect \(\mathcal{M}(q-2,2)^{ss}\).

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 [17] that the unique minimal Ekedahl-Oort stratum in \(\alpha\) has the Dieudonné module \[\label{eq:Malpha} M_\alpha := \bigoplus_{l=1}^r M_{m_l, n_l}.\tag{29}\] The Dieudonné module \(M_\alpha\) corresponds to a permutation \(\omega_\alpha \in \mathbf{W}_q\), which we determine in the results below.

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 \(w(m,n)=m+n\);

• \((Fw)(m,n) = \max\{0,w(m,n)-n\}\);

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

Recall the definition of the mod-\(p\) Dieudonné module \(M_\alpha\) from Equation 29 . Combining this with 8 gives, for any word \(w\) in the alphabet \(\{F,V^{-1}\}\), the formula \[w(M_\alpha) = w \left( \bigoplus_{l=1}^r M_{m_l,n_l} \right) = \bigoplus_{l=1}^r w(M_{m_l,n_l}) = \bigoplus_{l=1}^r W^{m_l,n_l}_{w(m_l,n_l)}.\] Now, let \(W^\alpha_\bullet\) be a final filtration of \(M_\alpha\) and define the function \(\eta_\alpha(j) = \dim \left( W^\alpha_j \cap M_\alpha[F] \right).\) We have the following restrictions on \(\eta_\alpha\): \[\begin{align} \eta_\alpha\left(\sum_{l=1}^r w(m_l,n_l)\right) &= \dim_\mathbx k\left( \left(\bigoplus_{l=1}^r W^{m_l,n_l}_{w(m_l,n_l)} \right) \cap M_\alpha [F] \right) \\ &= \dim_\mathbx k\left( \bigoplus_{l=1}^r W^{m_l,n_l}_{w(m_l,n_l)} \cap W^{m_l,n_l}_{n_l} \right) \\ &= \sum_{l=1}^r \min \{n_l, w(m_l,n_l) \}. \end{align}\] 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 [11].

References↩︎

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1. In the notation of [7], Inequality 4 actually implies \(wx\operatorname{\preccurlyeq}w'x\). In our work, as in [1], we drop the \(x\) from the notation and simply write \(w\operatorname{\preccurlyeq}w'\).↩︎