ON THE SUPERSINGULAR LOCUS IN HILBERT-BLUMENTHAL 4-FOLDS

CHIA-FU YU

Abstract. We study the supersingular locus of Hilbert-Blumenthal four-folds modulo p when p is inert in the totally real field. The dimension, local moduli spaces, number of the irreducible components, and a description of intersections of these components are given. We also show that each irreducible component is a smooth algebraic stack which is a quotient of a ruled surface over P^{1} by a finite group.

Contents

1. Introduction 1 2. Notations and definitions 3 3. a-number stratification 4 4. Normal forms 6 5. Cayley-Hamilton for a = 1 9 6. Preliminary results for g=4 12 7. Canonical isogenies 13 8. Construction of models 15 9. Superspecial points as class numbers 23 10. Cayley-Hamilton for higher a-numbers 26 11. Local moduli spaces I 26 12. Intersections 28 13. Local moduli spaces II 31 14. Appendix: Deformation via the Hecke correspondence 33 References 35

1. Introduction

We attempt to extend the work of K.-Z. Li and F. Oort [17] to Hilbert-Blumenthal modular varieties, and would like to answer the following questions:

1. What is the dimension of the supersingular locus?

2. How many irreducible components does the supersingular locus have?

3. Does the generic point of any irreducible component have a-number one? We limit ourselves to the case where the moduli space has good reduction. This happens

exactly when the rational prime p is unramified in the totally real field F [6, Thm 2.2]. In particular, we assume that the prime p is inert in F. This assumption does not lose the generality for answering the question 1 and 3 due to the theorem of Serre-Tate. For four-folds we give a fairly complete description of the supersingular locus. In particular, we obtain the following results:

1. The dimension of the supersingular locus is one half of the total space.

2. The number of the irreducible components is related to a class number.

3. There are some components whose generic points have a-number two. The answer 3. gives rise to a natural question: what is the “generic” a-number for Hilbert-Blumenthal varieties analogue to “a = 1” in Siegel modular varieties? We give a simple combinatorial description for generic a-numbers (called spaced in [9]), and give some evidence, see Appendix.

The main tools are the Cartier-Dieudonné theory [38], deformations via displays [22, 23, 39], the method of Cayley-Hamilton from Oort’s paper [29], and canonical isogenies (see Section 7, cf. [17, 16]).

Let F be a totally real field and let O_{F} be its ring of integers. Let A_{g}^{O_{F}} be the moduli space of principally polarized O_{F} -abelian varieties of dimension g = [F : ℚ]. A polarized O_{F} -abelian variety, (A, λ, ι), consists of an abelian variety A, a polarization λ : A → A^{t} and an embedding ι : O_{F} → End(A) such that λ ι(a) = ι(a)^{t}λ, ∀ a ∈ O_{F}. Such a polarization is called O_{F} -linear. Let p be a rational prime inert in F and k an algebraically closed field of characteristic p. For n ≥ 3, (n, p) = 1, let A_{g,n}^{O_{F}} denote the moduli space of principally polarized O_{F} -abelian varieties of dimension g with symplectic n-level structure (with a choice of an n-th root of unity). It is a fine moduli scheme over Spec ℤ[ζ_{n}, \frac{1}{n}] and each fibre is geometrically connected. Denote by 𝒮_{n} the supersingular locus of the fine moduli scheme A_{g,n}^{O_{F}} ⊗_{ℤ[ζ_{n}, \frac{1}{n}]} k. It is a Zariski closed subset of A_{g,n}^{O_{F}} ⊗_{ℤ[ζ_{n}, \frac{1}{n}]} k, therefore we can regard it as a reduced closed subscheme of A_{g,n}^{O_{F}} ⊗ k. The Galois group H_{n} of A_{g,n}^{O_{F}} ⊗ k → A_{g}^{O_{F}} ⊗ k preserves the close subscheme 𝒮_{n}. Let 𝒮 be the algebraic stack [𝒮_{n}/H_{n}], the supersingular locus of A_{g}^{O_{F}} ⊗ k. This definition of 𝒮 does not depend on the choice of n.

Assume that g = 4. Let (A, λ, ι) be a superspecial polarized O_{F} -abelian variety over k such that the polarization λ has the degree p^{4} the type (p, 1, p, 1) (see the definition in Section 2). An abelian variety is superspecial if it is isomorphic to a product of supersingular elliptic curves. The automorphism group Aut_{O_{F}} (A, λ), consisting of automorphisms which preserve the O_{F} -action and the polarization, is finite. We regard G = Aut_{O_{F}} (A, λ) as an algebraic group over Spec ℤ. For any commutative ring R, the group of its R-points is

G(R) = {ϕ ∈ (End_{O_{F}} (A) ⊗ R)^{×}|ϕ′ϕ = 1},

where the map ϕ → ϕ′ is the Rosati involution induced by λ. We summarize our main results in the theorem below; see Proposition 6.3, Theorem 9.7, 11.1, 11.2, and Proposition 8.2 and 8.10:

Main Theorem.

(1) The supersingular locus 𝒮 is equi-dimensional and dim 𝒮 = 2.

(2) The supersingular locus 𝒮 has 6h irreducible components, where h is the class number of G.

(3) The formal completion 𝒮̂_{x} at a superspecial point x is isomorphic to Spf k[[t_{0}, t_{1}, t_{2}, t_{3}]]/I, where I = ∩_{i=0}^{3}(t_{i}, t_{i+1} + t_{i−1}^{p^{2}}) ∩ (t_{0}, t_{2}) ∩ (t_{1}, t_{3}).

(4) Every irreducible component of 𝒮 is smooth as an algebraic stack.

(5) Every irreducible component is isomorphic to a finite quotient of X, where X is either P^{1} × P^{1} or a P^{1}-fibration over P^{1}.

The following is an outline of the contents of this paper. In Section 2 we list the notations used in our work. In Section 3 we introduce the stratification induced by a-numbers and describe the change of the strata under the specialization and generalization. The analysis of the a-number stratification on the supersingular locus, 𝒮, is the key step in studying 𝒮. A classification of Dieudonné modules in question is given in Section 4. In Section 5, we introduce the method of “Cayley-Hamilton” due to F. Oort, and use it to compute the local moduli at the point of usual a-number one. In Section 6, we combine the slope stratification and the a-number stratification, and obtain the dimension of each slope stratum. An result states that the generic point of some component of 𝒮 has usual a-number two, and not just one. In order to describe the geometry of 𝒮 globally, we attach to each component of 𝒮 with some isogeny level structure, which we call canonical isogenies. This idea of that may be traced back to the Manin’s thesis [18]. An explicit form was constructed first in the work of Oda and Oort [25]. We follow Li and Oort [17] closely to construct such a model in Section 7–9. Some improvement of the Cayley-Hamilton method in the Hilbert-Blumenthal case is made in Section 10. This improvement, together with the knowledge of the a-number stratification, helps us to compute all the local moduli spaces of the supersingular locus. We give an explicit recipe of computing these local moduli spaces in Section 11. The intersections of the irreducible components near a point can be obtained directly from their local moduli spaces. We use the result of Section 9 to describe the intersections globally. Local and global descriptions of the intersections are discussed in Section 12. Some computation of local moduli is placed in Section 13 for interested readers. In Appendix, we show that the points of generic a-numbers form an open dense subset in each slope stratum.

The setting of this paper is for good reduction of Hilbert-Blumenthal modular varieties, although explicit information is given only for g = 4. We will extend our results to higher dimensions in our sequent paper [37]. The paper grows out of the author’s thesis at University of Pennsylvania, where the main results were done in 1998. This work contains some results which were done in the work by E. Goren and F. Oort [9]. We don’t try to eliminate our original proofs of the results they have obtained first. Rather we cite their results in an appropriate place.

Acknowledgments. The author would like express his deep gratitude to his thesis advisor Ching-Li Chai for many years of guidance and encouragement. This work would have never been done without his discussions and suggestions. My intelligent debts to K.-Z. Li and Frans Oort are too obvious. It is a great pleasure to thank them for stimulating discussions and encouragement. The revision of this paper was done during my staying at Max-Planck-Institut für Mathematik in Bonn. I would like to thank the Institute for the kind hospitality and the good working condition.

2. Notations and definitions

In the paper, we will fix a prime p and assume that p is inert in F, where F is a fixed totally field. Let f be the residue degree of p, hence f = g = dim A. Let k be a perfect field containing 𝔽_{q} (q = p^{f}), W := W (k) be the ring of Witt vectors, and σ be the absolute Frobenius morphism on W. We label the embeddings Hom(O_{F} , W) = {σ_{0}, σ_{1}, …, σ_{f−1}} in a way that σ_{i+1} = σσ_{i} for i ∈ ℤ/fℤ. For a polarized O_{F} -abelian variety (A, λ, ι) over k, the covariant Dieudonné module M := M(A) of A has the following properties:

• M admits a ℤ/fℤ-grading: M = M^{0} ⊕ M^{1} ⊕ ⋯ ⊕ M^{f−1}, where M^{i} := { x ∈ M | a · x = σ_{i}(a)x } is a W -module of rank 2. We make the convention that M^{f} := M^{0}, and M^{−1} := M^{f−1}.

• the polarization induces a non-degenerate alternating pairing on M

⟨ , ⟩ : M × M → W

which is non-degenerate when restricted to M^{i} and ⟨M^{i}, M^{j}⟩ = 0 if i ≠ j.

• The operators F and V shift by degree 1 and degree −1, respectively. For each 0 ≤ i < f, there exists a basis X_{i}, Y_{i} ∈ M^{i} such that ⟨X_{i}, Y_{i}⟩ = p^{n_{i}}. The number n_{i} does not depend on the choice of bases. The polarization type of (A, λ, ι) is defined to be (p^{n_{0}}, p^{n_{1}}, …, p^{n_{f−1}}).

If L is a k-module or W -module equipped linear O_{F} -action, then the i-th component of L we denote by

L^{i} := { x ∈ L | a · x = σ_{i}(a)x }.

We define the a-number of an O_{F} -abelian variety A to be

a(A) = a(M) := (a_{0}(M), a_{1}(M), …, a_{f−1}(M)), a_{i}(M) := dim_{k}(M/(F, V)M)^{i},

where M is the covariant Dieudonné module of A.

If the Lie algebra condition holds for (A, λ, ι), i.e. Lie(A) is a free O_{F} ⊗ k-module, then each a_{i} = 0, or 1 since a_{i} ≤ dim Lie(A)^{i} = 1. In this case we define its a-index to be

τ_{A} = τ_{M} := {0 ≤ i < f | a_{i}(M) = 1 }.

When there is no confusion, write a := |a(M)| = |τ_{M}|. This is the usual a-number defined in [20].

An a-number (a_{0}, …, a_{f−1}) is called generic if a_{i}a_{i+1} = 0, for all i ∈ ℤ/fℤ, and we define similarly for τ if there are no consecutive integers in τ. This is called spaced in [9]. We say τ_{1} is higher than τ_{2} if τ_{1} ⊃ τ_{2}, and similarly for a-numbers. We will see that the a-indices induce a stratification of our moduli space, and the type of the stratum jumps up under the specialization.

In Section 3 we give the definition of the substack M_{τ} of M := A_{g}^{O_{F}} ⊗ 𝔽_{p} and we often identify the notations M\underline{a} and M_{τ}, e.g. M_{1010} = M_{\{0,2\}} = M_{0,2}. Although we are working on algebraic stacks, sometimes it is convenient to adapt the notation of moduli functors: M_{τ}(k) means the objects in M_{τ} over k.

Let 𝒮_{τ} denote the intersection of 𝒮 and M_{τ}, and let 𝒮̄_{τ} denote the Zariski closure of 𝒮_{τ} in 𝒮.

Let (A_{1}, λ_{1}, ι_{1}) and (A_{2}, λ_{2}, ι_{2}) be polarized O_{F} -abelian varieties. An isogeny ϕ : A_{1} → A_{2} between O_{F} -abelian varieties is O_{F} -linear if ι_{2}(a)ϕ = ϕι_{1}(a), ∀ a ∈ O_{F}. Call such an isogeny O_{F} -isogeny. By definition, the polarization of an polarized O_{F} -abelian variety is O_{F} -linear. An isogeny ϕ : A_{1} → A_{2} is polarized O_{F} -linear if it is O_{F} -linear and ϕ^{∗}λ_{2} = λ_{1}. Let R be a commutative ring. An O_{F} -isogeny ϕ : A_{1} → A_{2} is R-polarized if ϕ^{∗}λ_{2} = a λ_{1} for some a ∈ R^{×}.

In Section 7 we also use the classical contravariant Dieudonné modules, denoted by M^{∗}(A). We apologize for this inconsistence and any confusion it may cause.

We often write x^{(n)} for x^{σ^{n}} for short.

3. a-number stratification

In this section we study a stratification induced by the a-numbers of abelian varieties, which is the same as the one induced by their p-torsion subgroups, see [30, Thm 15.6]. This stratification is also studied in a joint paper of Goren and Oort [9], now called the Ekedahl-Oort stratification. The Ekedahl-Oort stratification is finer than a-number stratification for other modular varieties in general.

Let M := A_{g}^{O_{F}} ⊗𝔽_{p}. For any non-empty subset τ of \{0, 1, …, f − 1\}. Define a substack \overline{M}_{τ} over 𝔽_{q} whose objects are (A/S, λ, ι) ∈ Ob(M_{S}) such that the morphism F : Lie(A^{t})^{(p),i} → Lie(A^{t})^{i} is zero for all i ∈ τ. It is clear that we have the following properties

(1) \overline{M}_{τ} is representable by a closed subscheme of M.

(2) \overline{M}_{τ_{1}} is closed in \overline{M}_{τ_{2}} if τ_{1} ⊃ τ_{2}.

(3) The scheme-theoretic intersection of \overline{M}_{τ_{1}} and \overline{M}_{τ_{2}} is \overline{M}_{τ_{1}∪τ_{2}}

(4) \overline{M}_{τ} is proper over 𝔽_{q} since there is no intersection with the boundary of M.

Theorem 3.1. The inclusion \overline{M}_{τ} → M is a local complete intersection of codimension |τ| and \overline{M}_{τ}/𝔽_{q} is smooth.

Proof. This result was first obtained by Goren and Oort in a slightly different form, op. cit.. First, we will show that locally \overline{M}_{τ} is a closed subscheme of M defined by |τ| equations. It amounts to show that:

For any principally polarized O_{F} -abelian scheme (A/S, λ, ι), any s ∈ S there exists an affine neighborhood U of s and sections \{f_{i}\}_{i∈τ} of \underline{Hom}_{O_{F}} (H^{1}(A, O_{A})^{(p)}, H^{1}(A, O_{A})) over U such that U factors through \overline{M}_{τ} if and only if these sections vanish.

However, it is obvious because once one chooses an affine chart trivializing the coherent sheaves H^{1}(O_{A})^{(p)} and H^{1}(O_{A}), the Frobenius F is simply a multiplication by functions (f_{i}).

Now we will show that the tangent space of \overline{M}_{τ} at any point has dimension f − |τ|. This ensures that dim_{x} \overline{M}_{τ} = dim \overline{M}_{τ} everywhere. Let (A, λ, ι) ∈ \overline{M}_{τ}(k). Set M := H_{cris}^{1}(A/k), M^{(p)} := H_{cris}^{1}(A^{(p)}/k) = M ⊗_{k,F r} k. We have Hodge filtrations:

0 ⟶ F il^{(p)} ⟶ M^{(p)} ⟶ Q^{(p)} ⟶ 0
↓^{F}
0 ⟶ F il ⟶ M ⟶ Q ⟶ 0

Let {x_{i}, y_{i}} be a basis of M such that y_{i} ∈ F il^{i} ⊂ M^{i} and x_{i} ∈ M^{i}. One can choose them such that F (x_{i} ⊗ 1) = −y_{i} for all i ∈ τ. The differential forms x_{i}, y_{i} have unique liftings in M̃ := H_{cris}^{1}(A/R) with respect to the Gauss-Manin connection, where R is any PD-extension of k. We will denote the liftings by x_{i}, y_{i} again.

Let R := k[[\underline{t}]]/(\underline{t})^{2} be the first order universal deformation ring of (A, λ, ι). The first order universal deformation (Ã, λ̃, ι̃) gives rise to

0 ⟶ F̃ il^{(p)} ⟶ M̃^{(p)} ⟶ Q̃^{(p)} ⟶ 0
↓^{F}
0 ⟶ F̃ il ⟶ M̃ ⟶ Q̃ ⟶ 0

where F̃ il = span⟨y_{i}+t_{i}x_{i}⟩, F̃ il^{(p)} = span⟨(y_{i}+t_{i}x_{i})⊗1⟩ = span⟨y_{i}⊗1⟩, Q̃ = span⟨[x_{i}]⟩, Q̃^{(p)} = span⟨[x_{i} ⊗ 1]⟩.

For each i ∈ τ, F ([x_{i} ⊗ 1]) = [−y_{i}] = t_{i}[x_{i}]. It follows that t_{i} must vanish in order to lift (A, λ, ι) inside \overline{M}_{τ}. Therefore R/(t_{i})_{i∈τ} is the first order deformation ring in \overline{M}_{τ}, and the dimension of \overline{M}_{τ} is f − |τ|.

Second proof. Beginning with crystalline theory, we check the smoothness directly by using the deformation theory. Let I → R′ → R → k be artinian k-extensions such that I𝔪_{R′} = 0. We need to show that \overline{M}_{τ}(R′) → \overline{M}_{τ}(R) is surjective. Let (A/R, λ, ι) ∈ \overline{M}_{τ}(R) and set M := H_{cris}^{1}(A/R), F il := H^{0}(A, Ω_{A}). One can choose a basis x_{i}, y_{i} such that y_{i} ∈ F il^{i}, x_{i} ∈ M^{i} and F (x_{i} ⊗ 1) = −y_{i} ∀i ∈ τ. Set M′ := H_{cris}^{1}(A/R′) for R′ a PD-extension of R with the natural PD-structure. We lift the basis x_{i}, y_{i} over R′ by the Gauss-Manin connection and denote them by x_{i}, y_{i} again, but F (x_{i} ⊗ 1) = (−1 + a_{i})y_{i} + b_{i}x_{i} for some a_{i}, b_{i} ∈ I for each i ∈ τ.

Any deformation (A′, λ′, ι′) of (A/R, λ, ι) over R′ is given by F il′ = span⟨y_{i} + t_{i}x_{i}⟩ for some t_{i} ∈ I and all deformations arise in this way. Take t_{i} = −b_{i} ∀i ∈ τ. Then for each i ∈ τ,

\begin{aligned}F ([x_{i} ⊗ 1]) & = [(−1 + a_{i})y_{i} + b_{i}x_{i}] \\ & = [(−1 + a_{i})y_{i} + (−1 + a_{i})t_{i}x_{i}] + (t_{i} + b_{i})[x_{i}] = 0 \end{aligned}

where [− ⊗ 1], [−] denote the class in M′^{(p)}/F il′^{(p)}, M′/F il′ respectively. Thus we get a lift in \overline{M}_{τ}(R′).

Let M_{τ} be the substack of \overline{M}_{τ} whose k-points have a-index τ, with the reduced structure. Since \overline{M}_{τ} is reduced, M_{τ} is an open substack of \overline{M}_{τ} and its Zariski closure is \overline{M}_{τ}. This justifies the notation \overline{M}_{τ}. Using Cartier-Dieudonné theory, it is easy to deform any point in M_{τ_{1}} to a point in M_{τ_{2}} as long as τ_{1} is higher than τ_{2}. Conversely, it is true that we can specialize any point of lower a-index to a higher one. Indeed, we know that \overline{M}_{τ} is complete and that every stratum M_{τ} is quasi-affine, op. cit..

4. Normal forms

From now on and in the rest of this paper, the base field k is algebraically closed of characteristic p. In this section, we will give a normal form (Proposition 4.1) of the Dieudonné modules in question and its variant (Proposition 4.2). Let (A, λ, ι) be a principally polarized O_{F} -abelian variety over k, and let M be its covariant Dieudonné module with a-index τ = τ_{M} = {n_{1}, n_{2}, …, n_{a}} .

Proposition 4.1. There exists a symplectic basis {X_{i}, Y_{i}}_{0≤i<f} for M such that

(i) X_{i}, Y_{i} ∈ M^{i} and Y_{i} ∈ (V M)^{i}, (ii) the pairings are zero except ⟨X_{i}, Y_{i}⟩ = 1 for all i, and

(iii) F X_{i} = \begin{cases}X_{i+1} & if i + 1 ∉ τ \\ −Y_{i+1} + a_{i+1}pX_{i+1} & if i + 1 ∈ τ (∗) \end{cases}

for some a_{i+1} ∈ W (k),

F Y_{i} = \begin{cases}pY_{i+1} & if i + 1 ∉ τ \\ pX_{i+1} & if i + 1 ∈ τ \end{cases}

Proof. We prove the following inductive statement (I_{n}): There exists a W - basis {X_{i}, Y_{i}}_{0≤i<f} for M such that

(i) X_{i}, Y_{i} ∈ M^{i} and Y_{i} ∈ (V M)^{i}. (ii) the pairings are zero mod p^{n} except ⟨X_{i}, Y_{i}⟩ ≡ 1 mod p^{n} for all i. (iii) F X_{i} = X_{i+1} if i + 1 ∉ τ.

F Y_{i} = \begin{cases}pY_{i+1} & if i + 1 ∉ τ. \\ pX_{i+1} & if i + 1 ∈ τ. \end{cases}

Note that this would automatically imply F X_{i} = −Y_{i+1} + a_{i+1}pX_{i+1} for some a_{i+1} ∈ W (k) when i + 1 ∈ τ, namely, (∗) of (iii).

Let {X_{n_{j}} ∈ M^{n_{j}}} be a set of generators of M as a Dieudonné module. There is a W -basis {X_{i}, Y_{i}} of M such that (1) F X_{i} = X_{i+1} if i + 1 ∉ τ, (2) V Y_{i} = Y_{i−1} if i ∉ τ, and (3) V X_{i} = Y_{i−1} if i ∈ τ. For example, when f = 6 and τ = (0, 3, 5), the basis is

(\begin{matrix}X_{0} & X_{1} & X_{2} & X_{3} & X_{4} & X_{5} \\ Y_{0} & Y_{1} & Y_{2} & Y_{3} & Y_{4} & Y_{5} \end{matrix}) = (\begin{matrix}X_{0} & F X_{0} & F^{2}X_{0} & X_{3} & F X_{3} & X_{5} \\ V^{3}X_{3} & V^{2}X_{3} & V X_{3} & V^{2}X_{5} & V X_{5} & V X_{0} \end{matrix})

Begin with a set of generators X_{n_{j}} of M, extend a W -basis as above. Suppose ⟨X_{n_{j}}, Y_{n_{j}}⟩ ≡ α_{j} mod p. Let X′_{n_{j}} := λ_{j}X_{n_{j}}, then conditions

⟨X′_{n_{j}}, Y′_{n_{j}}⟩ ≡ 1 mod p ∀ j = 1, …, a

give the equations

λ̄_{j}λ̄_{j+1}^{p^{n_{j} −n_{j+1}}}α_{j} = 1 ∀ j = 1, …, a

where λ_{a+1} = λ_{1}, n_{a+1} = n_{1} + f and λ̄ = λ mod p. Raising to some power of p, we get equations

λ_{j}^{p^{n_{j+1}−n_{j}}} λ_{j+1}β_{j} = 1 ∀ j = 1, …, a

where β_{j} is a some power of α_{j}. Since k is algebraically closed these equations have solutions. Adjust X_{n_{j}} by X′_{n_{j}} and extend the basis again as above, hence (I_{1}) is done.

Now assume (I_{k}). Suppose ⟨X_{n_{j}}, Y_{n_{j}}⟩ ≡ 1 + α_{j}p^{k} mod p^{k+1}. Let X′_{n_{j}} = (1 + λ_{j}p^{k})X_{n_{j}}. We may regard λ_{j} ∈ k, for simplicity. Then

⟨X′_{n_{j}}, Y′_{n_{j}}⟩ ≡ 1 + (λ_{j} + λ_{j+1}^{p^{n_{j} −n_{j+1}}} + α_{j})p^{k} mod p^{k+1} ∀ j = 1, …, a.

A similar procedure for replacing X_{n_{j}} by X′_{n_{j}} will yield the statement (I_{k+1}).

It might be simpler not to deal with signs, hence we provide the following variant.

Proposition 4.2. Let M, τ be as in Proposition 4.1. Then there exists a basis {X_{i}, Y_{i}}, i = 0, …, f − 1, of M such that

(i) X_{i}, Y_{i} ∈ M^{i}, Y_{i} ∈ (V M)^{i}, ∀ i = 0 … f − 1, (ii) ⟨X_{i}, Y_{i}⟩ = u_{i}, a unit, and all other pairings are zero, and

(iii) F X_{i} = \begin{cases}X_{i+1} & if i + 1 ∉ τ \\ Y_{i+1} + a_{i+1}pX_{i+1} & if i + 1 ∈ τ \end{cases}

for some a_{i+1} ∈ W (k)

F Y_{i} = \begin{cases}pY_{i+1} & if i + 1 ∉ τ \\ pX_{i+1} & if i + 1 ∈ τ \end{cases}

Proof. Let {X_{i}, Y_{i}} be a symplectic basis satisfying (i)(ii)(iii) of Proposition 4.1. Let X′_{n_{i}} = λ_{i}X_{n_{i}}, λ_{i} ∈ W^{×}, for i = 1, …, a. Then

F^{n_{i+1}−n_{i}}X′_{n_{i}} = λ_{i}^{σ^{n_{i+1}−n_{i}}}(−Y_{n_{i+1}} + pa_{n_{i+1}}X_{n_{i+1}})
Y′_{n_{i+1}} = V^{n_{i+2}−n_{i+1}}X′_{n_{i+2}} = λ_{i+2}^{σ^{−(n_{i+2}−n_{i+1})}}Y_{n_{i+1}}

where n_{a+1} = n_{1} + g. Let λ_{i} ∈ W^{×}, i = 1, ⋯, a such that

−λ_{i}^{σ^{(n_{i+1}−n_{i})}} = λ_{i+2}^{σ^{−(n_{i+2}−n_{i+1})}}.

Then

F^{n_{i+1}−n_{i}}X′_{n_{i}} = Y′_{n_{i+1}} + pa′_{n_{i+1}}X′_{n_{i+1}}

where

a′_{n_{i+1}} = a_{n_{i+1}}λ_{i}^{σ^{(n_{i+1}−n_{i})}}λ_{i+1}^{−1}

.

Lemma 4.3. If M is superspecial, i.e. |a(M)| = f. Then (1) There exists a basis {X_{i}, Y_{i}}_{0≤i<f} of M such that

• X_{i}, Y_{i} ∈ M^{i}, Y_{i} ∈ (V M)^{i},

• F X_{i} = −Y_{i+1}, F Y_{i} = pX_{i+1},

• ⟨X_{i}, Y_{i}⟩ = 1, for all i = 0, …, f − 1.

(2) There exists a basis {X_{i}, Y_{i}} of M such that

• X_{i}, Y_{i} ∈ M^{i}, Y_{i} ∈ (V M)^{i},

• F X_{i} = Y_{i+1}, F Y_{i} = pX_{i+1},

• ⟨X_{i}, Y_{i}⟩ ∈ W (𝔽_{p^{r}})^{×}, where r = lcm(2, f), for all i = 0, …, f − 1.

Proof. (1) Suppose f is even. Let f = 2d, M^{∘} := {x ∈ M | F^{d}x = (−1)^{d}V x }. Since M is supersingular, we have F^{2}M^{∘} = pM^{∘} and M^{∘} ⊗_{W} (𝔽_{p^{f}})W (k) ≃ M. Choose X_{0}, Y_{0} ∈ (M^{∘})^{0} such that ⟨X_{0}, Y_{0}⟩ = 1. Define recursively

X_{i+1} = p^{−1}F Y_{i}, Y_{i+1} = −F X_{i}. (4.1)

Then X_{i+2} = \frac{−1}{p}F^{2}X_{i}, Y_{i+2} = \frac{−1}{p}F^{2}Y_{i}. One verifies that

X_{f} = (−1)^{d}p^{−d}F^{2d}X_{0} = X_{0},
Y_{f} = (−1)^{d}p^{−d}F^{2d}Y_{0} = Y_{0}.

and ⟨X_{i}, Y_{i}⟩ = 1 for all i = 0, …, f − 1. The basis {X_{i}, Y_{i}} is a desired one.

Suppose F is odd. Let f = 2d + 1 and M^{∘} = {x ∈ M | F^{2f} + p^{f}x = 0}. Choose X_{0} ∈ (M^{∘})^{0} \(V M)^{0} and let Y_{0} = (−1)^{d+1}p^{−d}F^{f}X_{0}. Define X_{i}, Y_{i} as (4.1). One verifies that

X_{f} = \frac{1}{p}F Y_{f−1} = (−1)^{d}p^{−d−1}F^{f}Y_{0} = X_{0},

and Y_{f} = Y_{0}. Let μ = ⟨X_{0}, Y_{0}⟩. From ⟨F^{f}X_{0}, F^{f}Y_{0}⟩ = ⟨(−1)^{d+1}p^{d}Y_{0}, (−1)^{d}p^{d+1}X_{0}⟩ we have μ^{(f)} = μ. Replace X_{0} by λX_{0}, then μ will change to λλ^{(f)}μ. Since B(𝔽_{p^{2f}})/B(𝔽_{p^{f}}) is a quadratic unramified extension, the norm map is surjective onto local units. Therefore, we can choose X_{0} such that ⟨X_{0}, Y_{0}⟩ = 1. The basis defined above is a desired one.

(2) We modify the proof of (1) as follows:

When f is even, one replaces M∘ by {x ∈ M | F^{f}x = V^{f}x} and (4.1) by

X_{i+1} = p^{−1}F Y_{i}, Y_{i+1} = F X_{i}. (4.2)

When f is odd, one replaces definition of Y_{0} by p^{−d}F^{f}X_{0} and (4.1) by (4.2).

5. Cayley-Hamilton for a = 1

A σ-linear version of “Cayley-Hamilton” was introduced by F. Oort. It is an effective method to compute the slopes of a Dieudonné module. Hence it provides a way to compute the local moduli spaces of a slope stratum. In [29] F. Oort used this method to (re)solve several problems of Manin, Grothendieck, and Koblitz. This section is completely inspired by his work.

The key point of this method is the following observation:

Proposition 5.1. Let M be a Dieudonné module of height h over a perfect field k. Suppose there exists an element X ∈ M such that

(i) the W -submodule W [F]X is of full rank, and (ii) X satisfies an equation

F^{h}X + p_{1}F^{h−1}X + ⋯ + p_{h−1}F X + p_{h}X = 0, p_{i} ∈ W. (5.1)

Then the Newton polygon of M is the same as that of the polynomial

x^{h} + p_{1}x^{h−1} + ⋯ + p_{h−1}x + p_{h}.

Proof. See a proof in op. cit..

The equation (5.1) is called a Cayley-Hamilton equation. A Cayley-Hamilton equation does dependent on the choice X and may not give the correct slopes if (i) is not satisfied. Most of the elements in M will satisfy this property. However, in order to obtain the coefficients p_{i} as simple as possible, the element X has to be carefully chosen. It will boil down to an explicit local calculation of Dieudonné modules.

The next step is to describe how to compute the closed subscheme of a given slope inside a deformation space, once one has obtained a “family” of Cayley-Hamilton equations.

Let (A_{0}, λ_{0}) be a polarized abelian variety over k such that its p-divisible group G is locallocal. Let (M_{0}, ⟨ , ⟩) be the covariant quasi-polarized Dieudonné module of (A_{0}, λ_{0}). In the rest of this section R will denote a complete Noetherian local ring with residue field k.

Suppose that we have a deformation (G, λ) of (G_{0}, λ_{0}) over R, as formal groups together with the biextension [20]. Then the n^{th}-order deformations (G_{n}, λ_{n}) over Spec R/𝔪^{n+1} are formal p-divisible groups. Hence (G, λ) induces a compatible system of deformations of pdivisible groups, thus a p-divisible group over Spf R. On the other hand, by Serre-Tate, it also gives a compatible system of polarized abelian schemes (A_{n}, λ_{n}), thus a polarized formal abelian scheme (Ā, λ̄) over Spf R. By Grothendieck’s existence theorem, there is a polarized abelian variety (A, λ) over Spec R whose formal 𝔪-adic completion is isomorphic to (Ā, λ̄).

For any prime ideal 𝔭 ∈ Spec R, let K(𝔭) be the perfect closure of the residue field k(𝔭). If A_{k(𝔭)} has p-rank 0, then A_{K(𝔭)}[p^{∞}] is a formal p-divisible group over a perfect field K(𝔭) and its covariant Dieudonné module

D_{∗}(A_{K(𝔭)}) ≃ D_{∗}(G) ⊗_{Cart_{p}(R)} Cart_{p}(K(𝔭))

where D_{∗} is the covariant functor in Cartier-Dieudonné theory. This is a finite W (K(𝔭))- module with F and V action. With the notion of display [22][23], everything can be explicitly written down: First, we display the Dieudonné module (M_{0}, ⟨ , ⟩) by choosing a basis with the relations

\begin{aligned}F X_{j} & = ∑_{i} a_{ij}X_{i}, j = 1, …, g, \\ X_{j} & = V ∑_{i} a_{ij}X_{i}, j = g + 1, …, 2g. \end{aligned}

Suppose that M is a deformation of M_{0} over R such that the biextension induced by ⟨ , ⟩ extends. By the theorem [22, Thm 1], M can be realized as a Cart_{p}(R)-module with generators {X_{i}} and relations

\begin{aligned}F X_{j} & = ∑_{i} ã_{ij}X_{i}, j = 1, …, g, \\ X_{j} & = V ∑_{i} ã_{ij}X_{i}, j = g + 1, …, 2g. \end{aligned}

where ã_{ij} is an element in W (R) ⊂ Cart_{p}(R) whose image in Cart_{p}(k) is a_{ij}.

For 𝔭 ∈ Spec R, the Cartier module M ⊗_{Cart_{p}(R)} Cart_{p}(K(𝔭)) is a Dieudonné module over K(𝔭) with the generators and relations:

\begin{aligned}F X_{j} & = ∑_{i} ã_{ij}(𝔭)X_{i}, j = 1, …, g, \\ X_{j} & = V ∑_{i} ã_{ij}(𝔭)X_{i}, j = g + 1, …, 2g. \end{aligned}

where ã_{ij}(𝔭) is the image of ã_{ij} in Cart_{p}(K(𝔭)). Denote this module by M_{𝔭}. This gives a way to compute (the local moduli of) a slope stratum with reduced structure.

Proposition 5.2. Let (M_{0}, ⟨ , ⟩), (M_{,}⟨ , ⟩) as before, γ the Newton polygon of M_{0}. Suppose there exists an element X ∈ {X_{i}} such that

(i) There exists an (Cayley-Hamilton) equation F^{2g}X +p_{1}F^{2g−1}X +⋯+p_{2g−1}F X +p_{2g} = 0, where p_{i} = (p_{i0}, p_{i1}, …) ∈ W (R).

(ii) For all 𝔭 ∈ Spec R, the W (K(𝔭))-submodule W (K(𝔭))[F]X of M_{𝔭} is of full rank 2g.

Then the closed subscheme with Newton polygon γ or higher is Spec R/\sqrt{I}, where I is the ideal generated by p_{ij} ∈ R for all (i, j) strictly below γ in the Newton polygon diagram.

Proof.

𝔭 ∈ Spec R such that M_{𝔭} has the Newton polygon γ, or higher.

⟺ The Cayley-Hamilton equation, after base change to W (K(𝔭)),

has the Newton polygon γ, or higher.

⟺ p_{ij} = 0 in K(𝔭) ∀ (i, j) below γ.
⟺ p_{ij} ∈ 𝔭 ∀ (i, j) strictly below γ.
⟺ I ⊂ 𝔭, i.e. 𝔭 ∈ Spec R/\sqrt{I}.

Example. Let γ is a Newton polygon as follows.

As the Newton polygons of abelian varieties are symmetry, we may look at just the points (i, j) with i ≤ g. In this case, the points strictly below γ are (1, 0), (2, 0), (3, 0), (4, 0), (4, 1). Therefore I = (p_{10}, p_{20}, p_{30}, p_{40}, p_{41}).

Theorem 5.3 (Oort). Let (M_{0}, ⟨ , ⟩) as before. If the polarization is separable and a(M_{0})=1, then there is an X for the universal deformation (M, ⟨ , ⟩) and an explicit Cayley-Hamilton equation such that the conditions in the previous proposition hold.

We apply this method, in the Hilbert case, to compute each slope stratum in the open subset a = 1. First note that the Dieudonné module of a point on M has slopes a/f and 1 − a/f with multiplicity f. This can be easily shown by using Katz’s sharp slope estimate. Let us denote M^{(m)} (for m < (f/2)+1) the substack of M, with the reduced structure, whose points have the lowest slope m/f.

Let τ = 10 … 0 and M = M(A) with (A, λ, ι) ∈ M_{τ}(k). We can choose a basis {X_{0} = X, X_{i}, Y_{0} = Y, Y_{i}} of M such that

\begin{matrix}F X_{i} = X_{i+1} (i ≠ f − 1), & F X_{f−1} = pa_{0}X − Y \\ F Y_{i} = pY_{i+1} (i ≠ f − 1), & F Y_{f−1} = pX. \end{matrix}

Iterating F f times, we have

F^{f}X = pa_{0}X − Y, F^{f}Y = p^{f}X.

It is easy to see that F^{2f}X − pa_{0}^{(f)}F^{f}X + p^{f}X = 0. This shows that the lowest slope of M is min(\frac{1}{f}(ord(a_{0}) + 1), 1/2).

Proposition 5.4. Let τ = 10 … 0 and M the Dieudonné module of a point x = (A, λ, ι) ∈ M_{τ}^{(m)}(k). Then the local moduli space M̂_{x}^{(m)} = M̂_{τ,x}^{(m)} is isomorphic to Spf k[[t_{m}, …, t_{f−1}]].

Proof. This is Theorem 5.3.3 in [9]. Choose a basis of M as above. The covariant Dieudonné module of the universal deformation of M inside M_{τ} is

\begin{matrix}F X_{i} = X_{i+1} (i ≠ f − 1), & F X_{f−1} = pa_{0}X − Y \\ F Y_{i} = pY_{i+1} + pT_{i+1} X_{i+1} (i ≠ f − 1), & F Y_{f−1} = pX. \end{matrix}

where T_{i} is the Teichmüller lift of t_{i}. Iterating F f-times, we obtain

F^{f}X = pa_{0}X − Y, F^{f}Y = (p^{f} + pa_{0}u)X − uY

where u = p^{f−1}T_{f−1}^{(1)} + p^{f−2}T_{f−2}^{(2)} + ⋯ + pT_{1}^{(f−1)}. It is easy to check that

F^{2f}X + (u − pa_{0}^{(f)})F^{f}X + p^{f}X = 0

The coefficient u − pa_{0}^{(f)} = (0, t_{1}^{p^{f}} , …, t_{m−1}^{p^{f}}, …), hence \sqrt{I} = (t_{1}, …, t_{m−1}) and M̂_{x}^{(m)} ≃ Spf k[[t_{1}, …, t_{f−1}]]/(t_{1}, …, t_{m−1}) ≃ Spf k[[t_{m}, …, t_{f−1}]].

6. Preliminary results for g=4

In this section g = f is 4.

Lemma 6.1. If (A, λ, ι) ∈ M_{1010}(k) then A is supersingular .

Proof. Let M be the Dieudonné module of (A, λ, ι). M can be written as in Proposition 4.1

\begin{matrix}F X_{0} = X_{1}, & F Y_{0} = pY_{1} \\ F X_{1} = −Y_{2} + pa_{2}X_{2}, & F Y_{1} = pX_{2} \\ F X_{2} = X_{3}, & F Y_{2} = pY_{3} \\ F X_{3} = −Y_{0} + pa_{0}X_{0}, & F Y_{3} = pX_{0}, \end{matrix}

for some a_{0}, a_{2} ∈ W. We iterate the Frobenius map four times

\begin{aligned}F^{4} & : X_{0} → −p^{2}X_{0} + a_{2}^{(2)}p(−Y_{0} + a_{0}pX_{0}) \\ F^{4} & : Y_{0} → p^{2}(−Y_{0} + a_{0}pX_{0}), \end{aligned}

or

\begin{aligned}p^{−2}F^{4} & : pX_{0} → −(pX_{0}) + a_{2}^{(2)}(−Y_{0} + a_{0}(pX_{0})) \\ p^{−2}F^{4} & : Y_{0} → (−Y_{0} + a_{0}(pX_{0})). \end{aligned}

Therefore p^{−2}F^{4} sends the submodule N generated by pX_{0}, Y_{0} to itself. Since N is a submodule of the same rank as M, the slopes of M and those of N are the same. So the slopes of M are 1/2.

Lemma 6.2. Every point (A, λ, ι) ∈ 𝒮_{1100}(k) can be deformed to a point of 𝒮_{0100} and also to a point of 𝒮_{1000}.

Proof. Let x = (A, λ, ι) ∈ 𝒮_{1100}(k) and M be its Dieudonné module. Let N ⊂ M ⊗ F racW (k) such that V N = M. Then there is (B, λ′, ι′) ∈ 𝒮_{0110}(k) and O_{F} -morphism φ : (A, λ, ι) → (B, λ′, ι′) such that φ^{∗}λ′ = pλ and N = M(B). Choose a basis {X_{i}, Y_{i}} of N as in Proposition 4.2, i.e.

F X_{i} = \begin{cases}X_{i+1} & if i ∉ \{0, 1\}, \\ Y_{i+1} + pa_{i+1}X_{i+1} & if i ∈ \{0, 1\}. \end{cases}

F Y_{i} = \begin{cases}pY_{i+1} & if i ∉ \{0, 1\}, \\ pX_{i+1} & if i ∈ \{0, 1\}. \end{cases}

Let M_{t} = span⟨Ỹ_{i}, X̃_{i}⟩, where Ỹ_{0} = Y_{0} + T X_{0}, Ỹ_{i} = Y_{i}(i > 0), X̃_{i} = pX_{i}(∀i) and T = [t] is the Teichmüller lift of t ∈ k. It is easy to check that M_{t} is a Dieudonné module inside N and ⟨ , ⟩_{N} = p⟨ , ⟩_{M_{t}}. This construction gives a one-parameter family in 𝒮. Finally it is straightforward to check a_{1}(M_{t}) = 0 when t ≠ 0, therefore we deform to a point of 𝒮_{1000}. Similarly, we can deform the point to a point of 𝒮_{0100}.

Proposition 6.3.

(1) The generic point of M is ordinary.

(2) For |τ| = 1, the generic point of each irreducible component of M_{τ} has slope 1/4 and the supersingular locus 𝒮_{τ} is a one-dimensional smooth divisor of M_{τ}.

(3) For |τ| > 2, every point on \overline{M}_{τ} is supersingular .

(4) Every point on M_{1010} or M_{0101} is supersingular .

(5) 𝒮_{1100} ⊂ 𝒮̄_{1000} ∩ 𝒮̄_{0100}.

(6) The generic point of each irreducible component of M_{1100} has slope 1/4.

(7) Every component of the supersingular locus has dimension 2. There are 6 types of them, namely 𝒮_{τ} with generic a-index τ.

(8) The slope stratification behaves well: \overline{M^{(2)}} ⊂ \overline{M^{(1)}} ⊂ \overline{M^{(0)}} and each irreducible component of \overline{M^{(m)}} has codimension m in M. Here \overline{M^{(m)}} is the Zariski closure of M^{(m)} in M.

Proof. (1) follows from Theorem 3.1. (2) is a consequence of Cayley-Hamilton for a = 1. (3) follows from (4). (4) is Lemma 6.1. (5) is Lemma 6.2. (6) follows from (5). (7) follows from (5). (8) follows from (7) and (2).

7. Canonical isogenies

Let k be a algebraically closed field of characteristic p > 0. Let O_{F} be the ring of integers of a totally real field F of degree f, and assume that p is inert in O_{F}. Let H be an α-group over k with an (left) O_{F} -action. By definition, an α-group over S (of characteristic p) is a commutative finite flat group scheme G → S such that the Frobenius F_{G/S} : G → G^{(p)} and Verschiebung V_{G/S} : G^{(p)} → G are zero. Denote by M^{∗}(H) the classical contravariant Dieudonné module of H, which is simply a k-vector space with a k-linear action by O_{F}. The action gives a decomposition of M^{∗}(H) into eigenspaces. The only invariant other than its dimension is the multiplicity of each one-dimensional representation σ_{i}. We say an α-group H is of type (n_{0}, …, n_{f−1}) if n_{i} is the multiplicity of σ_{i} on M^{∗}(H). Denote by α_{p}^{(n_{0},...,n_{f−1})} the α-group of type (n_{0}, …, n_{f−1}). When H is an α-group equipped with an O_{F} -action over a connected noetherian scheme S, we define n_{i}(H) to be the rank of the i-th component of the α-sheaf of H defined in [17], Section 2. Locally in Zariski topology H is isomorphic to α_{p}^{(n_{0},...,n_{f−1})} × S.

The Cartier dual H^{t} is again an α-group with a natural right O_{F} -action. Since O_{F} is commutative, we can regard it as a left action. According to the definition, we have the canonical pairing

( , ) : H^{t} × H → G_{m} (7.1)

and

(a f, x) = (f, a x) ∀a ∈ O_{F} , f ∈ H^{t}, x ∈ H. (7.2)

The non-degeneracy of the canonical pairing forces the same multiplicity of each eigenspace in M^{∗}(H) and M^{∗}(H^{t}). Therefore H and H^{t} have the same representation type. The relation between the O_{F} -action and duality as (7.1) and (7.2) still holds for any finite flat groups with an O_{F} -action.

Let (A, ι) be an O_{F} -abelian variety of dimension g over k, M its covariant Dieudonné module. M is canonically isomorphic to M^{∗}(A)^{t}, the (twisted) W -linear dual of M^{∗}(A). Let (A^{t}, ι^{t}) be its dual O_{F} -abelian variety. The Dieudonné module M(A^{t}) is canonically isomorphic to the dual Dieudonné module M^{t}. The α-group α(A) := ker F ∩ ker V^{(p^{−1})}, which is the smallest α-subgroup of A containing all α_{p} in A, is of type a(A^{t}). However, the invariant a(A^{t}) is not equal to a(A) in general. They are related by the following lemma.

Lemma 7.1. Let (A, ι), (A^{t}, ι^{t}), M, M^{t} as before. Assume that the Lie algebra Lie(A) = M/V M has the type (n_{0}, n_{1}, …, n_{f−1}), ∑ n_{i} = g and the a-number of A is (a_{0}, a_{1}, …, a_{f−1}), a_{i} ≤ n_{i}. Let (n_{0}^{t}, n_{1}^{t}, …, n_{f−1}^{t}) be the type of Lie(A^{t}), and (a_{0}^{t}, a_{1}^{t}, …, a_{f−1}^{t}) be the a-number of A^{t}. Then n_{i}^{t} = 2r − n_{i}, a_{i}^{t} = a_{i} + n_{i−1} − n_{i} for all i ∈ ℤ/fℤ, where r = g/f.

Proof. Let M̄ := M/pM, M̄^{t} := M^{t}/pM^{t}. The exact sequence

0 → V M̄ → M̄ →^{F} F M̄ → 0

yields the exact sequence

0 → V M̄^{i+1} → M̄^{i} →^{F} F M̄^{i} → 0.

The Lie algebra representation gives dim F M̄^{i} = n_{i}, dim V M̄^{i+1} = 2r − n_{i}. In the k-vector space M̄^{i}, there are 2 subspaces F M̄^{i−1}, V M̄^{i+1} of dimension n_{i−1}, 2r − n_{i} respectively. Let W^{i} := F M̄^{i−1} ∩ V M̄^{i+1}. From

dim W^{i} + dim(F M̄^{i−1} + V M̄^{i+1}) = dim F M̄^{i−1} + dim V M̄^{i+1}

we get dim W^{i} + (2r − a_{i}) = n_{i−1} + 2r − n_{i}, or dim W^{i} = a_{i} + n_{i−1} − n_{i}.

The pairing M̄^{t} × M̄ → k induces a non-degenerate pairing M̄^{t,i} × M̄^{i} → k. The subspace (F M̄^{i−1})^{⊥} consists of elements killed by V, hence it is F M̄^{t,i−1} because of the short exact sequence

0 → F M̄^{t,i−1} → M̄^{t,i} →^{V} V M̄^{t,i} → 0.

Similarly we can show that (V M̄^{i+1})^{⊥} = V M̄^{t,i+1}. Thus (W^{i})^{⊥} = F M̄^{t,i−1} + V M̄^{t,i+1}. By definition, a_{i}^{t} = 2r − dim(F M̄^{t,i−1} + V M̄^{t,i+1}) = dim W^{i} = a_{i} + n_{i−1} − n_{i}.

For any Dieudonné submodule N of M of index p^{h}, there is a unique isogeny φ : B → A of degree p^{h} up to isomorphism with M(B) = N. The submodule N is stable under the O_{F} -action if and only if there is an O_{F} -action on B such that φ is O_{F} -linear. And we can take the pull-back polarization for B so that φ is compatible with polarizations. The quotient M/N is canonically isomorphic to M^{∗}((ker φ)^{t}) because

1 → ker φ → B → A → 1
1 → (ker φ)^{t} → A^{t} → B^{t} → 1
0 → M^{∗}(B^{t}) → M^{∗}(A^{t}) → M^{∗}((ker φ)^{t}) → 0.

For any Dieudonné over-module N of M of index p^{h}, there is a unique isogeny φ : A → B of degree p^{h} up to isomorphism with M(B) = N. The Dieudonné module N is stable under the O_{F} -action if and only if there is an O_{F} -action on B such that φ is O_{F} -linear. The polarization λ can descend to B if and only if ⟨N, N⟩ ⊂ W; and p^{n}λ does if n is large enough. In any case λ descends to a fractional polarization λ_{B} on B, i.e. n λ_{B} is a polarization for some n ∈ ℤ. The quotient N/M is canonically isomorphic to M^{∗}((ker φ)^{t}). This is sort of a covariant version of the finite group ker φ although it does not make sense to talk about this.

To each M, the Dieudonné module of a polarized O_{F} -abelian variety, one can associate a unique Dieudonné lattice chain

⋯ ⊂ M_{2} ⊂ M_{1} ⊂ M_{0} ⊂ M_{−1} ⊂ M_{−2} ⋯

such that M_{0} = M and M_{n+1} = (F, V)M_{n} for all integer n. This Dieudonné lattice chain induces a chain of polarized O_{F} -isogenies

φ_{n} : (A_{n+1}, λ_{n+1}, ι_{n+1}) → (A_{n}, λ_{n}, ι_{n}) ∀ n

where (A_{0}, λ_{0}, ι_{0}) = (A, λ, ι), λ_{n} is a fractional polarization such that φ_{n}^{∗}λ_{n} = λ_{n+1}. We call this chain of isogenies the canonical isogenies attached to (A, λ, ι).

Remark. (1) The polarized O_{F} -abelian varieties (A_{n}, λ_{n}, ι_{n}) may not satisfy the freeness condition for their Lie algebras.

(2) The canonical isogenies can be also defined for polarized O_{B}-abelian varieties where O_{B} is an order of a more general semi-simple algebra B over ℚ with a positive involution.

8. Construction of models

In this section we compute the canonical isogenies of the “generic” principally polarized O_{F} - abelian varieties in question (g = f = 4). Using the information derived from these canonical isogenies, we construct a model attached to each component of the supersingular locus 𝒮 as in [17]. This gives a global description of each component. In particular, it gives another proof of the Main theorem (1).

Let S_{1} be the set of isomorphism classes of polarized O_{F} -abelian varieties (A, λ, ι) over k = k̄ of dimension 4 with a(A) = 0202 of polarization type (p, 1, p, 1). Let S_{2} be the set of isomorphism classes of polarized O_{F} -abelian varieties (A, λ, ι) over k of dimension 4 with a(A) = 0202 and the polarization type (p^{2}, p, p^{2}, p). Let Λ_{i} be a complete set of representatives of S_{i} for i = 1, 2. These sets are finite since there are finitely many ways to impose polarizations of fixed degree on a superspecial abelian variety and many ways to impose O_{F} -action on a fixed polarized abelian variety. There is a natural bijection between S_{1} and S_{2}, sending λ to (1/p)λ. These sets will correspond to the irreducible components and its cardinality can be realized as class numbers.

As the points in 𝒮_{τ} with generic a-number τ form an open dense subset in 𝒮 (Proposition 6.3), we only need to determine the information from the canonical isogenies of the these points. Let (A, λ, ι) ∈ M_{1010}(k), and let M be its Dieudonné module. Choose a presentation for M as in Proposition 4.1. A straightforward computation gives

\begin{aligned}M_{0} = M & = span \{ \begin{matrix}X_{0} & X_{1} & X_{2} & X_{3} \\ Y_{0} & Y_{1} & Y_{2} & Y_{3} \end{matrix} \} \\ M_{1} = (F, V)M_{0} & = span \{ \begin{matrix}pX_{0} & X_{1} & pX_{2} & X_{3} \\ Y_{0} & Y_{1} & Y_{2} & Y_{3} \end{matrix} \} \\ M_{2} = (F, V)M_{1} & = span \{ \begin{matrix}pX_{0} & pX_{1} & pX_{2} & pX_{3} \\ Y_{0} & pY_{1} & Y_{2} & pY_{3} \end{matrix} \} . \end{aligned}

We see that there is a polarized O_{F} -abelian variety (B, λ_{B}, ι_{B}) and a polarized O_{F} -isogeny φ : B → A of degree p^{2} such that φ_{∗}(M(B)) = M_{1} and ker φ ≃ α_{p}^{1010}. Clearly, (B, λ_{B}, ι_{B}) ∈ S_{1}. Let (A, λ, ι) ∈ 𝒮_{1000}(k), and let M be its Dieudonné module. Choose a presentation for M as in Proposition 4.1. Since A is supersingular, a_{0} ≡ 0 mod p. A straightforward computation gives

\begin{aligned}M_{0} = M & = span \{ \begin{matrix}X_{0} & X_{1} & X_{2} & X_{3} \\ Y_{0} & Y_{1} & Y_{2} & Y_{3} \end{matrix} \} \\ V M_{0} & = span \{ \begin{matrix}pX_{0} & pX_{1} & pX_{2} & pX_{3} \\ Y_{0} & Y_{1} & Y_{2} & Y_{3} \end{matrix} \} \\ M_{1} = (F, V)M_{0} & = span \{ \begin{matrix}pX_{0} & X_{1} & X_{2} & X_{3} \\ Y_{0} & Y_{1} & Y_{2} & Y_{3} \end{matrix} \} \\ V M_{1} & = span \{ \begin{matrix}pX_{0} & pX_{1} & pX_{2} & pX_{3} \\ Y_{0} & Y_{1} & Y_{2} & pY_{3} \end{matrix} \} \\ M_{2} = (F, V)M_{1} & = span \{ \begin{matrix}pX_{0} & pX_{1} & X_{2} & X_{3} \\ Y_{0} & Y_{1} & Y_{2} & pY_{3} \end{matrix} \} \\ V M_{2} & = span \{ \begin{matrix}p^{2}X_{0} & pX_{1} & pX_{2} & pX_{3} \\ Y_{0} & Y_{1} & pY_{2} & pY_{3} \end{matrix} \} \\ M_{3} = (F, V)M_{2} & = span \{ \begin{matrix}p^{2}X_{0} & pX_{1} & pX_{2} & X_{3} \\ Y_{0} & Y_{1} & pY_{2} & pY_{3} \end{matrix} \} \\ (∵ F X_{3} = & − Y_{0} + pa_{0}X_{0} ∈ span\{Y_{0}, p^{2}X_{0}\}) \\ V M_{3} & = span \{ \begin{matrix}p^{2}X_{0} & p^{2}X_{1} & pX_{2} & pX_{3} \\ Y_{0} & pY_{1} & pY_{2} & p^{2}Y_{3} \end{matrix} \} \\ M_{4} = (F, V)M_{2} & = span \{ \begin{matrix}p^{2}X_{0} & p^{2}X_{1} & pX_{2} & pX_{3} \\ Y_{0} & Y_{1} & pY_{2} & p^{2}Y_{3} \end{matrix} \} . \end{aligned}

From the result above and using Lemma 7.1, we obtain

a(M_{0}) = 1000, Lie(A_{0}) = k · 1111, a(M_{0}^{t}) = 1000, pol.type = (1, 1, 1, 1)
a(M_{1}) = 0101, Lie(A_{1}) = k · 0112, a(M_{1}^{t}) = 2000, pol.type = (p, 1, 1, 1)
a(M_{2}) = 1020, Lie(A_{2}) = k · 1021, a(M_{2}^{t}) = 1101, pol.type = (p, p, 1, p)
a(M_{3}) = 0202, Lie(A_{3}) = k · 0202, a(M_{3}^{t}) = 2020, pol.type = (p^{2}, p, p^{2}, p).

(8.1)

The symbol k · 0112 is an abbreviation for 0 ⊕ k ⊕ k ⊕ k^{2}, the decomposition with respect to the O_{F} -action.

Next step is to use this information to construct a model and to describe each irreducible component. We only need to deal with the cases τ = 1010 and τ = 1000, due to the symmetry.

Definition. (1) Let τ = 1010. An abelian scheme over S of τ-level structure consists of data (φ : (A_{1}, λ_{1}, ι_{1}) → (A_{0}, λ_{0}, ι_{0})) where

(i) Each (A_{i}, λ_{i}, ι_{i}) is a polarized O_{F} -abelian scheme over S and φ is a polarized O_{F} -isogeny with degree p^{2}.

(ii) (A_{1}, λ_{1}, ι_{1}) ≃ η × S for some η ∈ S_{1}. (iii) The subgroup ker φ is an α-group of type 1010. Two abelian schemes of τ-level structure (φ : (A_{1}, λ_{1}, ι_{1}) → (A_{0}, λ_{0}, ι_{0})) and (φ′ : (A′_{1}, λ′_{1}, ι′_{1}) → (A′_{0}, λ′_{0}, ι′_{0})) are isomorphic if there exist isomorphisms θ_{i} : A′_{i} → A_{i} for i = 0, 1, as polarized O_{F} -abelian varieties.

(2) Let τ = 1000. An abelian scheme over S of τ-level structure consists of data (φ_{n} : (A_{n+1}, λ_{n+1}, ι_{n+1}) → (A_{n}, λ_{n}, ι_{n}); n = 0, 1, 2) where (i) Each (A_{i}, λ_{i}, ι_{i}) is a polarized O_{F} -abelian scheme over S and each φ_{i} is a polarized O_{F} -

isogeny. (ii) (A_{3}, λ_{3}, ι_{3}) ≃ η × S for some η ∈ S_{2}. (iii) The subgroups ker φ_{2}, ker φ_{1}, and ker φ_{0} are α-groups of type 1020, 0101 and 1000,

respectively. We define similarly for two abelian schemes of τ-level structure being isomorphic.

Case τ = 1010. Let η ∈ Λ_{1}. We define

N_{τ,η}(S) = \{ \begin{matrix}isomorphism classes of abelian schemes over S \\ of τ-level structure s.t. (A_{1}, λ_{1}, ι_{1}) = η × S \end{matrix} \} ,

the moduli space parameterizing the abelian varieties of τ-level structure with the source η. Two objects (φ : (A_{1}, λ_{1}, ι_{1}) → (A_{0}, λ_{0}, ι_{0})) and (φ′ : (A′_{1}, λ′_{1}, ι′_{1}) → (A′_{0}, λ′_{0}, ι′_{0})) are isomorphic if there exists an isomorphism θ_{i} : A_{i} → A′_{i} of polarized O_{F} -abelian varieties for each i = 0, 1 such that

[figure: A_{1}; θ_{1}; φ_{0}; A_{0}; θ_{0}; A′_{1}; φ′_{0}; A′_{0}]

Note that we rigidify the source, hence there is no non-trivial automorphism in N_{τ,η}.

Lemma 8.1. N_{τ,η} is representable by P^{1} × P^{1}.

Proof. Every abelian scheme of τ-level structure is uniquely determined by ker φ, which is an α-subgroup of type 1010 in α(η) = α_{p}^{2020} × S. Therefore there is one-one correspondence between N_{τ,η}(S) and α-subgroups of type 1010 in α_{p}^{2020} × S. The latter are isomorphism classes of rank two quotient bundles of O_{S}^{2} ⊕ 0 ⊕ O_{S}^{2} ⊕ 0 (O_{F} -decomposition) of type 1010, which is representable by P^{1} × P^{1}.

Since N_{τ,η} is irreducible, the last projection pr : N_{τ,η} → M induces a morphism

∐_{η∈Λ_{1}} N_{τ,η} → 𝒮̄_{τ},

which is proper and dominant. For each abelian variety x in 𝒮̄_{τ}(k) there is a unique abelian variety y of τ-level structure, up to isomorphism, such that pr(y) = x. Therefore we have a morphism

∐_{η∈Λ_{1}} N_{τ,η}/G_{η} → 𝒮̄_{τ},

which restricts to each component N_{τ,η}/G_{η} is an isomorphism, where G_{η} is the automorphism group of η. We have thus shown

Proposition 8.2. The algebraic stack \overline{M}_{1010} = 𝒮̄_{1010} has |S_{1}| irreducible components and each component is isomorphism to a quotient of P^{1} × P^{1} by a finite group.

We will describe the preimage of higher a-number strata via pr. We first need a lemma.

Lemma 8.3. Let (B, λ_{B}, ι_{B}) ∈ S_{1} and N be its Dieudonné module. Then there is a basis {X_{i}, Y_{i}}_{0≤i<4} for N such that • X_{i}, Y_{i} ∈ N^{i}.

• ⟨X_{i}, Y_{i}⟩ = p⟨X_{i+1}, Y_{i+1}⟩ = p for i = 0, 2.

• \begin{matrix}F X_{0} = −pY_{1}, & F X_{1} = −Y_{2}, & F X_{2} = −pY_{3}, & F X_{3} = −Y_{0} \\ F Y_{0} = pX_{1}, & F Y_{1} = X_{2}, & F Y_{2} = pX_{3}, & F Y_{3} = X_{0}. \end{matrix}

Proof. By Proposition 8.2 there is a superspecial principally polarized O_{F} -abelian variety (A, λ, ι) and an O_{F} -isogeny B → A such that ker φ ≃ α_{p}^{1010}. By Lemma 4.3 we can choose a basis {X′_{i}, Y′_{i}}_{0≤i<4} for the Dieudonné module M of A such that F X′_{i} = −Y′_{i+1}, F Y′_{i} = pX′_{i+1} and all the pairings are zero except ⟨X_{i}, Y_{i}⟩ = 1 . Since M/N = k ⊕ 0 ⊕ k ⊕ 0, N is generated by X_{i} := pX′_{i} (i = 0, 2), X_{i} = X′_{i} (i = 1, 3) and Y_{i} = Y′_{i} (∀ 0 ≤ i < 4). This basis {X_{i}, Y_{i}} is the desired one.

Let Z be an irreducible component of \overline{𝒮}_{1010}. Let η = (B, λ_{B}, ι_{B}) ∈ Λ_{1} such that pr : P^{1} × P^{1} = N_{τ,η} → Z. We can choose a presentation for N as in Lemma 8.3. Then

N_{−1} = (F, V)^{−1}N = span \{ \begin{matrix}\frac{1}{p}X_{0} & X_{1} & \frac{1}{p}X_{2} & X_{3} \\ \frac{1}{p}Y_{0} & Y_{1} & \frac{1}{p}Y_{2} & Y_{3} \end{matrix} \} .

For any point p = ((x_{0}, y_{0}), (x_{2}, y_{2})) ∈ P^{1} × P^{1}(k), the Dieudonné module of the image pr(p) in Z is

M_{p} = ⟨N, \frac{x̃_{0}}{p} X_{0} + \frac{ỹ_{0}}{p} Y_{0}, \frac{x̃_{2}}{p} X_{2} + \frac{ỹ_{2}}{p} Y_{2}⟩

where x̃_{i} and ỹ_{i} are any lifts of x_{i} and y_{i}. Note that M_{p} is independent of the choice of the liftings. For simplicity, we write the affine coordinates x = x_{0}/y_{0}, y = x_{2}/y_{2}:

M_{(x,y)} = span \{ \begin{matrix}X_{0} & X_{1} & X_{2} & X_{3} \\ V_{0} & Y_{1} & V_{2} & Y_{3} \end{matrix} \} (8.2)

where V_{0} := \frac{x̃}{p}X_{0} + \frac{1}{p}Y_{0}, V_{2} := \frac{ỹ}{p}X_{2} + \frac{1}{p}Y_{2}. A simple computation shows that

a_{0}(M_{(x,y)}) = a_{2}(M_{(x,y)}) = 1
((F, V)M̄_{(x,y)})^{1} = span{−x^{p}Y_{1} + X_{1}, y^{p^{−1}}Y_{1} − X_{1}}
((F, V)M̄_{(x,y)})^{3} = span{−y^{p}Y_{3} + X_{3}, x^{p^{−1}}Y_{3} − X_{3}}.

Therefore a_{1}(M_{(x,y)}) = 1 ⟺ x^{p^{2}} = y and a_{3}(M_{(x,y)}) = 1 ⟺ x = y^{p^{2}}. We deduce the points of P^{1} × P^{1} whose images in 𝒮 have higher a-numbers:

Proposition 8.4. Let pr : N_{τ,η} = P^{1} × P^{1} → Z be as above.

(1) The point in P^{1} ×P^{1} whose image lies in (Z ∩\overline{M}_{1110}) is exactly in \{(x, x^{p^{2}})\} ⊂ P^{1} ×P^{1}; the point in P^{1} × P^{1} whose image lies in (Z ∩ \overline{M}_{1011}) is exactly in \{(y^{p^{2}}, y)\} ⊂ P^{1} × P^{1}. In particular, Z ∩ \overline{M}_{1110} (or Z ∩ \overline{M}_{1011}) is a finite quotient of P^{1}, whose geometric quotient is isomorphic to P^{1}.

(2) The stratum \overline{M}_{1110} (or \overline{M}_{1011}) has |S_{1}| connected components and the each connected component is a finite quotient of P^{1}.

(3) The set of superspecial points on P^{1} × P^{1} is {(x, x^{p^{2}}) |x ∈ 𝔽_{p^{4}}} ≃ P^{1}(𝔽_{p^{4}}).

Next we describe the action of G_{η} on P^{1} × P^{1}. Since G_{η} ⊂ Aut_{O_{F}} (B) ⊂ Aut_{O_{F}} (N), we first compute End_{O_{F}} (N). The module N is generated by X_{1}, Y_{1}, X_{3}, Y_{3}. Write g ∈ End_{O_{F}} (N) by

g · (\begin{matrix}X_{1} \\ Y_{1} \end{matrix}) = (\begin{matrix}a_{1} & b_{1} \\ c_{1} & d_{1} \end{matrix}) (\begin{matrix}X_{1} \\ Y_{1} \end{matrix}) , g · (\begin{matrix}X_{3} \\ Y_{3} \end{matrix}) = (\begin{matrix}a_{3} & b_{3} \\ c_{3} & d_{3} \end{matrix}) (\begin{matrix}X_{3} \\ Y_{3} \end{matrix})

From the relation F^{2}X_{1} + pX_{3} = 0 and F^{2}Y_{1} + pY_{3} = 0, we obtain

(\begin{matrix}a_{3} & b_{3} \\ c_{3} & d_{3} \end{matrix}) = (\begin{matrix}a_{1} & b_{1} \\ c_{1} & d_{1} \end{matrix})^{(2)} .

Let g ∈ G_{η} and ρ = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ Aut_{O_{F}} (N/pN, ⟨ , ⟩) = SL_{2}(𝔽_{p^{4}}) be the image of g. Then

g · (x, y) = (ρx, ρ^{(2)}y), ∀ (x, y) ∈ P^{1} × P^{1}.

Lemma 8.5. The G_{η}-orbits of the superspecial points P^{1}(𝔽_{p^{4}}) one-one correspond the superspecial points on Z.

Proof. To assert that we need to prove for any two morphisms α_{1} : (B, λ_{B}, ι_{B}) → (A_{1}, λ_{1}, ι_{1}) and α_{2} : (B, λ_{B}, ι_{B}) → (A_{2}, λ_{2}, ι_{2}) such that there exits an isomorphism β : A_{1} → A_{2} of polarized O_{F} -abelian varieties, then the composition α_{2}^{−1}βα_{1} should be an isomorphism.

\begin{matrix}B & ⟶^{α_{1}} & A_{1} \\ ↓^{?} & & ↓^{β} \\ B & ⟶^{α_{2}} & A_{2} \end{matrix}

In our situation the degree of α_{2} is a power of p, hence it suffices to verify whether the quasi-isogeny map α_{2}^{−1}βα_{1} induces a morphism on their Dieudonné modules. Let

\begin{matrix}N & ⟶^{α_{1}} & M_{1} \\ ↓^{?} & & ↓^{β} \\ N & ⟶^{α_{2}} & M_{2} \end{matrix}

be the corresponding Dieudonné modules that M_{1} and M_{2} are submodules of N_{−1}. Since β is Dieudonné , it induces an isomorphism form M_{1}/(F, V)M_{1} to M_{2}/(F, V)M_{2}. Both of them are isomorphic to k · 1111. The image N̄ of N in either vector space is k · 1010. Since the morphism β is also compatible with the O_{F} -action, it preserves N̄, hence N.

From the smoothness of 𝒮̄_{1010}, 𝒮̄_{1010} is a disjoint union of irreducible components. Every superspecial point lies in exact one component of 𝒮̄_{1010}. The number is superspecial points on 𝒮 is

∑_{η∈Λ_{1}} #(P^{1}(𝔽_{p^{4}})/G_{η}). (8.3)

Remark. A classification of possible G_{η} will help compute the number of superspecial points on 𝒮. It seems to me a delicate problem, so we leave this aside in this work.

Case τ = 1000. Let η ∈ Λ_{2}. We define

N_{τ,η}(S) = \{ \begin{matrix}isomorphism classes of abelian schemes over S \\ of τ-level structure s.t. (A_{3}, λ_{3}, ι_{3}) = η × S \end{matrix} \} ,

the moduli space parameterizing the abelian varieties of τ-level structure with the source η.

Two objects (A_{3} →^{φ_{2}} A_{2} →^{φ_{1}} A_{1} →^{φ_{0}} A_{0}) and (A′_{3} →^{φ′_{2}} A′_{2} →^{φ′_{1}} A′_{1} →^{φ′_{0}} A′_{0}) are isomorphic if there

exists an isomorphism θ_{i} : A_{i} → A′_{i} of polarized O_{F} -abelian varieties for each i = 0, 1, 2, 3 such that

[figure: A_{3}; θ_{3}; φ_{2}; A_{2}; φ_{1}; θ_{2}; A_{1}; φ_{0}; θ_{1}; A_{0}; θ_{0}; A′_{3}; φ′_{2}; A′_{2}; φ′_{1}; A′_{1}; φ′_{0}; A′_{0}.]

Note that we rigidify the source, hence there is no non-trivial automorphism in N_{τ,η}.

Lemma 8.6. N_{τ,η} is representable by a projective scheme over k.

Proof. Consider the truncated τ-level structure: for m = 0, 1, 2, denote by N_{τ,η}^{(m)} the moduli space parameterizing a chain of isogenies (A_{3} → A_{2} → ⋯ → A_{m}) of a τ-level structure (A_{3} → A_{2} → ⋯ → A_{0}). By the same proof in Lemma 8.1, N_{τ,η}^{(2)} is representable by P^{1}. Suppose N_{τ,η}^{(m)} is representable by a projective scheme S_{m} and let (φ_{i} : A_{i+1} → A_{i})_{i≥m} be the universal family. The fibre of the truncated morphism t : N_{τ,η}^{(m−1)} → N_{τ,η}^{(m)} is given by the α-subgroups of A_{m}[F] ×_{S_{m}} T of a prescribed type, hence representable by a relatively projective scheme, cf [17], Section 3.5.

The last projection pr gives a proper morphism N_{τ,η} → 𝒮. If N_{τ,η} is irreducible, then the last projection pr induces a morphism

∐_{η∈Λ_{2}} N_{τ,η} → 𝒮̄_{τ},

which is proper and dominant.

We will prove the irreducibility of N_{τ,η} by representing it as a P^{1}-fibration over P^{1}, see Corollary 8.9 below.

Lemma 8.7. Let (A_{3} →^{φ_{2}} A_{2} →^{φ_{1}} A_{1} →^{φ_{0}} A_{0}) be an abelian variety of τ-level structure over k. Then a(A_{1})=01x1, a(A_{1}^{t})=20x0, a(A_{2}) = 1020 and a(A_{2}^{t}) = 1101, where x = 0 or 1.

Proof. Let M_{i} be the corresponding Dieudonné modules, and let M_{i} = M_{i}^{0} ⊕ ⋯ ⊕ M_{i}^{3} be the decomposition with respect to O_{F} -action. Since ker φ_{2} = α_{p}^{1020}, M_{2}/M_{3} = k ⊕ 0 ⊕ k^{2} ⊕ 0. So M_{2}^{0}/M_{3}^{0} = k, M_{2}^{1} = M_{3}^{1}, M_{2}^{2} = (1/p)M_{3}^{2} and M_{2}^{3} = M_{3}^{3}. Since a(M_{3}) = 0202 and ((F, V)M_{2})^{0} = F M_{3}^{3} +V M_{3}^{1} = M_{3}^{0}, hence a(A_{2}) = 1020. And V M_{2} = M_{3}^{0} ⊕M_{3}^{1} ⊕M_{3}^{2} ⊕V M_{2}^{0}, hence Lie(A_{2}) = 1021, so a(A_{2}^{t}) = 1101.

Since the polarization type of M_{1} is (p, 1, 1, 1), we have a(A_{1}) =01x1. We also check that Lie(M_{1}) = 0112, thus a(A_{1}^{t}) =20x0.

Corollary 8.8. The stratum 𝒮_{1100} is empty.

Proof. We will show that any (A_{0}, λ_{0}, ι_{0}) ∈ 𝒮_{1100}(k) does not admit a τ-level structure. Let (A_{i}) be an abelian variety of τ-level structure with a(A_{0}) = 1100 and M_{i} be the corresponding Dieudonné modules. From M_{0}/M_{1} = k · 1000 and M_{1}/M_{2} = k · 0101, we compute that

M_{0} = span \{ \begin{matrix}X_{0} & X_{1} & X_{2} & X_{3} \\ Y_{0} & Y_{1} & Y_{2} & Y_{3} \end{matrix} \}

M_{1} = span \{ \begin{matrix}pX_{0} & X_{1} & X_{2} & X_{3} \\ Y_{0} & Y_{1} & Y_{2} & Y_{3} \end{matrix} \}

M_{2} = span \{ \begin{matrix}pX_{0} & pX_{1} & X_{2} & X_{3} \\ Y_{0} & Y_{1} & Y_{2} & pY_{3} \end{matrix} \}

(F, V)M_{2} = span \{ \begin{matrix}p^{2}X_{0} & pX_{1} & pX_{2} & X_{3} \\ pY_{0} & Y_{1} & pY_{2} & pY_{3} \end{matrix} \} .

and a(M_{2}) = 2020. This contradicts with Lemma 8.7. By Lemma 6.2, this completes the corollary.

Corollary 8.9.

(1) For any (A, λ, ι) ∈ 𝒮̄_{τ}(k) there is a unique abelian variety (φ_{n}, (A_{n}, λ_{n}, ι)) of τ-level structure, up to isomorphism, such that (A_{0}, λ_{0}, ι_{0}) ≃ (A, λ, ι).

(2) N_{τ,η} is a P^{1}-fibration over P^{1}.

Proof. For (1), it is immediate just to compare the a-numbers of ker φ_{n} and A_{n}.

(2) First, N_{τ,η}^{(2)} = P^{1}. By Lemma 8.7, the map N_{τ,η}^{(1)}(k) → N_{τ,η}^{(2)}(k) is bijective and for each point Spec k → N_{τ,η}^{(1)} the fibre N_{τ,η}^{(0)} ×_{N_{τ,η}^{(1)}} Spec k is isomorphic to P^{1}. So the morphism N_{τ,η}^{(0)} → N_{τ,η}^{(2)} gives the desired fibration structure.

Proposition 8.10. The algebraic stack 𝒮̄_{τ} has |S_{2}| irreducible components. Each irreducible component is isomorphic to N_{τ,η}/G_{η} for a unique η.

Proof. We proved (Corollary 8.9) that the projection pr induces a bijection between N_{τ,η}(k)/G_{η} and Z_{τ,η}(k), where Z_{τ,η} is the irreducible component of 𝒮̄_{τ} corresponding to η. It is easy to check that pr is generically étale by a tangent space calculation. Therefore pr : N_{τ,η}/G_{η} → Z_{τ,η} is an isomorphism.

We do the tangent space calculation in the following lemma.

Lemma 8.11. The projection pr above is generically étale.

Proof. It is enough to check the point where a(M_{0}) = 1000. Use the bases for M_{i} appeared in the canonical isogenies. Let F̃ il_{i} ⊂ M̃_{i} be the first order universal deformations. We have

F̃ il_{0} = ⟨Y_{i} + t_{i0}X_{i}⟩
F̃ il_{1} = ⟨pX_{0}, Y_{0}, Y_{1} + t_{11}X_{1}, Y_{2} + t_{21}X_{2}⟩
F̃ il_{2} = ⟨Y_{0} + t_{02}pX_{0}, pX_{1}, Y_{1}, pY_{3} + t_{32}X_{3}⟩
F̃ il_{3} = ⟨p^{2}X_{0}, Y_{0}, pX_{2}, pY_{2}⟩.

From F̃ il_{i+1} ⊂ F̃ il_{i} we get the relations t_{00} = 0, t_{10} = t_{11}, t_{20} = t_{21}, t_{11} = 0, t_{32} = 0, and t_{02} = 0. The projection map induces an isomorphism on the tangent spaces.

S. Shatz asked which ruled surface the model N_{τ,η} is in the classification of algebraic surfaces. The answer turns out to be F_{2}, a Hirzebruch surface [1, Chapter IV]. We thank him for this interesting question.

Proposition 8.12. The model N_{τ,η} is isomorphic to the Hirzebruch surface F_{2}.

Proof. Let M_{3} be the Dieudonné module of η. Write N_{τ,η}^{(2)} = P^{1} = {[x_{0}, x_{1}]}. Let U_{0} = {[x_{0}, x_{1}]; x_{0} ≠ 0} and x = x_{1}/x_{0} be its coordinate. Let U_{1} = {[x_{0}, x_{1}]; x_{1} ≠ 0} and x′ = x_{0}/x_{1} be its coordinate. Choose a presentation of M_{3} as Lemma 8.3. For [x_{0}, x_{1}] ∈ P^{1}(k), the Dieudonné modules corresponding to this point are

M_{3} = span \{ \begin{matrix}X_{0} & X_{1} & X_{2} & X_{3} \\ Y_{0} & Y_{1} & Y_{2} & Y_{3} \end{matrix} \}

M_{2} = span \{ \begin{matrix}X_{0}, Y_{0} & X_{1} & \frac{1}{p}X_{2} & X_{3} \\ V_{0} & Y_{1} & \frac{1}{p}Y_{2} & Y_{3} \end{matrix} \} ,

where V_{0} = x̃_{0}(\frac{1}{p}X_{0}) + x̃_{1}(\frac{1}{p}Y_{0}), x̃_{0} and x̃_{1} are any liftings of x_{0} and x_{1} in W, for example, the Teichmüller ones. Then

M_{1} = span \{ \begin{matrix}X_{0}, Y_{0} & X_{1}, Y_{1} & \frac{1}{p}X_{2} & X_{3}, \frac{1}{p}V V_{0} \\ V_{0} & \frac{1}{p}F V_{0} & \frac{1}{p}Y_{2} & Y_{3} \end{matrix} \} .

Let π : N_{τ,η} → P^{1} be the projection map. Any point in the fibre over [x_{0}, x_{1}] corresponds to a subspace M_{0}^{0}

M_{1}^{0} ⊂ M_{0}^{0} ⊂ ((F, V)^{−1}M_{1})^{0},

and ((F, V)^{−1}M_{1})^{0} = span\{\frac{1}{p}X_{0}, \frac{1}{p}Y_{0}, \frac{1}{p}V_{0}\}. Set X := X_{0} and Y := Y_{0}. then we have M_{1}^{0} = span{X, Y, x̃_{0}X + x̃_{1}Y }.

Over U_{1}: M_{1}^{0} = span\{X, Y′\}, where Y′ = x̃′ \frac{1}{p}X + \frac{1}{p}Y. We have an isomorphism α_{1} :

U_{1} × P^{1} →^{∼} N_{τ,η} by sending (x′, [y′_{0}, y′_{1}]) to

M_{0}^{0} = span\{X, Y′, ỹ′_{0}(\frac{1}{p}X) + ỹ′_{1}(\frac{1}{p}Y′)\}.

Over U_{0}: M_{1}^{0} = span\{X′, Y \}, where X′ = \frac{1}{p}X +x̃\frac{1}{p}Y. We have an isomorphism α_{0} : U_{0}×P^{1} →^{∼} N_{τ,η} by sending (x, [y_{0}, y_{1}]) to

M_{0}^{0} = span\{X′, Y, ỹ_{0}(\frac{1}{p}X′) + ỹ_{1}(\frac{1}{p}Y)\}.

Over U_{0} ∩ U_{1}, we have

M_{1}^{0} = span{X, Y′} = span{X′, Y }

with relation Y′ = x̃′X′ and X = pX′ −x̃Y. Using this relation, the image of (x′, [y′_{0}, y′_{1}]) ∈ U_{1} in N_{τ,η} is

span\{X, Y′, ỹ′_{0}(\frac{1}{p}X) + ỹ′_{1}(\frac{1}{p}Y′)\} = span\{X′, Y, x̃′ỹ_{0}(\frac{1}{p}X′) − x̃ỹ_{1}(\frac{1}{p}Y)\}.

Therefore N_{τ,η} = U_{0} × P^{1} ∐ U_{1} × P^{1}/ ∼, where (x, [y_{0}, y_{1}]) ∼ (x′, [y′_{0}, y′_{1}]) if and only if xx′ = 1 and y_{0}y′_{0}x + y_{1}y′_{1}x′ = 0. Choose another trivialization of N_{τ,η}|_{U_{0}} by precomposing the isomorphism (x, [y_{0}, y_{1}]) → (x, [y_{0}, −y_{1}]). We have N_{τ,η} = U_{0} × P^{1} ∐ U_{1} × P^{1}/ ∼, where (x, [y_{0}, y_{1}]) ∼ (x′, [y′_{0}, y′_{1}]) if and only if xx′ = 1 and y_{0}y′_{0}x − y_{1}y′_{1}x′ = 0. Set y := y_{1}/y_{0} and y′ = y′_{1}/y′_{0}. The equivalent relation above is xx′ = 1 and y′ = x^{2}y. Hence N_{τ,η} is isomorphic to the Hirzebruch surface F_{2}.

9. Superspecial points as class numbers

In this section we interpret the number of components, |S_{1}|, which appears in the previous section, as a class number. The main result is Theorem 9.6. The ground field k is an algebraically closed field of characteristic p.

Proposition 9.1. Let (A_{1}, λ_{1}, ι_{1}) and (A_{2}, λ_{2}, ι_{2}) be two supersingular polarized O_{F} -abelian varieties. Then there exists an O_{F} -linear isogeny φ : A_{1} → A_{2} such that φ^{∗}λ_{2} = qλ_{1} for some q ∈ ℚ^{×}, i.e. φ is a ℚ-polarized O_{F} -linear isogeny (see the definition in Section 2).

Proof. Since they are supersingular, we can choose an isogeny φ : A_{1} → A_{2}. The action of O_{F} on A_{2} induces an action ι′_{1} of O_{F} on “A_{1} ⊗ ℚ”, an abelian variety up to isogeny, by

A_{1} ⟶^{g} A_{1} ⟶^{φ} A_{2}
↓^{ι_{1}(a)}  ↓^{ι′_{1}(a)}  ↓^{ι_{2}(a)}
A_{1} ⟶^{g} A_{1} ⟶^{φ} A_{2}

ι′_{1}(a) := φ^{−1}ι_{2}(a)φ ∈ End(A_{1}) ⊗ ℚ for a ∈ O_{F}. Therefore we have two morphisms ι_{1}, ι′_{1} : F → End(A_{1}) ⊗ ℚ. Since the center of End(A_{1}) ⊗ ℚ is ℚ, by Noether-Skolem’s theorem, there exists g ∈ End(A_{1}) ⊗ ℚ such that ι_{1} = g^{−1}ι′_{1}g. The element g may be chosen in End(A_{1}). Replacing φ by φg, we obtain an O_{F} -linear isogeny between A_{1} and A_{2}.

Next we can adjust φ to be compatible with the polarizations up to a rational number. Let ∗_{1} be the Rosati involution induced by λ_{1} and ∗′_{1} the one induced by λ′_{1} := φ^{∗}λ_{2}. The involutions ∗_{1} and ∗′_{1} are positive on End(A_{1}) ⊗ ℚ. By Noether-Skolem’s theorem, there is a positive element c ∈ End(A_{1}) such that ∗_{1} = c ∗′_{1} c^{−1}.

Lemma 9.2. The element c can be chosen in End_{O_{F}} (A_{1}).

Proof. We can restrict ∗_{1} and ∗′_{1} on End_{O_{F}} (A_{1}) ⊗ ℚ since both polarizations λ_{1} and λ′_{1} are O_{F} -linear. Then there exists c ∈ End_{O_{F}} (A_{1}) such that ∗_{1} = c ∗′_{1} c^{−1} on End_{O_{F}} (A_{1}) ⊗ ℚ. So they are the same, as involutions on End(A_{1}) ⊗ ℚ, up to an inner conjugation by an integral element commuting with all elements in End_{O_{F}} (A_{1}), which is in O_{F} so as in End_{O_{F}} (A_{1}). Therefore c can be chosen in End_{O_{F}} (A_{1}).

We recall some basic properties

• If φ is O_{F} -linear isogeny then φ^{∗}λ is O_{F} -linear ⟺ λ is O_{F} -linear.

• If λ is an O_{F} -linear polarization then the Rosati involution preserves the subalgebra End_{O_{F}} (A) ⊗ ℚ.

• If two polarizations λ_{1} and λ_{2} induce the same Rosati involution on End(A) ⊗ ℚ, then λ_{2} = qλ_{1} for some q ∈ ℚ.

• Let ϕ ∈ End_{O_{F}} (A). Let ∗_{1} and ′ be the involutions induced by λ and ϕ^{∗}λ, respectively. Then x′ = (ϕ^{∗_{1}}ϕ)^{−1}x^{∗_{1}}(ϕ^{∗_{1}}ϕ).

We check the last statement in the following:

\begin{aligned}x′ & = (ϕ^{∗}λ)^{−1}x^{t}(ϕ^{∗}λ) = ϕ^{−1}λ^{−1}(ϕ^{t})^{−1}x^{t}ϕ^{t}λϕ \\ & = ϕ^{−1}(ϕxϕ^{−1})^{∗_{1}}ϕ = (ϕ^{∗_{1}}ϕ)^{−1}x^{∗_{1}}(ϕ^{∗_{1}}ϕ). \end{aligned}

The question is reduced to asking: is the map a → a^{∗_{1}}a surjective from End_{O_{F}} (A_{1}) to the positive elements of itself? In our case, the semi-simple algebra is End_{O_{F}} (A_{1}) ⊗ ℚ = F ⊗End(E), where E is a supersingular elliptic curve. The positive element c = c^{∗_{1}} is a totally positive element in F. The assertion of the surjectivity of the norm map from F ⊗ End(E) to F^{+} is a well-known consequence of the Hasse principle. See a proof in the lemma below.

Let g ∈ End_{O_{F}} (A_{1}) such that c = g^{∗_{1}}g, and replace φ by φg. Therefore we produce an O_{F} -linear isogeny φ from A_{1} to A_{2} such that φ^{∗}λ_{2} and λ_{1} induce the same Rosati involution on End(A_{1}) ⊗ ℚ, hence φ^{∗}λ_{2} = qλ_{1} for some q ∈ ℚ. This proves the proposition.

Lemma 9.3. Let D = End(E) ⊗ ℚ. The reduced norm map form F ⊗ D to F^{+} is surjective.

Proof. Let c ∈ F^{+}. Consider the short exact sequence of reductive algebraic groups

1 → H ⟶ (F ⊗ D)^{×} ⟶^{N} F^{×} → 1

and we have the diagram of pointed sets

(F ⊗ D)^{×}  ⟶^{N}  F^{×}  ⟶^{δ} H^{1}(ℚ, H)
↓  ↓^{r}  ↓^{r_{∞}}
(ℍ^{×} × ⋯ × ℍ^{×}) ⟶^{det} (ℝ^{×} × ⋯ × ℝ^{×}) ⟶^{δ_{∞}} H^{1}(ℝ, H)

The algebraic group Hℚ̄ ≃ SL_{2} × ⋯ × SL_{2} (f copies) is semi-simple and simply connected, hence H^{1}(ℚ_{p}, H) = 0 for all primes p and r_{∞} is bijective. Since r(c) is totally positive, which is an image of the map det, therefore r_{∞}(δc) = δ_{∞}r(c) = 0 ⇒ δc = 0 and c is an image of N.

Corollary 9.4. Let (A_{i}, λ_{i}, ι_{i}), for i = 1, 2, be two supersingular polarized O_{F} -abelian varieties. Then there is an element φ in Hom_{O_{F}} (A_{1}, A_{2}) ⊗ ℚ such that φ^{∗}λ_{2} = λ_{1}.

Proof. By Proposition 9.1, there is an element φ_{1} in Hom_{O_{F}} (A_{1}, A_{2}) such that φ^{∗}λ_{2} = qλ_{1} for some q ∈ ℚ^{×}. Since λ_{i} are induced from ample line bundles, q > 0. By Lemma 9.3, there is α ∈ End_{O_{F}} (A_{1}) ⊗ ℚ such that α^{∗_{1}} · α = q^{−1}, where the map α → α^{∗_{1}} is the Rosati involution induced by λ_{1}. Then (φ_{1} ∘ α)^{∗}λ_{2} = α^{t}qλ_{1}α = qλ_{1}α^{∗_{1}}α = λ_{1}.

Lemma 9.5. Let (A_{1}, λ_{1}, ι_{1}) and (A_{2}, λ_{2}, ι_{2}) be polarized O_{F} -abelian varieties such that the degree of each λ_{i} is a power of p. Then their ℓ-adic Tate modules are isomorphic , as O_{F} ⊗ℤ_{ℓ}- modules with alternating pairings, for all ℓ ≠ p.

Proof. The Tate module M := T_{ℓ}(A) is a rank 2 free O_{F} ⊗ℤ_{ℓ}-module together with a self-dual alternating ℤ_{ℓ}-valued pairing such that

⟨a x, y⟩ = ⟨x, a y⟩ ∀ a ∈ O_{F} , x, y ∈ M.

We may assume O_{F} ⊗ ℤ_{ℓ} is a domain by decomposing M. Since the degree of the pairing is 1 and O_{F} ⊗ ℤ_{ℓ} is a PID, we are done.

Recall that S_{1} the set of isomorphism classes of polarized O_{F} -abelian varieties (A, λ, ι) over k such that a(A) = 0202 and deg λ = p^{4}. We fix an object (A_{0}, λ_{0}, ι_{0}) as a reference. From Lemma 8.3, 9.5, the ℓ-divisible groups A(ℓ) and A_{0}(ℓ) (ℓ can be p) are isomorphic for each (A, λ, ι) ∈ S_{1}. Let G the the automorphism group of (A_{0}, λ_{0}, ι_{0}), regarded as an algebraic group over Spec ℤ. That is, for any commutative ring R, the group of R-points of G is

G(R) = {ϕ ∈ (End_{O_{F}} (A) ⊗ R)^{×}|ϕ′ϕ = 1}

where the map ϕ → ϕ′ is the Rosati involution induced by λ.

Theorem 9.6. There is an one-one correspondence between the pointed set S_{1} and the double coset G(ℚ)\G(𝔸_{f})/G(ℤ̂).

Proof. For any (A, λ, ι) ∈ S_{1} there exists a ℚ-polarized O_{F} -linear isogeny ϕ : (A_{0}, λ_{0}, ι_{0}) → (A, λ, ι). Such ϕ gives rise to an element in G(𝔸_{f}) up to a right translation of G(ℤ̂): for each prime ℓ (ℓ can be p) there exists ϕ_{ℓ} ∈ G(ℚ_{ℓ}) such that ϕϕ_{ℓ} ∈ Isom(A_{0}(ℓ), A(ℓ)) compatible with additional structure (the polarization and endomorphism). The element ϕ_{ℓ} lies in G(ℤ_{ℓ}) for almost all ℓ, therefore (ϕ_{ℓ}) ∈ G(𝔸_{f}). The element (ϕ_{ℓ}) is uniquely determined by ϕ up to a right translation of G(ℤ̂), since ϕϕ_{ℓ}, ϕϕ′_{ℓ} ∈ Isom(A_{0}(ℓ), A(ℓ)) implies ϕ′_{ℓ} ∈ ϕ_{ℓ}G(ℤ_{ℓ}).

However, ϕ is not uniquely determined by (A, λ, ι). Suppose that ϕ_{1} is another ℚ-polarized O_{F} -linear isogeny from (A_{0}, λ_{0}, ι_{0}) to the same (A, λ, ι), then ϕ_{1} ∈ ϕG(ℚ). That is, the elements (ϕ_{ℓ})G(ℤ̂) determined by different ϕ’s of a fixed (A, λ, ι) belong to an orbit by the left translation of G(ℚ). Therefore we have a well-defined map from S_{1} to G(ℚ)\G(𝔸_{f})/G(ℤ̂).

Proof of injectivity: Let α : (A_{0}, λ_{0}, ι_{0}) → (A, λ_{A}, ι_{A}) and β : (A_{0}, λ_{0}, ι_{0}) → (B, λ_{B}, ι_{B}) be two ℚ-polarized O_{F} -linear isogenies. Suppose that the images of α and β in the double coset are the same, then there exist ϕ ∈ G(ℚ) and (ϕ_{ℓ}) ∈ G(𝔸_{f}) such that αϕ_{ℓ} ∈ Isom(A_{0}(ℓ), A(ℓ)) and βϕϕ_{ℓ} ∈ Isom(A_{0}(ℓ), B(ℓ)) for all ℓ. The polarized O_{F} -quasi-isogeny βϕα^{−1} = βϕϕ_{ℓ}(αϕ_{ℓ})^{−1} : A → B is isomorphic at every place, hence an isomorphism.

Proof of surjectivity: Given [(ϕ_{ℓ})] ∈ G(ℚ)\G(𝔸_{f})/G(ℤ̂), then there is a positive integer N such that Nϕ_{ℓ} : A_{0}(ℓ) → A_{0}(ℓ) is an isogeny for all ℓ. Let H be the product of all ker(Nϕ_{ℓ}). Since each Nϕ_{ℓ} is O_{F} -linear, H is stable under the action of O_{F}. Take A := A_{0}/H, then there is a natural O_{F} -action, ι, on A such that the projection π : A_{0} → A is an O_{F} - linear morphism. The degree of π is N^{2g}. The polarization Nλ_{0} descends to a fractional polarization λ on A. Since we have the induced isomorphism A(ℓ) → A_{0}(ℓ) w.r.t. the polarizations and endomorphisms, λ is ℓ-adic integral for all ℓ, hence a polarization on A. Comparing the degrees of Nλ_{0} and π^{∗}λ, λ is principal. The polarized O_{F} -linear quasi-isogeny N^{−1}π : (A_{0}, λ_{0}, ι_{0}) → (A, λ, ι) gives the class [(ϕ_{ℓ})] by the construction.

Remark. (1) In the proof of the theorem above, we use a special property of supersingular abelian varieties: Hom(A, B) ⊗ ℤ_{p} ≃ Hom(M(A), M(B)) and Hom(A, B) ⊗ ℤ_{ℓ} ≃ Hom(T_{ℓ}(A), T_{ℓ}(B)) for all ℓ ≠ p.

(2) What we have proved for Theorem 9.6 is more general. Let (A_{0}, λ_{0}, ι_{0}) be a supersingular polarized O_{F} -abelian variety over k (p is not necessarily unramified in F, [F : ℚ]|g, and the degree of the polarization can be arbitrary). Then the double coset for the automorphism group of (A_{0}, λ_{0}, ι_{0}) is in bijection corresponding the set of isomorphism classes of supersingular polarized O_{F} -abelian varieties (A, λ, ι) over k such that A_{0}(ℓ) and A(ℓ) are isomorphic as quasi-polarized ℓ-divisible groups with the O_{F} ⊗ ℤ_{ℓ}-actions for all ℓ.

(3) We can consider the statement (2) for the “basic” points in the moduli space of polarized O_{B}-abelian varieties , e.g. see [35]. The statement (2) holds if the same statements for (1) and for Proposition 9.1 hold. In this way one can relate the irreducible components of the basic locus of Shimura varieties of PEL type and the class numbers of certain groups.

Theorem 9.7. Let G be as in Theorem 9.6. The supersingular locus 𝒮 has 6h irreducible components, where h is the class number of G, i.e. the cardinality of the double coset G(ℚ)\G(𝔸_{f})/G(ℤ̂).

Proof. This is nothing but the collection of Proposition 8.2, 8.10 and Theorem 9.6.

10. Cayley-Hamilton for higher a-numbers

The method of Cayley-Hamilton seems getting more complicated when the usual a-number is higher. Fortunately, for our case the situation is not too complicated even dealing with the higher a-number Dieudonné modules. Indeed, it turns out that we only have to deal with 2 or 3 cases separately.

Let M be the Dieudonné module of a point (A, λ, ι) ∈ M(k). Let X, Y be any W (k)-basis of M^{0}. Suppose that

\begin{aligned}F^{f}X & = aX + bY \\ F^{f}Y & = cX + dY \end{aligned}

where a, b, c, d ∈ W. Case 1: If b = c = 0 then slopes of M are ord(a)/f and ord(d)/f. Case 2: If b ≠ 0 then X generates a full-rank submodule and we obtain a Cayley-Hamilton equation

bF^{2f}X − (b^{(f)}d + ba^{(f)})F^{f}X + b^{(f)}(ad − bc)X = 0. (10.1)

Therefore the lowest slope of M is min(\frac{1}{f}(ord(b^{(f)}d + ba^{(f)}) − ord(b)), \frac{1}{2}). Case 3: If c ≠ 0 then we have the similar situation for Y. The lowest slope of M is min(\frac{1}{f}(ord(c^{(f)}a + cd^{(f)}) − ord(c)), \frac{1}{2}).

We will write p_{0}F^{2f}X − p_{1}F^{f}X + p_{2}X = 0 for the equation (10.1) in the rest of the paper.

11. Local moduli spaces I

In this section we compute the local moduli space 𝒮̂_{x} at any given supersingular point x. If N is a stratum of M and x is a point on N, then we denote by N_{x} = Spec R where N̂_{x} = Spf R, the local moduli space at x inside N. We say N_{x} the ordinary scheme associated to the local moduli space N̂_{x}. By Grothendieck’s existence theorem, there is an abelian scheme over N_{x}. As a topological space, N_{x} has more (non-closed) points than its formal completion. This enables us to test if a point 𝔭 of N_{x} lies in the supersingular locus by making the base change Spec k(𝔭) → Spec R.

For an a-index τ or the corresponding a-number, recall that M_{τ} denote the locally closed substack of M whose points have the a-index τ; 𝒮_{τ} = 𝒮 ∩ M_{τ}; and 𝒮̄_{τ} the Zariski closure of 𝒮_{τ} in 𝒮.

The idea to computing 𝒮̂_{x} is as follows:

First, the supersingular locus 𝒮 decomposes into 6 pieces (Proposition 6.3):

𝒮 = ⋃_{τ} 𝒮̄_{τ}, τ : generic a-indices

Hence it suffices to compute the possible components 𝒮̄_{i,x} and 𝒮̄_{(i,i+2),x} in 𝒮_{x}, and then take the union of them. We know when 𝒮_{x} contains a component 𝒮̄_{(i,i+2),x} from the a-number of x. Also since 𝒮̄_{i,i+2} = \overline{M}_{i,i+2} is smooth, there is at most one irreducible component of such type passing through x, and this component is isomorphic to Spec O/(t_{i}, t_{i+2}), where O = k[[t_{0}, t_{1}, t_{2}, t_{3}]]. Therefore, it remains to check if 𝒮̄_{i,x} ⊂ 𝒮_{x} and compute the defining equations.

Suppose that we want to compute 𝒮̄_{τ,x}, where τ = 1000, then it suffices to look at the

supersingular locus, Z, in \overline{M}_{τ,x}. Compute the formal universal deformation over ˆ \overline{M}_{τ,x} = Spf R_{τ}. In fact, using the explicit deformation theory of displays and the Cartier-Dieudonné theory, we directly write down the deformation over Spec R_{τ}. Compute the coefficients p_{0}, p_{1} (defined at the end of the previous section) of the Cayley-Hamilton equation by iterating the Frobenius map. The next step, roughly speaking, is to “catch” each irreducible component of Z from the closed congruence condition “ord(p_{1}) − ord(p_{0}) ≥ 2”. The following is an explicit recipe.

Write p_{i} = (p_{i0}, p_{i1}, …), the coordinates in the Witt ring W (R_{τ}). Denote by p_{i}(𝔭) the image of p_{i} in W ((R_{τ}/𝔭)^{pf}). Note that p_{i}(𝔭) ≡ 0 (mod p^{n+1}) iff 𝔭 ∈ V (p_{i0}, p_{i1}, …, p_{i,n}). Therefore, the reduced subscheme V = V (p_{i0}, p_{i1}, …, p_{i,n}) is characterized by the property that 𝔭 ∈ V iff p_{i}(𝔭) ≡ 0 (mod p^{n+1}). So we will say V is defined by the congruence condition p_{i} ≡ 0 (mod p^{n+1}). The supersingular locus Z in \overline{M}_{τ,x} is contained in the reduced subscheme, Z_{1}, of \overline{M}_{τ,x} defined by p_{1} ≡ 0 mod p^{2}, or equivalently, by p_{11} = 0 (The condition p_{10} = 0 is satisfied since we are deforming inside \overline{M}_{τ}). Look at the irreducible components of Z_{1}, and then throw away the components which cause non-generic a-numbers locus since these are not whole components of Z. The remaining components are our candidates, and the supersingular locus Z is contained in the union of them. Let V_{1} be one of these candidates. If V_{1} is not contained in the reduced subscheme B_{1} defined by p_{0} ≡ 0 (mod p), or equivalently by p_{00} = 0, then V_{1} is a whole component of Z.

If V_{1} is contained in B_{1}, the another condition p_{1} ≡ 0 mod p^{3} must be satisfied. The new equation p_{12} = 0 defines a reduced subscheme Z_{2}, and we throw away the bad components of Z_{2} again. An irreducible component of Z contained in V_{1} must be contained in V_{1} ∩ V_{2} for some good component V_{2} of Z_{2}. If V_{1} ∩ V_{2} is not contained in the reduced subscheme B_{2} defined by p_{0} ≡ 0 mod p^{2}, then it is a whole component. If it is in the other case, then another higher congruence condition of p_{1} must be satisfied. Continuing this process until finding out one irreducible component of Z.

Now we are ready to compute 𝒮̄_{τ,x} for |τ| = 1. Because of symmetry, we may assume a(x) = 1111, 1110, 1100, 1010 or 1000. The case a(x) = 1000 is done in Section 5. We see that a(x) = 1100 does not exist (Corollary 8.8). We will do the case a(x) = 1111 in this Section and put the remaining cases in Section 13.

Let x ∈ M_{1111}(k) be a superspecial point, M its Dieudonné module. We can choose a basis {X_{i}, Y_{i}} of M such that F X_{i} = −Y_{i+1}, F Y_{i} = pX_{i+1} for i = 0, …, 3. The universal deformation of M over \overline{M}_{1000,x} = Spec O/(t_{0}) in \overline{M}_{1000} is given by the relation

F X_{0} = −Y_{1} − T_{1}X_{1}
F X_{1} = −Y_{2} − T_{2}X_{2}
F X_{2} = −Y_{3} − T_{3}X_{3}
F X_{3} = −Y_{0},

where T_{i} is the Teichmüller lifting of t_{i}. Iterating the Frobenius map, we obtain

F^{4}X_{0} = −pF^{2}X_{2} − T_{1}^{(3)}F^{3}X_{1}
F^{3}X_{1} = pY_{0} − T_{2}^{(2)}F^{2}X_{2}
F^{2}X_{2} = −pX_{0} + T_{3}^{(1)}Y_{0}
F^{4}Y_{0} = pF^{3}X_{1}

We compute that

a = p(p − T_{1}^{(3)}T_{2}^{(2)})
b = T_{1}^{(3)}T_{2}^{(2)}T_{3}^{(1)} − p(T_{3}^{(1)} + T_{1}^{(3)})
c = p^{2}T_{2}^{(2)}
d = p(p − T_{2}^{(2)}T_{3}^{(1)})

p_{0} = b = T_{1}^{(3)}T_{2}^{(2)}T_{3}^{(1)} + higher p-power terms
p_{1} = (b^{(f)} d + b a^{(f)})
= −pT_{2}^{(2)}T_{3}^{(1)}T_{1}^{(7)}T_{2}^{(6)}(T_{3}^{(5)} + T_{1}^{(3)}) + higher p-power terms.

The condition p_{1} ≡ 0 mod p^{2} defines a reduced subscheme t_{1}t_{2}t_{3}(t_{3}^{p^{2}} + t_{1}) = 0. The components t_{1} = 0 and t_{3} = 0 cause non-generic a-numbers, so they are out. The component t_{2} = 0 is just the component corresponding to a-number 1010. The condition p_{0} ≡ 0 mod p defines the reduced scheme B_{1} by the equation t_{1}t_{2}t_{3} = 0. Since the component t_{3}^{p^{2}} + t_{1} = 0 is not contained in the bad locus B_{1}, it is a component of the supersingular locus. The generic fibre has the a-number 1000, hence 𝒮̄_{1000,x} ≃ Spec O/(t_{0}, t_{3}^{p^{2}} + t_{1}). Similarly, we obtain that 𝒮̄_{i,x} = Spec O/(t_{i}, t_{i−1}^{p^{2}} + t_{i+1}), for i ∈ ℤ/4ℤ.

Theorem 11.1. Suppose x = (A, λ, ι) ∈ M_{1111}(k). Then the local moduli space 𝒮̂_{x} ≃ Spf k[[t_{0}, t_{1}, t_{2}, t_{3}]]/I, where I is the intersection of ideals (t_{0}, t_{2}), (t_{1}, t_{3}) and (t_{i}, t_{i−1}^{p^{2}} +t_{i+1}) for i = 0, …, 3.

Let x ∈ 𝒮_{1110}(k). From Section 13, one concludes that \overline{M}_{1000,x}∩𝒮 ⊂ 𝒮̄_{1010} and \overline{M}_{0010,x}∩𝒮 ⊂ 𝒮̄_{1010}. Therefore there are two irreducible components of 𝒮̄_{x}:

(11.1)  𝒮̄_{1010,x} ≃ Spec O/(t_{0}, t_{1})
(11.2)  𝒮̄_{0100,x} ≃ Spec O/(t_{1}, t_{2} + t_{0}^{p^{2}} + ā_{1}^{p}t_{2}t_{0}^{p^{2}}),

where ā_{1} is the coefficient a_{1} in (13.1) mod p.

Let x ∈ 𝒮_{1010}(k). From Section 13, we have the local moduli spaces

(11.3)  𝒮_{x} ∩ \overline{M}_{1000,x} ≃ Spec O/(t_{2}, t_{0}(t_{3} − ā_{2}^{p}))
(11.4)  𝒮_{x} ∩ \overline{M}_{0010,x} ≃ Spec O/(t_{0}, t_{2}(t_{1} − ā_{0}^{p})),

where ā_{0} and ā_{2} the coefficients a_{0} and a_{2} in (13.3) mod p. These local moduli spaces are irreducible or not depending on the divisibility of a_{0} and a_{2} by p. Nevertheless, each irreducible component is smooth at x.

From the local moduli spaces 𝒮̄_{τ,x} for all generic a-numbers τ and all points x, we get

Theorem 11.2. Every irreducible component of 𝒮 is smooth as an algebraic stack.

12. Intersections

In this section, we will describe the intersection of two irreducible components. We content ourselves by describing the intersections in the supersingular locus with an auxiliary level structure. In this setting one avoids the issue whether the operations the quotient and intersection in our case commute. Therefore we change our notations a bit.

Let n be an integer such that n ≥ 3 and (n, p) = 1. Let M be the moduli space of principally polarized O_{F} -abelian varieties in characteristic p with symplectic n-level structure, i.e. M := A_{g,n}^{O_{F}} ⊗_{ℤ[ζ_{n},1/n]} 𝔽̄_{p}. Denote by 𝒮 the supersingular locus of M. For each a-index (or a-number) τ, we denote 𝒮_{τ} the reduced subscheme of 𝒮 whose points have the a-index (or a-number) τ, and 𝒮̄_{τ} the Zariski closure of 𝒮_{τ} in 𝒮. Finally, let S_{i}, Λ_{i} denote the sets as in Section 8 except the abelian varieties there with symplectic n-level structure.

From (8.3) and Theorem 9.7, one deduces that the supersingular locus has 6h_{n} irreducible components and #P^{1}(𝔽_{p^{4}}) · h_{n} superspecial points. Here the number h_{n} is

#G(ℚ)\G(𝔸_{f})/K_{n} = [G(ℤ̂) : K_{n}] (∑_{c} \frac{1}{|Γ_{c}|}) , (12.1)

where K_{n} is the kernel of the map G(ℤ̂) → G(ℤ̂/nℤ̂). The sum is taken over all the representatives of the double coset G(ℚ)\G(𝔸_{f})/G(ℤ̂), and Γ_{c} = G(ℚ) ∩ cG(ℤ̂)c^{−1} is the stabilizer of the coset cG(ℤ̂) in G(ℚ). Each irreducible component of 𝒮 is isomorphic to exact one model N_{τ,η} (defined in Section 8) and has #P^{1}(𝔽_{p^{4}}) superspecial points.

As each irreducible component contains a superspecial point, one describes the intersections near a superspecial point first.

Lemma 12.1. Let x be a superspecial point and O := k[[t_{0}, t_{1}, t_{2}, t_{3}]].

(1) 𝒮̄_{1010,x} ∩ 𝒮̄_{0101,x} ≃ Spec O/(t_{0}, t_{1}, t_{2}, t_{3}).

(2) 𝒮̄_{1010,x} ∩ 𝒮̄_{1000,x} ≃ Spec O/(t_{0}, t_{2}, t_{1} + t_{3}^{p^{2}}).

(3) 𝒮̄_{1010,x} ∩ 𝒮̄_{0100,x} ≃ Spec O/(t_{0}, t_{1}, t_{2}).

(4) 𝒮̄_{1000,x} ∩ 𝒮̄_{0100,x} ≃ Spec O/(t_{0}, t_{1}, t_{2}, t_{3}^{p^{2}}).

(5) 𝒮̄_{1000,x} ∩ 𝒮̄_{0010,x} ≃ Spec O/(t_{0}, t_{1}, t_{2}, t_{3}).

Proposition 12.2.

(1) The intersection 𝒮̄_{1010} ∩ 𝒮̄_{0101} consists of all superspecial points with multiplicity one.

(2) The intersection 𝒮̄_{1010} ∩ 𝒮̄_{1000} is a smooth complete curve whose generic points have anumber 1010.

(3) The intersection 𝒮̄_{1010} ∩ 𝒮̄_{0100} is \overline{M}_{1110}. Moreover, the intersection of any component of 𝒮̄_{1010} and any component of 𝒮̄_{0100}, if not empty, is isomorphic to P^{1}.

(4) The intersection 𝒮̄_{1000} ∩ 𝒮̄_{0100} consists of all superspecial points with multiplicity p^{2}.

(5) The intersection 𝒮̄_{1000} ∩ 𝒮̄_{0010} consists of all superspecial points with multiplicity one and some other points with multiplicity one.

Proof. (1) is clear and (3) follows from Proposition 8.4.

(2) We have checked that the intersection is smooth at superspecial points. Let x ∈ M_{1010}∩ 𝒮̄_{1000}. From the previous section, we know that 𝒮̄_{1000,x} = Spec O/(t_{0}, t_{3}), hence the intersection of 𝒮̄_{1010} and 𝒮̄_{1000} at x has local equations t_{0}, t_{2}, t_{3}.

(4) Let x ∈ 𝒮̄_{1000} ∩ 𝒮̄_{0100}. If a(x) = 1110, then 𝒮̄_{1000,x} = ϕ by (13.2). Similarly, a(x) ≠ 1101. By Corollary 8.8, a(x) = 1111.

(5) Let x ∈ 𝒮̄_{1000} ∩ 𝒮̄_{0010}, then x ∈ 𝒮̄_{1010}. By (13.2), a(x) ≠ 1110 or 1011. If a(x) = 1111, we have Lemma 12.1 (5). If a(x) = 1010, it follows from (13.4) that 𝒮̄_{1000,x} ∩ 𝒮̄_{0010,x} = Spec O/(t_{0}, t_{1}, t_{2}, t_{3}).

We will determine that the complete smooth curve in (2) is isomorphic to a disjoint union of P^{1}’s. We need a lemma.

Lemma 12.3. Let M = M^{0} ⊕ ⋯ ⊕ M^{3} be a Dieudonné module given by the relation

\begin{matrix}F X_{0} = X_{1}, & F Y_{0} = pY_{1} \\ F X_{1} = −Y_{2} + pa_{2}X_{2}, & F Y_{1} = pX_{2} \\ F X_{2} = X_{3}, & F Y_{2} = pY_{3} \\ F X_{3} = −Y_{0} + pa_{0}X_{0}, & F Y_{3} = pX_{0} \end{matrix}

for some a_{0}, a_{2} ∈ W (k). Then

(i) a_{2} ≡ 0 mod p ⟺ F^{4}(M^{0}) ⊂ p^{2}M^{0}. (ii) a_{0} ≡ 0 mod p ⟺ F^{4}(M^{2}) ⊂ p^{2}M^{2}.

Proof. Set \overline{M} := M/p^{2}M and all elements are considered as their images in \overline{M}.

F \overline{M} = span\{X_{1}, Y_{2}, X_{3}, Y_{0}, pY_{1}, pX_{2}, pY_{3}, pX_{0}\}

\begin{matrix}(∵ & span\{−Y_{2} + pa_{2}X_{2}, pX_{2}\} = span\{Y_{2}, pX_{2}\}, \\ span\{−Y_{0} + pa_{0}X_{0}, pX_{0}\} = span\{Y_{0}, pX_{0}\}) \end{matrix}

F^{2}\overline{M} = span\{−Y_{2} + pa_{2}X_{2}, pY_{3}, −Y_{0} + pa_{0}X_{0}, pY_{1}, pX_{3}, pX_{1}\}
F^{3}\overline{M} = span\{−pY_{3} + pa_{2}^{(1)}X_{3}, −pY_{1} + pa_{0}^{(1)}X_{1}, −pY_{0}, −pY_{2}\}
F^{4}\overline{M} = span\{pa_{2}^{(2)}Y_{0}, pa_{0}^{(2)}Y_{2}\}.

Therefore we are done.

Proposition 12.4. The set of k-points of N_{1010,η} = P^{1} ×P^{1} which belong to 𝒮̄_{1000} is {(x, y) ∈ P^{1} × P^{1}(k) | x ∈ P^{1}(𝔽_{p^{4}})}

Proof. Use the affine coordinates as (8.2), for any point (x, y), the Dieudonné module corresponded is written by

M_{(x,y)} = span \{ \begin{matrix}X_{0}, & X_{1}, & X_{2}, & X_{3} \\ V_{0}, & Y_{1}, & V_{2}, & Y_{3} \end{matrix} \} ,

where V_{0} = x̃/pX_{0} + 1/pY_{0} and V_{2} = ỹ/pX_{2} + 1/pY_{2}. We have F V_{0} = −x̃^{(1)}/pY_{1} + X_{1} and F V_{2} = −ỹ^{(1)}/pY_{3} + 1/pX_{3}.

The point belongs to 𝒮̄_{1000} iff the condition F^{4}M_{(x,y)}^{0} ⊂ p^{2}M_{(x,y)}^{0} is satisfied. Iterating the Frobenius map F, we get

X_{0} →^{F} −pY_{1} →^{F} −pX_{2} →^{F} p^{2}Y_{3}
V_{0} →^{F} −x̃^{(1)}Y_{1} + X_{1}

→^{F} −x̃^{(2)}X_{2} − Y_{2}

→^{F} x̃^{(3)}pY_{3} − pX_{3}

→^{F} x̃^{(4)}pX_{0} + pY_{0} = (x̃^{(4)} − x̃)pX_{0} + p^{2}V_{0}.

The condition gives x̃^{(4)} − x̃ ≡ 0 mod p, i.e. x^{p^{4}} − x = 0.

Corollary 12.5. Each connected component of the intersection 𝒮̄_{1010} ∩ 𝒮̄_{1000} is isomorphic to P^{1}.

Remark. The intersection of one component of 𝒮̄_{1010} and one of component of 𝒮̄_{1000} may not be connected. By Proposition 8.4 (3) and 12.4, every connected component of the intersection contains only one superspecial point. If the intersection is connected, then the number h_{n} of components of 𝒮̄_{1000} must be at least |P^{1}(𝔽_{p^{4}})|, the number of superspecial points on one component of 𝒮̄_{1010}. This is not true in general.

13. Local moduli spaces II

In this section we compute the local moduli space 𝒮̂_{x} for a(x) = 1110 and a(x) = 1010, the remaining cases. We denote x^{(n)} = x^{σ^{n}} for simplicity.

Let x ∈ 𝒮_{1110}(k) and M be its Dieudonné module. Take a basis of M such that

\begin{matrix}F X_{0} = −Y_{1} + pa_{1}X_{1}, & F Y_{0} = pX_{1} \\ F X_{1} = −Y_{2} + pa_{2}X_{2}, & F Y_{1} = pX_{2} \\ F X_{2} = X_{3}, & F Y_{2} = pY_{3} \\ F X_{3} = −Y_{0} + pa_{0}X_{0}, & F Y_{3} = pX_{0}. \end{matrix} (13.1)

The universal deformation of M over M_{x} is given by

\begin{matrix}F X_{0} = −Y_{1} + (pa_{1} − T_{1})X_{1}, & F Y_{0} = pX_{1} \\ F X_{1} = −Y_{2} + (pa_{2} − T_{2})X_{2}, & F Y_{1} = pX_{2} \\ F X_{2} = X_{3}, & F Y_{2} = pY_{3} + pT_{3}X_{3} \\ F X_{3} = −Y_{0} + (pa_{0} − T_{0})X_{0}, & F Y_{3} = pX_{0}. \end{matrix}

where T_{i} is the Teichmüller lift of t_{i}. Iterating the Frobenius map, one has

F^{4}X_{2} = −pF X_{1} + (pa_{0}^{(2)} − T_{0}^{(2)})F^{2}X_{0}
F^{2}X_{0} = −pX_{2} + (pa_{1}^{(1)} − T_{1}^{(1)})F X_{1}
F X_{1} = −Y_{2} + (pa_{2} − T_{2})X_{2}
F^{4}Y_{2} = p^{2}F^{2}X_{0} + pT_{3}^{(3)}F^{4}X_{2}.

Case 1: t_{0} = 0, i.e. considering the components of 𝒮_{x} in \overline{M}_{1000,x}. We compute that

p_{0} = b = p(1 + a_{0}^{(2)}T_{1}^{(1)} − pa_{0}^{(2)}a_{1}^{(1)})
p_{1} = p^{2}(1 + a_{0}^{(2)}T_{1}^{(1)})T_{2}^{(4)}(1 + a_{0}^{(6)}T_{1}^{(5)}) + p^{3}(⋯)

Note that ord(p_{0}) = 1, and the condition p_{1} ≡ 0 mod p^{3} gives the equation t_{2} = 0. The supersingular locus in \overline{M}_{1000,x} is given by equations t_{0} = t_{2} = 0, and it is 𝒮̄_{1010,x}.

Case 2: t_{1} = 0, i.e. considering the components of 𝒮_{x} in \overline{M}_{0100,x}. We compute that

p_{0} = b = p(1 + a_{1}^{(1)}T_{0}^{(2)} − pa_{0}^{(2)}a_{1}^{(1)})
p_{1} = p^{2}(1 + a_{1}^{(1)}T_{0}^{(2)})(T_{0}^{(6)} + a_{1}^{(5)}T_{0}^{(6)}T_{2}^{(4)}) + p^{3}(⋯)

Note that ord(p_{0}) = 1, and the condition p_{1} ≡ 0 mod p^{3} gives the equation t_{0}^{p^{2}} +t_{2} +ā_{1}^{p}t_{0}^{p^{2}}t_{2} = 0. The supersingular locus in \overline{M}_{1000,x} is given by equations t_{1} = t_{0}^{p^{2}} + t_{2} + ā_{1}^{p}t_{0}^{p^{2}}t_{2} = 0, and it is 𝒮̄_{0100,x}.

Case 3: t_{2} = 0, i.e. considering the components of 𝒮_{x} in \overline{M}_{0010,x}. We compute that

p_{0} = −T_{0}^{(2)}T_{1}^{(1)} + p(1 + a_{1}^{(1)}T_{0}^{(2)} + a_{0}^{(2)}T_{1}^{(1)} − pa_{0}^{(2)}a_{1}^{(1)})
p_{1} = pT_{0}^{(2)}T_{1}^{(1)}T_{0}^{(6)}(T_{1}^{(5)}T_{3}^{(3)} − T_{1}^{(5)}T_{2}^{(4)} − 1) + p^{2}(⋯).

The congruence condition p_{1} ≡ 0 mod p^{2} gives the equation t_{0}t_{1} =. The component t_{1} = 0 will cause a non-generic a-number, so we throw it away. The only possible component t_{0} = 0 has the a-number 1010, which is supersingular . Hence we get one component whose defining equations are t_{0} = t_{2} = 0.

We conclude that \overline{M}_{1000,x} ∩ 𝒮 ⊂ 𝒮̄_{1010} and \overline{M}_{0010,x} ∩ 𝒮 ⊂ 𝒮̄_{1010}. Therefore there are two irreducible components of 𝒮̄_{x}:

𝒮̄_{1010,x} ≃ Spec O/(t_{0}, t_{2})
𝒮̄_{0100,x} ≃ Spec O/(t_{1}, t_{2} + t_{0}^{p^{2}} + ā_{1}^{p}t_{2}t_{0}^{p^{2}}),

(13.2)

where O := k[[t_{0}, t_{1}, t_{2}, t_{3}]].

Let x ∈ 𝒮_{1010}(k), and let M be its Dieudonné module. Take a basis of M such that

\begin{matrix}F X_{0} = X_{1}, & F Y_{0} = pY_{1} \\ F X_{1} = −Y_{2} + pa_{2}X_{2}, & F Y_{1} = pX_{2} \\ F X_{2} = X_{3}, & F Y_{2} = pY_{3} \\ F X_{3} = −Y_{0} + pa_{0}X_{0}, & F Y_{3} = pX_{0}. \end{matrix} (13.3)

The universal deformation of M over \overline{M}_{0010,x} is

\begin{matrix}F X_{0} = X_{1}, & F Y_{0} = pY_{1} + pT_{1}X_{1} \\ F X_{1} = −Y_{2} + pa_{2}X_{2}, & F Y_{1} = pX_{2} \\ F X_{2} = X_{3}, & F Y_{2} = pY_{3} + pT_{3}X_{3} \\ F X_{3} = −Y_{0} + (pa_{0} − T_{0})X_{0}, & F Y_{3} = pX_{0}. \end{matrix}

where T_{i} is the Teichmüller lift of t_{i}. Iterating the Frobenius map, one has

F^{4}X_{0} = −F^{2}Y_{2} + pa_{2}^{(2)}F X_{3}
F^{2}Y_{2} = p^{2}X_{0} + pT_{3}^{(1)}F X_{3}
F X_{3} = −Y_{0} + (pa_{0} − T_{0})X_{0}
F^{4}Y_{0} = p^{2}F X_{3} + pT_{1}^{(3)}F^{4}X_{0}

We compute that

a = −pT_{0}^{(0)}(−T_{3}^{(1)} + a_{2}^{(2)})
d = −p^{2}(1 − T_{1}^{(3)}T_{3}^{(1)} + T_{1}^{(3)}a_{2}^{(2)}).

Since p^{2}|d and p_{1} = b^{(f)}d+ba^{(f)}, the slope will go up to 1/2 if and only if p^{2}|a, which gives the equation t_{0}(t_{3} − ā_{2}^{p}) = 0. Hence the supersingular locus in \overline{M}_{0010,x} is defined by the equations t_{2} = t_{0}(t_{3} − ā_{2}^{p}) = 0.

We conclude that

𝒮_{x} ∩ \overline{M}_{0010,x} ≃ Spec O/(t_{2}, t_{0}(t_{3} − ā_{2}^{p})),
𝒮_{x} ∩ \overline{M}_{1000,x} ≃ Spec O/(t_{0}, t_{2}(t_{1} − ā_{0}^{p})).
𝒮̄_{0010,x} ≠ ϕ ⟺ ā_{2} = 0 ⟹ 𝒮̄_{0010,x} ≃ Spec O/(t_{2}, t_{3})
𝒮̄_{1000,x} ≠ ϕ ⟺ ā_{0} = 0 ⟹ 𝒮̄_{1000,x} ≃ Spec O/(t_{0}, t_{1}).

(13.4)

14. Appendix: Deformation via the Hecke correspondence

Let N be the moduli stack parameterizing the triple ((A_{1}, λ_{1}, ι_{1}), (A_{2}, λ_{2}, ι_{2}), φ) where (A_{1}, λ_{1}, ι_{1}) and (A_{2}, λ_{2}, ι_{2}) are in M and φ : A_{1} → A_{2} is an O_{F} -linear isogeny morphism such that φ^{∗}λ_{2} = pλ_{1}. The algebraic stack N is a Hecke correspondence of M, or called the p-parahoric level structure of M. Let p_{1}, p_{2} be the two projection morphisms to M. The idea to deform a higher a-number point to a lower one along a slope stratum is the follows: for a point (A, λ, ι) of higher a-number, find a B ∈ N such that p_{1}(B) = A. Then we compute the fibre p_{2}^{−1}(p_{2}(B)) of p_{2}(B) and determine the a-numbers of the points in the fibre.

Given M, the covariant Dieudonné module of a point (A, λ, ι) ∈ M(k). Let N be the Dieudonné module containing M such that V N = M. We have pN ⊂ M ⊂ N and τ_{N} = {n_{1} + 1, …, n_{a} + 1} if τ_{M} = {n_{1}, …, n_{a}}. Let {X_{i}, Y_{i}} be a basis of N as in Proposition 4.2. We have the relation:

F X_{i} = \begin{cases}X_{i+1} & if i ∉ τ_{M}, \\ Y_{i+1} + pa_{i+1}X_{i+1} & if i ∈ τ_{M}. \end{cases}  V X_{i} = \begin{cases}pX_{i−1} & if i ∉ τ_{N}, \\ Y_{i−1} & if i ∈ τ_{N}. \end{cases}

F Y_{i} = \begin{cases}pY_{i+1} & if i ∉ τ_{M}, \\ pX_{i+1} & if i ∈ τ_{M}. \end{cases}  V Y_{i} = \begin{cases}Y_{i−1} & if i ∉ τ_{N}, \\ pX_{i−1} − pa_{i+1}^{σ^{−1}}pY_{i−1} & if i ∈ τ_{N}. \end{cases}

where a_{i} are elements of W (k) for a suitable i. For the Dieudonné module N, we have the following

i ∉ τ_{N} F X_{i−1} = X_{i}  i + 1 ∉ τ_{N} V X_{i+1} = pX_{i}
i ∈ τ_{N} F X_{i−1} = Y_{i} + a_{i}pX_{i}  i + 1 ∈ τ_{N} V X_{i+1} = Y_{i}
i ∉ τ_{N} F Y_{i−1} = pY_{i}  i + 1 ∉ τ_{N} V Y_{i+1} = Y_{i}
i ∈ τ_{N} F Y_{i−1} = pX_{i}  i + 1 ∈ τ_{N} V Y_{i+1} = pX_{i} − a_{i}^{σ^{−1}}pY_{i}

For the Dieudonné module M = span⟨Y_{i}, pX_{i}⟩, we have the following

i − 1 ∉ τ_{M} F (pX_{i−1}) = pX_{i}  i ∉ τ_{M} V (pX_{i+1}) = p(pX_{i})
i − 1 ∈ τ_{M} F (pX_{i−1}) = pY_{i} + a_{i}p(pX_{i})  i ∈ τ_{M} V (pX_{i+1}) = pY_{i}
i − 1 ∉ τ_{M} F Y_{i−1} = pY_{i}  i ∉ τ_{M} V Y_{i+1} = Y_{i}
i − 1 ∈ τ_{M} F Y_{i−1} = (pX_{i})  i ∈ τ_{M} V Y_{i+1} = (pX_{i}) − a_{i}^{σ^{−1}}pY_{i}

Consider a family of maximal isotropic subspaces of N̄ = N/p stable under the action of O_{F}: \{0 ⊂ \overline{M}_{t} = span⟨Y_{i} + t_{i}X_{i}⟩ ⊂ \overline{N}; t_{i} ∈ k\}. This is an affine open neighborhood of M/pN in the (compact) Lagrangian subvariety compatible with the O_{F}. The preimage M_{t} is span⟨Y_{i} + T_{i}X_{i}, pX_{i}⟩ , where T_{i} = [t_{i}] is the Teichmüller lifting of t_{i}. The W -module M_{t} is Dieudonné if and only if the variables t_{i} satisfy the relations F M̄_{t} ⊂ M̄_{t} and V M̄_{t} ⊂ M̄_{t}. But in fact both relations provide the same equations and we just apply F on M̄_{t}.

i ∉ τ_{M}, F : Y_{i} + t_{i}X_{i} → t_{i}^{p}X_{i+1} ⇒ t_{i}^{p} = 0
i ∈ τ_{M}, F : Y_{i} + t_{i}X_{i} → t_{i}^{p}Y_{i+1} = t_{i}^{p}(Y_{i+1} + t_{i+1}X_{i+1}) − t_{i}^{p}t_{i+1}X_{i+1}
⇒ t_{i}^{p}t_{i+1} = 0

This is the local equations of the fibre mention above.

As we will “physically” deform M (t_{i} is taken the values in k), we may take the reduced structure of the deformation space.

Example.

1. When a_{M} = (1, 0, 1, 0, 0), the defining equations are t_{0}t_{1}, t_{1}, t_{2}t_{3}, t_{3}, t_{4}, and the deformation space is Spec k[t_{0}, t_{2}].

2. When a_{M} = (1, 1, 0, 1, 1, 1, 0), the defining equations are t_{0}t_{1}, t_{1}t_{2}, t_{2}, t_{3}, t_{3}t_{4}, t_{4}t_{5}, t_{5}t_{6}, t_{6}, and the deformation space is Spec k[t_{0}, t_{1}, t_{3}, t_{4}, t_{5}]/(t_{0}t_{1}, t_{3}t_{4}, t_{4}t_{5})

3. In general, the deformation space is Spec k[t_{i}]_{i∈τ_{M}} /(t_{i}t_{i+1})_{i,i+1∈τ_{M}}

We rewrite the deformed Dieudonné module M_{t} as span⟨X̃_{i}, Ỹ_{i}⟩, where X̃_{i} = pX_{i}, Ỹ_{i} = Y_{i} + T_{i}X_{i}, where T_{i} = 0 if i ∉ τ_{M}. The relations in terms of this new basis are

i − 1 ∉ τ_{M} : F X̃_{i−1} = X̃_{i}
i − 1 ∉ τ_{M} : F Ỹ_{i−1} = pỸ_{i} − T_{i}X̃_{i}
i − 1 ∈ τ_{M} : F X̃_{i−1} = pỸ_{i} + (a_{i}p − T_{i})X̃_{i}
i − 1 ∈ τ_{M} : F Ỹ_{i−1} = T_{i−1}^{σ}Ỹ_{i} + (1 + a_{i}T_{i−1}^{σ} − c_{i−1})X̃_{i}
i ∉ τ_{M} : V X̃_{i+1} = pX̃_{i}
i ∉ τ_{M} : V Ỹ_{i+1} = Ỹ_{i} + T_{i+1}^{σ^{−1}} X̃_{i}
i ∈ τ_{M} : V X̃_{i+1} = pỸ_{i} − T_{i}X̃_{i}
i ∈ τ_{M} : V Ỹ_{i+1} = (T_{i+1}^{σ^{−1}} − a_{i}^{σ^{−1}}p)Ỹ_{i} + (1 + a_{i}^{σ^{−1}}T_{i} − c_{i}^{σ^{−1}})X̃_{i}

where c_{i} = T_{i}^{σ}T_{i+1}/p. In the following, we will deform by taking the values either t_{i} = 0 or t_{i+1} = 0, therefore we may assume c_{i} = 0.

We now look at the ith component ((F, V)M̄_{t})^{i} of (F, V)M_{t}/pM_{t}. Case i ∉ τ_{M}:

(i) i − 1 ∉ τ_{M}: ((F, V)M̄_{t})^{i} is generated by Ỹ_{i} + t_{i+1}^{p^{−1}} X̃_{i}, X̃_{i}. Then a_{i}(M_{t}) = 0 (ii) i − 1 ∈ τ_{M}: ((F, V)M̄_{t})^{i} is generated by Ỹ_{i} + t_{i+1}^{p^{−}1}X̃_{i} and t_{i−1}^{p}Ỹ_{i} + (1 + a_{i}t_{i−1}^{p})X̃_{i}. As the

discriminant δ = 1 + a_{i}t_{i−1}^{p} − t_{i+1}^{p^{−1}}t_{i−1}^{p} ≠ 0 for a general value of t_{i−1}, we have a_{i}(M_{t}) = 0.

Case i ∈ τ_{M}: ((F, V)M̄_{t})^{i} contains t_{i}X̃_{i} and t_{i+1}^{p^{−1}} Ỹ_{i} + (1 + a_{i}^{p^{−1}}t_{i})X̃_{i}, thus contains t_{i}X̃_{i} and t_{i+1}^{p^{−1}} Ỹ_{i} + X̃_{i}.

(i) i − 1 ∉ τ_{M}: ((F, V)M̄_{t})^{i} is generated by t_{i+1}^{p^{−1}} Ỹ_{i}, X̃_{i}.

(ii) i − 1 ∈ τ_{M}: ((F, V)M̄_{t})^{i} is generated by t_{i+1}^{p^{−1}} Ỹ_{i} + X̃_{i}, t_{i}X̃_{i}, and t_{i−1}^{p}Ỹ_{i} + (1 + a_{i}^{p}t_{i−1})X̃_{i}. If i + 1 ∉ τ_{M}, then t_{i+1} = 0 and it is generated by X̃_{i}, t_{i−1}^{p}Ỹ_{i}.

In the case i ∈ τ_{M}, we have the following subcase 1. i − 1, i + 1 ∉ τ_{M}: ((F, V)M̄_{t})^{i} is generated by X̃_{i} ⇒ a_{i}(M_{t}) = 1. subcase 2. i − 1 ∉ τ_{M}, i + 1 ∈ τ_{M}: let t_{i+1} ≠ 0, t_{i} = 0 ⇒ a_{i}(M_{t}) = 0. subcase 3. i − 1 ∈ τ_{M}, i + 1 ∉ τ_{M}: let t_{i−1} ≠ 0, t_{i} = 0 ⇒ a_{i}(M_{t}) = 0. subcase 4. i − 1 ∈ τ_{M}, i + 1 ∈ τ_{M}: take t_{i} = 0, t_{i−1}^{p} ≠ (1 + a_{i}^{p}t_{i−1})t_{i+1}^{p^{−1}} ⇒ a_{i}(M_{t}) = 0

Theorem 14.1. Let (A, λ, ι) ∈ M(k). Assume a_{i}(A) = a_{i+1}(A) = 1. Then there exist a deformation (A_{t}, λ_{t}, ι_{t}) keeping in the same slope such that either a_{i}(A_{t}) = 0, a_{i+1}(A_{t}) = 1 or a_{i}(A_{t}) = 1, a_{i+1}(A_{t}) = 0

Corollary 14.2. Let M^{(m)} be as in Section 5. Then \overline{M^{(m)}} = ∪_{τ} \overline{M_{τ}^{(m)}}. where τ runs over all generic a-numbers.

Corollary 14.3. If \overline{M_{τ}^{(m+1)}} ⊂ \overline{M_{τ}^{(m)}} in M_{τ} holds for all m + 1 ≥ |τ| and all generic anumbers τ. Then \overline{M^{(m+1)}} ⊂ \overline{M^{(m)}} for all m < (f/2).

Corollary 14.4. The a-number of every generic point of M^{(m)} is generic.

Conversely, we expect that the points of every generic a-numbers do form whole components of each slope stratum (if M_{τ}^{(m)} is not empty). That is, we expect that if τ is generic each component of M_{τ}^{(m)}, if not empty, has dimension g − m. This is a refinement of Conjecture 5.2.5 in [9].

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Department of Mathematics, Columbia University, New York, NY 10027, USA

  E-mail address: chiafu@math.columbia.edu