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 4 3. a-number stratification 5 4. Normal forms 6 5. Cayley-Hamilton for a = 1 9 6. Deformation via the Hecke correspondence 11 7. Preliminary results for g = 4 14 8. Canonical isogenies 15 9. Construction of models 17 10. Superspecial points and class numbers 25 11. Cayley-Hamilton for higher a-numbers 27 12. Local moduli spaces I 28 13. Intersections 31 14. Local moduli spaces II 33 References 35

1. Introduction

We attempt to extend the work of K.-Z. Li and F. Oort [19] on the supersingular locus of Siegel modular varieties to Hilbert-Blumenthal varieties in positive characteristic. We 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? Recall that the a-number of an abelian variety A of characteristic p is the dimension of the

vector space Hom(α_{p}, A). The supersingular locus is the closed subset of the moduli space which parameterizes supersingular abelian varieties. The work of Li and Oort answers to all these questions and the answer to 3 is affirmative.

We limit ourselves to the case that the moduli space has good reduction. This happens exactly when the rational prime p is unramified in the totally real field F [7, Thm 2.2]. In order to concentrate on the essential difficulty, we assume in this paper that the prime number p is inert in the totally real field F. This assumption does not lose the generality for answering the question 1 and 3, by the Serre-Tate theorem.

If one knows that the answer to 3 is affirmative, then one can use the method developed in [31] to handle the first question and use the method in [19] to solve the second one. A priori, we do not know this until we have enough information on 1 and 2. However, this is not the case for Hilbert-Blumenthal varieties. The first interesting cases are Hilbert-Blumenthal 4-folds.

In this paper we work out the description of the supersingular locus of Hilbert-Blumenthal 4-folds. The main part of our results will be summarized in a theorem below. In particular, we obtain the following results:

1. The dimension of the supersingular locus is two.

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

3. There are some irreducible components whose generic points have a-number two.

The answer 1 is expected from the conjectural purity of Newton strata, [6, 3, 10]. The answer 2 is suggested from the work of Li and Oort. The answer 3 causes a new difficult in analyzing the slope strata in Hilbert-Blumenthal varieties.

The answer 3 gives rise to a natural question: what is the “generic” a-number for Hilbert-Blumenthal varieties analogous to “a = 1” in Siegel modular varieties? The a-numbers here should be refined with respect to the action of the ring of integers of F (2.4). It seems that these (geometrically) generic a-numbers have a simple combinatorial description. We simply call an a-number with such a combinatorial property (combinatorially) generic (2.6). We show that two notions of generic a-numbers are the same for four-folds and expect that they are the same in general. For higher dimensional cases, we show that any geometrically generic a-number is (combinatorially) generic, see Section 6.

The main tools used in this paper are the Cartier-Dieudonné theory [40], deformations of displays [24, 25, 41], the method of Cayley-Hamilton following Oort [31], and canonical isogenies (see Section 8, cf. [19, 18]).

We describe now the main results of this paper. Let F be a totally real field and let O_{F} be its ring of integers. Recall that a polarized O_{F}-abelian variety is a triple (A, λ, ι), where A is an abelian variety, λ : A → A^{t} is a polarization and ι : O_{F} → End(A) is a ring monomorphism such that λ ∘ ι(a) = ι(a)^{t} ∘ λ, ∀ a ∈ O_{F}. Let A_{g}^{O_{F}} denote the moduli stack of principally polarized O_{F}-abelian varieties of dimension g = [F : ℚ]. 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 [7, Thm. 2.2]. Denote by 𝒮_{n} the supersingular locus of the 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 and we regard it as a reduced closed subscheme of A_{g,n}^{O_{F}} ⊗ k. The Galois group H_{n} for the finite étale cover A_{g,n}^{O_{F}} ⊗ k → A_{g}^{O_{F}} ⊗ k preserves the closed 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 degree p^{4} and type (p, 1, p, 1) (2.2). An abelian variety is superspecial if it is isomorphic to a product of supersingular elliptic curves. Let G be the group scheme over Spec ℤ so that 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 7.2, Theorem 10.6, 12.1, 12.3, and Proposition 9.3 and 9.11:

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, i.e. the cardinality of the double coset G(ℚ)\G(𝔸_{f})/G(ℤ̂).

(3) The formal completion 𝒮̂_{x} of 𝒮 at any 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 a ruled surface over P^{1}.

The following is an outline of the contents of this paper. In Section 2 we list the notations used in this work. In Section 3 we introduce the stratification induced by a-numbers and describe these strata. The analysis of the a-number stratification on the supersingular locus, 𝒮, is the key step in studying 𝒮. A classification of the Dieudonné modules in question is given in Section 4. In Section 5, we introduce the method of Cayley-Hamilton due to F. Oort [31]. Using this, one settles the slope stratification for the points of a-number one. In Section 6, we introduce one step isogeny deformation by Hecke correspondence. This reduces to considering only the points of generic a-numbers (2.6). In Section 7, we organize the results obtained from previous sections for four-folds. We show that the generic point of some component of 𝒮 has the 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 Manin’s thesis [20]. An explicit form was constructed first in the work of Oda and Oort [27]. In Section 9, we classify possible canonical isogenies (in this case there are two) and follow Li and Oort [19] to construct such a model for each type. In Section 10, we relate certain type of superspecial points with a class number. An improvement of the Cayley-Hamilton method for the Hilbert-Blumenthal case is made in Section 11. 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 12. The intersections of the irreducible components at a point can be obtained directly from its local moduli space. We use the results of Section 9 to describe the intersections globally. We discuss the local and global descriptions of the intersections in Section 13. The computation of all other local moduli is placed in the last section for interested readers.

Acknowledgments. The paper grew out of the author’s thesis at University of Pennsylvania. I would like express my deep gratitude to my 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 intellectual debt to K.-Z. Li and Frans Oort is too obvious. It is a great pleasure to thank them for stimulating discussions and encouragement. The revision of this paper was done during my stay at Max-Planck-Institut für Mathematik in Bonn. I would like to thank the Institute for the kind hospitality and the good working conditions. Finally, I would like to thank the referee for pointing out some unclear arguments and for the useful comments that improve the exposition of this paper significantly.

2. Notations and definitions

(2.1) Let F be a totally real field of degree g and let O_{F} be its ring of integers. Denote by A_{g}^{O_{F}} the moduli stack of principally polarized O_{F}-abelian varieties of dimension g. We fix a rational prime number p and assume that p is inert in F. Let ℳ := A_{g}^{O_{F}} ⊗𝔽_{p} be the reduction of our moduli space modulo p. Denote by 𝒮 the supersingular locus of ℳ. Let f be the residue degree of p, hence f = g = dim A, the dimension of the abelian varieties in question. Let k be a perfect field containing 𝔽_{q} (q = p^{f}), W := W (k) denote the ring of Witt vectors, B(k) denote the fractional field of W (k), and σ be the absolute Frobenius morphism on W. We label the embeddings Hom(O_{F}, W) = {σ_{i}}, i ∈ ℤ/fℤ in a way that σ_{i+1} = σσ_{i} for i ∈ ℤ/fℤ.

(2.2) For each polarized O_{F}-abelian variety (A, λ, ι) over k, the covariant Dieudonné module M := M(A) [40] 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, ∀ a ∈ O_{F} } is a W -module of rank 2.

• 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}}).

(2.3) If L is a k-module or W -module equipped with a linear O_{F}-action, then we set:

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

(2.4) We (re)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. This is an element in {0, 1, 2}^{ℤ/fℤ}. The sum |a(A)| = a_{0} + ⋯ + a_{f−1} is equal to the dimension of the vector Hom(α_{p}, A) over k, which we will refer it as the usual a-number of A in this paper.

(2.5) If the Lie algebra Lie(A) of A satisfies the freeness condition, i.e. Lie(A) is a free O_{F} ⊗kmodule, then each a_{i}(A) = 0, or 1 for all i ∈ ℤ/fℤ. It is so for any principally polarized O_{F}-abelian variety over k. In this case we define the a-index of the O_{F}-abelian variety A to be

τ_{A} = τ_{M} := {i ∈ ℤ/fℤ, | a_{i}(M) = 1 },

as a subset of ℤ/fℤ. The a-index of A, when it is defined, is determined by its a-number and vice versa. We say τ_{1} is higher than τ_{2} if τ_{1} ⊃ τ_{2}, and similarly for a-numbers: a ≥ b if a_{i} ≥ b_{i} for all i ∈ ℤ/fℤ. We will see (Section 3) that the a-indices (the same for a-numbers) induce a stratification of our moduli space, and the type of the stratum goes up under specialization.

(2.6) We call an a-number (a_{0}, …, a_{f−1}) generic if a_{i}a_{i+1} = 0, for all i ∈ ℤ/fℤ. We define similarly for a a-index τ to be generic if there are no consecutive integers in τ. This term “generic” is called spaced in [10]. The generic a-numbers will play the similar role in Hilbert-Blumenthal varieties as a-number one does in Siegel modular varieties.

(2.7) In Section 3 we give the definition of the substack ℳ_{τ} of ℳ := A_{g}^{O_{F}} ⊗𝔽_{p} for each a-index τ. As a-numbers and a-indices correspond in bijection, we will not distinguish the stratum of a-index and the corresponding a-number, e.g. ℳ_{1010} is ℳ_{{0,2}} (g = 4). Let 𝒮_{τ} denote the intersection of 𝒮 and ℳ_{τ}, and let 𝒮̄_{τ} denote the Zariski closure of 𝒮_{τ} in 𝒮.

(2.8) The set of slopes of each (A, λ, ι) ∈ ℳ(k) is either \{\frac{a}{f}, … \frac{a}{f}, \frac{f−a}{f}, …, \frac{f−a}{f}\} (with multiplicity f) for 0 ≤ a ≤ [f/2], a ∈ ℤ, or \{\frac{1}{2}, …, \frac{1}{2}\} (with multiplicity 2f) [10, Thm.5.2.1]. For each m ∈ ℤ with 0 ≤ m ≤ (f/2), we denote by ℳ^{(m)} the subset of ℳ that parameterizes the points of lowest slope m/f. When f = 2d + 1 is odd, we also write ℳ^{(d+1)} for 𝒮. We regard ℳ^{(m)} as a substack with the induced reduced structure. By the notation ℳ_{τ}^{(m)}, we mean the (set-theoretic) intersection of ℳ_{τ} and ℳ^{(m)}.

(2.9) 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.

(2.10) 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}. We call such an isogeny O_{F}-isogeny. By definition, the polarization of a 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}. For a commutative ring R, we call an O_{F}-isogeny ϕ : A_{1} → A_{2} R-polarized if ϕ^{∗}λ_{2} = a λ_{1} for some a ∈ R^{×}.

(2.11) In Section 8 we also use the classical contravariant Dieudonné modules [20], which we denote by M^{∗}(A). We need this to trace the Dieudonné modules of finite group schemes arising from isogenies and duality. We apologize for this inconsistency and any confusion it may cause.

(2.12) If X is a scheme and x is a point of X, we write X_{x} for Spec Ô_{X,x} and X̂_{x} for Spf Ô_{X,x}.

(2.13) We often write x^{(n)} for x^{σ^{n}} for short, where x is an element in a Witt ring and σ is the Frobenius endomorphism on the Witt ring .

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 [10, Thm. 3.2.8]. This stratification is also studied in a joint paper of Goren and Oort [10], 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 ℤ/fℤ, we denote by \overline{ℳ}_{τ} the substack of ℳ ⊗ 𝔽_{q} whose objects (A/S, λ, ι) satisfy the following condition: the Frobenius morphism F : Lie(A^{t})^{(p),i} → Lie(A^{t})^{i} is zero for all i ∈ τ. It is clear that this condition is closed and locally for Zariski topology \overline{ℳ}_{τ} is defined by finitely many equations. Therefore, \overline{ℳ}_{τ} is a closed substack of ℳ. We have the following properties

(1) (A, λ, ι) ∈ \overline{ℳ}_{τ}(k) if and only if τ(A) ⊃ τ.

(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}. Indeed, the points in \overline{ℳ}_{τ} have p-rank 0, so that they do not degenerate under specialization by Grothendieck’s specialization theorem.

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 is essentially Theorem 2.3.4 of [10] except that they define the stratification set-theoretically and take the reduced subscheme structure, while ours is scheme-theoretic. The statement can also be checked by a tangent space calculation, which is omitted.

Let ℳ_{τ} be the reduced algebraic substack of \overline{ℳ}_{τ} whose k-points have a-index τ. 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}. Let \underline{a} be the a-number corresponding to τ, we often write ℳ\underline{a} for ℳ_{τ}.

Working on fine moduli schemes, Goren and Oort show [10, Prop. 2.3.9] that ℳ_{τ} is quasiaffine. This implies (with (4)) that the Zariski closure of any irreducible component of ℳ_{τ} contains a point of a-index τ′ for any τ′ ⊃ τ, in particular, a superspecial point.

4. Normal forms

For 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 non-ordinary 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}} ⊂ ℤ/fℤ.

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). Indeed, write F X_{i−1} = aX_{i} + bY_{i} (i ∈ τ). It follows from ⟨F X_{i−1}, X_{i}⟩ = ⟨X_{i−1}, V X_{i}⟩^{σ} that b = −1. It follows from V^{2}(M^{i+1}) ⊂ pM that a = ⟨F X_{i−1}, Y_{i}⟩ = ⟨X_{i−1}, V^{2}Y_{i+1}⟩^{σ} ∈ pW.

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}, we set

(\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})

Note that {X_{i}, Y_{i}} forms a W -basis for M. This follows from that {X_{n_{j}}} generates M/pM over k[F, V] and that F M^{n_{j}−1} = V M^{n_{j}+1} mod pM.

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. We extend a W -basis {X′_{i}, Y′_{i}} for M as in the proof above, then we have

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

This completes the proof.

Lemma 4.3. If M is superspecial, i.e. |a(M)| = f [29]. 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 ∈ ℤ/fℤ.

(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 ∈ ℤ/fℤ.

Proof. (1) Suppose that f is even. Let f = 2d, M^{∘} := {x ∈ M | F^{d}x = (−1)^{d}V x }. Since M is superspecial, we have F^{2}M^{∘} = pM^{∘} and M^{∘} ⊗_{W (𝔽_{p^{f}})} W (k) ≃ M [19]. Choose X_{0}, Y_{0} ∈ (M^{∘})^{0}

such that ⟨X_{0}, Y_{0}⟩ = 1 and Y_{0} ∈ V M^{∘}. Define recursively

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

Then we have 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 that f is odd. Let f = 2d + 1 and M^{∘} = {x ∈ M | F^{2f}x + 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}⟩ ∈ W (k)^{×}. 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. If we 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 adjust X_{0} by choosing a suitable λ 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).

Lemma 4.4. Let N be the Dieudonné module of a superspecial polarized O_{F}-abelian variety (A, λ, ι) over k with a-number 0202 … 02 (f is even). Then

(1) The polarization type (2.2) is (p^{n+1}, p^{n}, …, p^{n+1}, p^{n}).

(2) There is a basis {X_{i}, Y^{i}} of N such that

(i) X_{i}, Y_{i} ∈ N^{i} for i ∈ ℤ/fℤ. (ii) ⟨X_{i}, Y_{i}⟩ = p⟨X_{i+1}, Y_{i+1}⟩ = p^{n+1} for all i even.

(iii) F X_{i} = \begin{cases}−pY_{i+1} & if i is even, \\ −Y_{i+1} & if i is odd, \end{cases} F Y_{i} = \begin{cases}pX_{i+1} & if i is even, \\ X_{i+1} & if i is odd. \end{cases}

Proof. (1) This follows from ⟨F X_{i}, F X_{i}⟩ = p⟨X_{i}, Y_{i}⟩^{σ} and the restriction of a-numbers. (2) The proof is the same as (1) of Lemma 4.3 except replacing (4.1) by the rule (iii).

5. Cayley-Hamilton for a = 1

In this section we recall the method of Cayley-Hamilton in Dieudonné modules according to F. Oort [31]. The key point of this method is the following observation [31, Prop. 3.6]:

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) the element 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}.

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. This amounts to an explicit classification of Dieudonné modules.

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}). Suppose that we have a deformation (G, λ) of (G_{0}, λ_{0}) over R, where R is a complete local Noetherian k-algebra with residue field k. By the Serre-Tate theorem and Grothendieck’s existence theorem, there is a polarized abelian scheme (A, λ) over Spec R whose associated formal group is isomorphic to (G, λ).

For any prime ideal 𝔭 ∈ Spec R, let K(𝔭) denote 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 is

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

where D_{∗} is the covariant functor in Cartier-Dieudonné theory (eg. in [40]) and Cart_{p}(R) is the Cartier ring (the ring W (R)[F][[V]] in [25], also see [4]). This is a finite free W (K(𝔭))-module with F and V action, which we will denote by M_{𝔭}.

The theory of display [24][25] allows us to compute the deformation of quasi-polarized Dieudonné modules. Suppose the Dieudonné module (M_{0}, ⟨ , ⟩ can be displayed by as follows

\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}

and (M, ⟨ , ⟩) is a deformation of (M_{0}, ⟨ , ⟩) over R in the displayed form

\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}. Then we get a deformation (G, λ) of (G_{0}, λ_{0}) over R and a polarized abelian scheme (A, λ) over R as above. For 𝔭 ∈ Spec R, then the Dieudonné module M_{𝔭} is

\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 W (K(𝔭)).

Proposition 5.2. Let (M_{0}, ⟨ , ⟩), (M_{,}⟨ , ⟩) as before, γ the Newton polygon of M_{0}. Suppose that 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. Let 𝔭 ∈ Spec R, we have 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) strictly 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 symmetric, 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}).

Applying this method, in the Hilbert case, one can compute the dimension of each slope stratum in the open subset a = 1. Note that the Dieudonné module of a point on ℳ has slopes a/f and 1 − a/f with multiplicity f [10, Thm. 5.2.1]. Let us denote ℳ^{(m)} (for m ≤ (f/2)) 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). By Proposition 4.1, 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 the Frobenius map 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). The following result is [10, Thm. 5.3.3], which is obtained from the method of Cayley-Hamilton.

Proposition 5.3. Let τ = 10 … 0 and M be 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}]].

6. Deformation via the Hecke correspondence

In this section we show that the points of generic a-numbers in the sense of (2.6) are dense in each slope stratum ℳ^{(m)}. The method is to construct a deformation by Hecke correspondence. The geometry of Hilbert-Blumenthal varieties of parahoric level structures was studied in [37, 33].

Let N be the moduli stack parameterizing the triples ((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}. Let p_{1}, p_{2} be the two projection morphisms to ℳ. The idea of deforming a point of higher a-number to a lower one along the slope stratum is the following: for a point x = (A, λ, ι) of higher a-number, find a point y ∈ N (k) such that p_{1}(y) = x. Then we compute the fibre p_{2}^{−1}(p_{2}(y)) and determine the a-numbers of the sources in this family.

Let (A, λ, ι) ∈ ℳ(k) be a non-ordinary point and M be the covariant Dieudonné module. Let N be the Dieudonné module containing M such that V N = M. A quasi-polarization ⟨ , ⟩_{N} on N is chosen such that ⟨ , ⟩_{M} = p⟨ , ⟩_{N}. Then there is a tripe y := ((A, λ, ι), (A_{1}, λ_{1}, ι_{1}), φ) ∈ N (k) such that M(A_{1}) = N. 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:

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}

Let X be the space of O_{F}-invariant maximal isotropic “Dieudonné ” subspaces of \overline{N} := N/pN over k

X (k) = \{0 ⊂ \overline{M′} ⊂ \overline{N}; dim \overline{M′} = f, O_{F}(\overline{M′}) ⊂ \overline{M′}, F \overline{M′} ⊂ \overline{M′}, V \overline{M′} ⊂ \overline{M′}, ⟨\overline{M′}, \overline{M′}⟩_{N} = 0\}

We regard X as a reduced subscheme over k. It is a closed subscheme of the Lagrangian Grassmanian LG(f, 2f), hence a projective variety. The preimage M′ of each \overline{M′} in N is a quasi-polarized Dieudonné submodule stable under the O_{F}-action and satisfies ⟨ , ⟩_{M′} = p⟨ , ⟩_{N}. It gives rise to a point in the fibre Y := p_{2}^{−1}p_{2}(y). Therefore we have a finite morphism pr : X → Y that sends \overline{M} := M/pN to y.

Let \overline{M}_{t} ∈ X be the k-subspace of \overline{N} generated by Y_{i} +t_{i}X_{i}, t_{i} ∈ k for i ∈ ℤ/fℤ. The points \overline{M}_{t} forms a Zariski open neighborhood of \overline{M} in X, which we denote by U. From F \overline{M}_{t} ⊂ \overline{M}_{t}, we get

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+1}X_{i+1}) − t_{i}^{p}t_{i+1}X_{i+1} ⇒ t_{i}^{p}t_{i+1} = 0

From V \overline{M}_{t} ⊂ \overline{M}_{t}, we get the same equations above. Therefore, we have

U = Spec k[t_{i}]_{i∈τ_{M}} /(t_{i}t_{i+1})_{i,i+1∈τ_{M}} .

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 U = 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 U = Spec k[t_{0}, t_{1}, t_{3}, t_{4}, t_{5}]/(t_{0}t_{1}, t_{3}t_{4}, t_{4}t_{5}).

We rewrite the 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 the Dieudonné module M 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)\overline{M}_{t})^{i} of (F, V)M_{t}/pM_{t}. Case i ∉ τ_{M}:

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

The only case that a_{i}(M) = 1 and a_{i}(M_{t}) = 1 is subcase 1. This happens when a_{i−1}(M) = a_{i+1}(M) = 0. We have

Theorem 6.1. Let (A, λ, ι) ∈ ℳ(k). Suppose that a_{i}(A) = a_{i+1}(A) = 1 for some i. Then there exists a deformation (A_{t}, λ_{t}, ι_{t}) of (A, λ, ι) with Newton polygon NP (A_{t}) = NP (A) such that either a_{i}(A_{t}) = 0 and a_{i+1}(A_{t}) = 1, or a_{i}(A_{t}) = 1 and a_{i+1}(A_{t}) = 0.

Corollary 6.2. If g ≤ 3, then the points of usual a-number ≤ 1 are dense in each slope stratum ℳ^{(m)}.

Corollary 6.3. We have \overline{ℳ^{(m)}} = ∪_{τ} \overline{ℳ_{τ}^{(m)}}. where τ runs over all generic a-numbers in the sense of (2.6).

Corollary 6.4. If \overline{ℳ_{τ}^{(m+1)}} ⊂ \overline{ℳ_{τ}^{(m)}} in ℳ_{τ} holds for all m + 1 ≥ |τ| and all generic a-numbers τ. Then \overline{ℳ^{(m+1)}} ⊂ \overline{ℳ^{(m)}} for all 0 ≤ m < (f/2).

Proof. By Theorem 6.1, it suffices to check that \overline{ℳ_{τ}^{(m+1)}} ⊂ \overline{ℳ_{τ}^{(m)}} for all m and all generic a-numbers τ. Then this follows from a theorem of Goren and Oort [10, Theorem 5.4.11] that the generic point of each irreducible component of ℳ_{τ} has lowest slope |τ|/f.

Corollary 6.5. The a-number of every generic point of ℳ^{(m)} is generic in the sense of (2.6).

Conversely, we expect that the points of every generic a-number (2.6) do form whole components of each slope stratum (if ℳ_{τ}^{(m)} is not empty). That is, we expect that if τ is generic, then each irreducible component of ℳ_{τ}^{(m)}, if not empty, has the right dimension g − m. This is a refinement of Conjecture 5.2.5 in [10].

7. Preliminary results for g = 4

In this section g = f is 4.

Lemma 7.1. If (A, λ, ι) ∈ ℳ_{1010}(k) ∪ ℳ_{0101}(k), then A is supersingular.

Proof. Let M be the Dieudonné module of (A, λ, ι). We may assume that a(M) = 1010 and 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 and get

\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} maps 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. Similarly, we can do the same computation for the case a(M) = 0101.

Proposition 7.2.

(1) The generic point of ℳ is ordinary.

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

(3) We have ℳ_{1010} = 𝒮_{1010} and ℳ_{0101} = 𝒮_{0101}. Therefore, \overline{S}_{1010} and \overline{S}_{0101} are smooth of pure dimension two.

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

(5) 𝒮_{1100} ⊂ \overline{𝒮}_{1000} ∩ \overline{𝒮}_{0100}.

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

(7) Every irreducible component of the supersingular locus has dimension 2. There are six types of them, namely 𝒮_{τ} with generic a-number 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 the method of Cayley-Hamilton for a = 1, Proposition 5.3. (3) follows from Lemma 7.1 and Theorem 3.1. (4) follows from (3). (5) is a special case of Theorem 6.1. (6) If ℳ_{1100} = 𝒮_{1100}, then ℳ_{1100} ⊂ \overline{𝒮}_{1000} ∩ \overline{𝒮}_{0100}. The latter has dimension at most one from (2), contradiction. (7) follows from (2), (3) , (4) and (5). (8) It suffices to show \overline{ℳ^{(2)}} ⊂ \overline{ℳ^{(1)}} due to (1). Any supersingular point can be deformed to one of usual a-number one, then this follows from the Cayley-Hamilton method.

8. 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 [20], 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 (a_{0}, …, a_{f−1}) if n_{i} is the multiplicity of σ_{i} on M^{∗}(H). Denote by α_{p}^{(a_{0},...,a_{f−1})} the α-group of type (a_{0}, …, a_{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 [19], Section 2. Locally in Zariski topology H is isomorphic to α_{p}^{(a_{0},...,a_{f−1})} × S.

The Cartier dual H^{t} of H 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 of Cartier dual, we have the canonical pairing

( , ) : H^{t} × H → G_{m} (8.1)

and

(a f, x) = (f, a x) ∀a ∈ O_{F}, f ∈ H^{t}, x ∈ H. (8.2)

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 (8.1) and (8.2) still holds for any finite flat group scheme with an O_{F}-action.

Let (A, ι) be an O_{F}-abelian variety of dimension g over k, M its covariant Dieudonné module. The 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 8.1. 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}).

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 integers 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 ∈ ℤ

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 semi-simple algebra B over ℚ with a positive involution. Therefore, this method can be applied to other Shimura varieties of PEL-type as well.

9. Construction of models

In this section, we work on four-dimensional O_{F}-abelian varieties. We compute first 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 irreducible component of the supersingular locus 𝒮 as in [19]. This gives a global description of each irreducible component. As a byproduct, it gives another proof of part (1) of the Main theorem.

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. Indeed, the set of the superspecial points in the moduli space A_{g,p^{n}} is 0-dimensional [19] and there are only finitely many ways to impose a compatible O_{F}-action on a polarized abelian variety [38, Lemma 1.2]. There is a natural bijection between the sets S_{1} and S_{2}, sending λ to pλ. These sets will correspond to the irreducible components of 𝒮 and its cardinality can be realized as a class number.

As the points in 𝒮_{τ} with generic a-number τ form an open dense subset of 𝒮 (Proposition 7.2), we only need to determine the information from the canonical isogenies of 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

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} \} .

(9.1)

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 (Section 5). A straightforward computation gives

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} \} .

(9.2)

From the result above and using Lemma 8.1, 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).

(9.3)

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

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

Definition. (1) Let τ = 1010 and S be a base scheme of characteristic p. An polarized O_{F}- abelian 4-fold over S with 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 4-fold 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 scheme ker φ is (locally isomorphic to) an α-group of type 1010.

An isomorphism between two polarized O_{F}-abelian 4-folds with 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})) consists of isomorphisms θ_{i} : A′_{i} → A_{i} for i = 0, 1, as polarized O_{F}-abelian varieties, such that θ_{0} ∘ φ = φ′ ∘ θ_{1}.

(2) Let τ = 1000 and S be a base scheme of characteristic p. An polarized O_{F}-abelian 4-fold over S with 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 4-fold 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 subgroup schemes ker φ_{2}, ker φ_{1}, and ker φ_{0} are (locally isomorphic to) α-groups of

type 1020, 0101 and 1000, respectively. We define similarly for isomorphisms between two polarized O_{F}-abelian 4-folds with τ-level structure.

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

N_{τ,η}(S) = \{ \begin{matrix}isomorphism classes of polarized O_{F}-abelian 4-folds over S \\ with τ-level structure s.t. (A_{1}, λ_{1}, ι_{1}) = η × S \end{matrix} \} ,

the moduli space parameterizing the polarized O_{F}-abelian 4-folds of τ-level structure and 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 the diagram

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

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

Lemma 9.1. The functor N_{τ,η} is representable by P^{1} × P^{1}.

Proof. Every polarized O_{F}-abelian 4-fold with 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

pr : ∐_{η∈Λ_{1}} N_{τ,η} → 𝒮̄_{τ},

which is proper and dominant. As 𝒮̄_{τ} is smooth (Proposition 7.2), it is a disjoint union of irreducible components: 𝒮̄_{τ} = ∐_{η∈Λ_{1}} Z_{η} with pr : N_{τ,η} → Z_{η}.

Lemma 9.2. The morphism pr is étale.

Proof. Let (A_{i}) be an polarized O_{F}-abelian 4-fold with τ-level structure. Then a(A_{0}) ≥ 1010. Let M_{i} be the Dieudonné module of A_{i}. Choose a basis {X_{i}, Y_{i}} for M_{0} as in Proposition 4.1. Let F̃ il_{i} ⊂ M̃_{i} be the first order universal deformations. From (9.1), We have

F̃ il_{0} = ⟨Y_{i} + t_{i}X_{i}⟩_{i∈ℤ/4ℤ}

F̃ il_{1} = ⟨pX_{0}, Y_{0}, pX_{2}, Y_{2}⟩

From F̃ il_{1} ⊂ F̃ il_{0}, we get t_{0} = t_{2} = 0. The morphism pr induces an injective map between tangent spaces and both source and target spaces are smooth of dimension two. Therefore, pr is étale.

For each abelian variety x in 𝒮̄_{τ}(k) there is a unique abelian 4-fold y with τ-level structure, up to isomorphism, such that pr(y) = x. It follows from this and Lemma 9.2 that the morphism pr induces the isomorphism

N_{τ,η}/G_{η} ≃ Z_{η},

for η ∈ Λ_{1}, where G_{η} is the automorphism group of η. We have thus shown

Proposition 9.3. The algebraic stack \overline{ℳ}_{1010} = 𝒮̄_{1010} has |S_{1}| irreducible components and each irreducible component is isomorphic to a quotient of P^{1} × P^{1} by a finite group.

We now describe the subset of N_{τ,η} = P^{1} × P^{1} whose points have higher a-numbers under the projection pr. Let η = (B, λ_{B}, ι_{B}) ∈ Λ_{1} and N be the Dieudonné module of B. By Lemma 4.4, we can choose a basis {X_{i}, Y_{i}} for the Dieudonné module N such that

\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}

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} \} (9.4)

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 following

Proposition 9.4. Let pr : N_{τ,η} = P^{1} × P^{1} → Z_{η} be as above.

(1) The subset of P^{1} × P^{1} whose image lies in (Z_{η} ∩ \overline{ℳ}_{1110}) is \{(x, x^{p^{2}})\} ⊂ P^{1} × P^{1}; the subset of P^{1} × P^{1} whose image lies in (Z_{η} ∩ \overline{ℳ}_{1011}) is \{(y^{p^{2}}, y)\} ⊂ P^{1} × P^{1}. In particular, Z_{η} ∩ \overline{ℳ}_{1110} (or Z_{η} ∩ \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}}). Here a point (A_{i}) is superspecial we mean A_{0} is superspecial.

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 9.5. The set of the G_{η}-orbits of the superspecial points P^{1}(𝔽_{p^{4}}) is in bijective correspondence with the set of the superspecial points on Z_{η}.

Proof. We need to prove: for two points 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 exists an isomorphism β : (A_{1}, λ_{1}, ι_{1}) ≃ (A_{2}, λ_{2}, ι_{2}), then the composition α_{2}^{−1}βα_{1} is an isomorphism.

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

Since the degree of α_{2} is a power of p, 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 so that N, M_{1} and M_{2} are submodules of N_{−1}. The map β induces an isomorphism from 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 O_{F}-linear, it preserves N̄, hence N.

Since 𝒮̄_{1010} is a disjoint union of irreducible components, every superspecial point lies in exactly one component of 𝒮̄_{1010}. The number of superspecial points on 𝒮 is

∑_{η∈Λ_{1}} #(P^{1}(𝔽_{p^{4}})/G_{η}). (9.5)

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 polarized O_{F}-abelian 4-folds over S \\ with τ-level structure s.t. (A_{3}, λ_{3}, ι_{3}) = η × S \end{matrix} \} ,

the moduli space parameterizing polarized O_{F}-abelian 4-folds with τ-level structure and 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 the diagrams

[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}.]

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

Lemma 9.6. The functor 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 9.1, 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 [19], Section 3.5.

The last projection pr gives a proper morphism N_{τ,η} → 𝒮 whose image contains 𝒮̄_{τ}. 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 9.9 below.

Lemma 9.7. Let (A_{3} →^{φ_{2}} A_{2} →^{φ_{1}} A_{1} →^{φ_{0}} A_{0}) be a polarized O_{F}-abelian 4-fold with 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 from Lemma 8.1.

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 from Lemma 8.1.

Corollary 9.8. The stratum 𝒮_{1100} is empty.

Proof. Let (A_{0}, λ_{0}, ι_{0}) ∈ 𝒮_{1100}(k). From 𝒮_{1100} ⊂ 𝒮̄_{1000} ((5) of Proposition 7.2), it admits a τ-level structure. Let (A_{i}) be polarized O_{F}-abelian 4-fold with 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 to Lemma 9.7.

Corollary 9.9.

(1) For any (A, λ, ι) ∈ 𝒮̄_{τ}(k) there is a unique abelian 4-fold (φ_{n}, (A_{n}, λ_{n}, ι_{n})) with τ-level structure, up to isomorphism, such that (A_{0}, λ_{0}, ι_{0}) ≃ (A, λ, ι).

(2) The scheme 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} from Lemma 9.7.

(2) First, N_{τ,η}^{(2)} = P^{1}. By Lemma 9.7, 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.

Lemma 9.10. 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 (9.2). Let F̃ il_{i} ⊂ M̃_{i} be the first order universal deformations. We have

F̃ il_{0} = ⟨Y_{i} + t_{i0}X_{i}⟩_{i∈ℤ/4ℤ}
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 tangent space is two dimensional and its image under pr is two-dimensional. It follows from the smoothness of 𝒮_{1000} (Proposition 5.3) that the projection map is étale over 𝒮_{1000}.

Proposition 9.11. 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 use a fact that each irreducible component Z of 𝒮̄_{τ} is smooth. This fact is proved from the computation of local moduli in Section 12.

Consider the moduli space A_{g,n}^{O_{F}} ⊗ k with a prime-to-p n-level structure (n ≥ 3), and choose an irreducible component Z_{n} of A_{g,n}^{O_{F}} ⊗ k over Z. There is a lifting p̃r : N_{τ,η} → Z_{n} of pr by choosing a suitable level structure for objects in N_{τ,η}. It follows from (1) of Corollary 9.9 and Lemma 9.10 that p̃r induces a bijection for the underlying sets and is generically étale. Therefore, p̃r is a finite birational morphism between smooth projective algebraic surfaces, hence it is an isomorphism. This completes the proof.

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} (= P(O_{P^{1}} ⊕O_{P^{1}}(2)) over P^{1}), a Hirzebruch surface [1, Chapter IV]. We thank him for this interesting question.

Proposition 9.12. 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 4.4. 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. One can see that this is a gluing relation for the projective space bundle P(O_{P^{1}} ⊕ O_{P^{1}}(2)) over P^{1}. Hence N_{τ,η} is isomorphic to the Hirzebruch surface F_{2}.

10. Superspecial points and class numbers

In this section we interpret the number of irreducible components, |S_{1}|, which appears in the previous section, as a class number. The main result is Theorem 10.5. The ground field k is an algebraically closed field of characteristic p.

Proposition 10.1. 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} up to isogeny by ι′_{1}(a) := φ^{−1}ι_{2}(a)φ ∈ End(A_{1}) ⊗ ℚ for a ∈ O_{F}.

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}

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 the Noether-Skolem 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 [42, Satz 1.1], there is a positive element c ∈ End(A_{1}) with c = c^{∗_{1}} such that ∗_{1} = c ∗′_{1} c^{−1}.

Lemma 10.2. 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 that commutes with all elements in End_{O_{F}}(A_{1}). This integral element is in O_{F}, hence 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. Note that End(E) ⊗ ℚ is a definite quaternion algebra over ℚ of discriminant p. The positive element c = c^{∗_{1}} is a totally positive element in F. The surjectivity of the norm map from F ⊗ End(E) to F^{+} is a consequence of the Hasse principle.

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.

Corollary 10.3. 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 10.1, 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. 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 10.4. 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 decompose M = ⊕M_{v} by the decomposition O_{F} ⊗ ℤ_{ℓ} = ⊕O_{v}. Since the discriminant of the pairing on M_{v} is 1 and O_{v} is a PID, there is only one isomorphism class.

Let x = (A_{0}, λ_{0}, ι_{0}) be a supersingular polarized O_{F}-abelian variety over k of dimension g. Let G_{x} denote the group scheme over Spec ℤ whose group of R-points, for each commutative ring R, is

G_{x}(R) = {ϕ ∈ (End_{O_{F}}(A_{0}) ⊗ R)^{×}; ϕ′ϕ = 1},

where the map ϕ ↦ ϕ′ is the Rasoti involution induced by λ_{0}. Let A_{0}(ℓ) (ℓ can be p) denote the ℓ-divisible group of A_{0}. As A_{0} is supersingular, we have G_{x}(ℤ_{ℓ}) = Aut(T_{ℓ}(A_{0})) = Aut(A_{0}(ℓ)) for ℓ ≠ p and G_{x}(ℤ_{ℓ}) = Aut(M(A_{0})) = Aut(A_{0}(ℓ)) [26]. Here the automorphisms are meant to preserve the O_{F}-action and the alternating pairing.

Let Λ_{x} denote the set of the isomorphism classes of supersingular polarized O_{F}-abelian varieties (A, λ, ι) of dimension g over k such that

(i) the Dieudonné module M(A) is isomorphic to M(A_{0}) as quasi-polarized Dieudonné modules with O_{F} ⊗ ℤ_{p}-actions,

(ii) the Tate module T_{ℓ}(A) is isomorphic to T_{ℓ}(A_{0}) as non-degenerate alternating O_{F} ⊗ ℤ_{ℓ}-

modules for all ℓ ≠ p, and (iii) there exists an element ϕ ∈ Hom_{O_{F}}(A_{0}, A) ⊗ ℚ such that ϕ^{∗}λ = λ_{0}.

Theorem 10.5. There is a natural bijection of pointed sets between Λ_{x} and G_{x}(ℚ)\G_{x}(𝔸_{f})/G_{x}(ℤ̂).

Proof. For any (A, λ, ι) ∈ Λ_{x}, there is an ℚ-polarized O_{F}-linear isogeny ϕ : (A_{0}, λ_{0}, ι_{0}) → (A, λ, ι), by definition. Then there is an ϕ_{ℓ} ∈ G_{x}(ℚ_{ℓ}) for all primes ℓ such that ϕϕ_{ℓ} ∈ Isom(A_{0}(ℓ), A(ℓ)) from (i) and (ii). The element ϕ_{ℓ} lies in G_{x}(ℤ_{ℓ}) for almost all ℓ and it is determined by ϕ up to a right translation of G_{x}(ℤ_{ℓ}). Therefore, ϕ gives an element (ϕ_{ℓ}) ∈ G_{x}(𝔸_{f})/G_{x}(ℤ̂). Let ϕ′ be another ℚ-polarized O_{F}-linear isogeny from A_{0} to A. Then we have ϕ′g = ϕ for some g ∈ G_{x}(ℚ) and (ϕ′_{ℓ})G_{x}(ℤ̂) = (gϕ_{ℓ})G_{x}(ℤ̂). Therefore, we get a well-defined map σ from Λ_{x} to G_{x}(ℚ)\G_{x}(𝔸_{f})/G_{x}(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 such that σ(α) = σ(β). Then there exist ϕ ∈ G_{x}(ℚ) and (ϕ_{ℓ}) ∈ G_{x}(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 an isomorphism at every place, hence it is an isomorphism.

Proof of surjectivity. Given an element [(ϕ_{ℓ})] ∈ G_{x}(ℚ)\G_{x}(𝔸_{f})/G_{x}(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 finite group ker(Nϕ_{ℓ}). Since each Nϕ_{ℓ} is O_{F}-linear, H is stable under the O_{F}- action. 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. Note that we have the induced isomorphism A(ℓ) → A_{0}(ℓ) of quasi-polarized ℓ-divisible groups. Therefore the polarization λ is ℓ-adic integral for all ℓ, hence it is a polarization on A. Therefore, N^{−1}π : (A_{0}, λ_{0}, ι_{0}) → (A, λ, ι) gives an element in Λ_{x} and it induces the element [(ϕ_{ℓ})].

Remark. We can consider the similar statement for the “basic” points in the moduli space of abelian varieties with a PEL-type structure, e.g. see [17, 35, 38]. 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.

Let b = (B, λ_{B}, ι_{B}) ∈ S_{1} and let G := G_{b} defined as before. It follows from Lemma 4.4, Corollary 10.3 and Lemma 10.4 that S_{1} = Λ_{b}.

Theorem 10.6. Let G be as before. 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 follows from Proposition 9.3, 9.11 and Theorem 10.5.

11. Cayley-Hamilton for higher a-numbers

It seems that Cayley-Hamilton is getting more complicated when the 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 either 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 the 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. (11.1)

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 (11.1) in the rest of the paper.

12. 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 will write N_{x} for Spec Õ_{N ,x} and write Ñ_{x} for Spf Õ_{N.x}, the local moduli space of N at x. By Grothendieck’s existence theorem, there is an abelian scheme over N_{x}. As a topological space, the scheme N_{x} has more (non-closed) points than its formal completion. This enables us to test whether 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} denotes 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 of computing the local moduli space 𝒮̂_{x} is as follows:

First, the supersingular locus 𝒮 is a union of six types of irreducible components (Proposition 7.2):

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

and each type consists of h irreducible components (Theorem 10.6). Hence it suffices to find 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 the component 𝒮̄_{{i,i+2},x} from the a-number of x: 𝒮̄_{{i,i+2},x} ⊂ 𝒮_{x} if and only if \{i, i + 2\} ⊂ τ(x). Moreover, since 𝒮̄_{\{i,i+2\}} = \overline{ℳ}_{\{i,i+2\}} is smooth (Proposition7.2), 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.

A priori, we do not know whether 𝒮̄_{{i},x}, if exists, is irreducible. However, we will see from computation that every 𝒮̄_{{i},x}, if not empty, is irreducible. From this we conclude that 𝒮̄_{τ} is a disjoint union of irreducible components.

Suppose that we want to compute 𝒮̄_{τ,x}, where τ = 1000. 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 can write down the deformation M_{τ} of the Dieudonné module M_{x} over R_{τ} directly. Compute the coefficients p_{0}, p_{1} of the Cayley-Hamilton equation (defined in the previous section) for the Dieudonné module M_{τ} by iterating the Frobenius map. Note that p_{i} are in the Witt ring W (R_{τ}). 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}, …) ∈ W (R_{τ}) with the coordinates p_{ij} in 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}) of Spec R_{τ} 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 a non-generic a-number locus, as these can not whole components of Z. The remaining irreducible 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 irreducible 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} which cause a non-generic a-number locus again. Any 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 irreducible component. If it is in the other case, then another higher congruence condition of p_{1} must be satisfied. Continue this process until we find out one irreducible component of Z.

Now we are ready to compute 𝒮̄_{τ,x} for |τ| = 1. Because of symmetry, we may assume that 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 9.8). We will do the case a(x) = 1111 in this section and put the computation of the remaining cases in Section 14.

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 ∈ ℤ/4ℤ. Let O = k[[t_{0}, t_{1}, t_{2}, t_{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 irreducible 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 an irreducible component of the supersingular locus. The generic point of this component has 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 12.1. Suppose x = (A, λ, ι) ∈ 𝒮_{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 ∈ ℤ/4ℤ.

In the following we state the results of our computation for other cases. See Section 13 for the detail computation.

Theorem 12.2. (1) Let x ∈ 𝒮_{1110}(k). Let a_{i} be the coefficients in the normal form of the associated Dieudonné module M_{x} in (14.1) and set ā_{i} = a_{i} mod p. Then 𝒮̄_{x} = 𝒮̄_{1010,x} ∪ 𝒮̄_{0100,x} and

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

(2) Let x ∈ 𝒮_{1010}(k). Let a_{i} be the coefficients in the normal form of the associated Dieudonné module M_{x} in (14.3) and set ā_{i} = a_{i} mod p. Then there are at most three components of 𝒮̄_{x} : 𝒮̄_{x} = 𝒮̄_{1010,x} ∪ 𝒮̄_{0010,x} ∪ 𝒮̄_{1000,x}, where 𝒮̄_{1010,x} ≃ O/(t_{0}, t_{2}) and the other two components may be empty. Moreover, we have

(12.3)  𝒮̄_{0010,x} ≠ ϕ ⟺ ā_{2} = 0 ⟹ 𝒮̄_{0010,x} ≃ Spec O/(t_{2}, t_{3})
(12.4)  𝒮̄_{1000,x} ≠ ϕ ⟺ ā_{0} = 0 ⟹ 𝒮̄_{1000,x} ≃ Spec O/(t_{0}, t_{1}).

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

Theorem 12.3. Every irreducible component of 𝒮 is smooth.

13. 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. 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 9 except the abelian varieties there with symplectic n-level structure.

From (9.5) and Theorem 10.6, 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}|}) , (13.1)

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 9) 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 13.1. 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}).

Proof. This follows from Theorem 12.1.

Proposition 13.2.

(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 irreducible component of 𝒮̄_{1010} and any irreducible 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 9.4.

(2) Let x ∈ 𝒮̄_{1010} ∩ 𝒮̄_{1000}. It follows from (1) of Theorem 12.2 that if a(x) = 1110 or a(x) = 1011 then 𝒮̄_{1000,x} is empty, contradiction. We have checked that the intersection is smooth at superspecial points. Let x ∈ ℳ_{1010} ∩ 𝒮̄_{1000}. From (2) of Theorem 12.2, we know that 𝒮̄_{1000,x} = Spec O/(t_{0}, t_{3}). Therefore, 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}. By the same reason as (2), a(x) ≠ 1110 or 1101. Therefore, a(x) = 1111 (Corollary 9.8).

(5) Let x ∈ 𝒮̄_{1000} ∩ 𝒮̄_{0010}, then x ∈ 𝒮̄_{1010}. By the same reason as (2), a(x) ≠ 1110 or 1011. If a(x) = 1111, we have Lemma 13.1 (5). If a(x) = 1010, then it follows from (2) of Theorem 12.2 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 13.3. 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}\}
(∵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}})
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 13.4. 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 (9.4), 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 13.5. 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 component of 𝒮̄_{1000} may not be connected. By Proposition 9.4 (3) and 13.4, 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}. The latter is not true in general.

14. 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} (14.1)

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} = 0. 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}}),

(14.2) 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} (14.3)

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}).

(14.4)

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

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