The Supersingular Loci and Mass

Formulas on Siegel Modular Varieties

Chia-Fu Yu

Received: October 2, 2006 Revised: December 4, 2006

Communicated by Takeshi Saito

Abstract. We describe the supersingular locus of the Siegel 3-fold with a parahoric level structure. We also study its higher dimensional generalization. Using this correspondence and a deep result of Li and Oort, we evaluate the number of irreducible components of the supersingular locus of the Siegel moduli space A_{g,1,N} for arbitrary g.

2000 Mathematics Subject Classification: Primary 11G18; Secondary 14K12. Keywords and Phrases: Siegel modular varieties, supersingular locus, superspecial locus, parahoric level structure, mass formula

1

1. Introduction

In this paper we discuss some extensions of works of Katsura and Oort [5], and of Li and Oort [8] on the supersingular locus of a mod p Siegel modular variety. Let p be a rational prime number, N ≥ 3 a prime-to-p positive integer. We choose a primitive N-th root of unity ζ_{N} in \overline{ℚ} ⊂ ℂ and an embedding \overline{ℚ} ↪ \overline{ℚ}_{p}. Let A_{g,1,N} be the moduli space over ℤ_{(p)}[ζ_{N}] of g-dimensional principally polarized abelian varieties (A, λ, η) with a symplectic level-N structure (See Subsection 2.1). Let A_{2,1,N,(p)} be the cover of A_{2,1,N} which parametrizes isomorphism classes of objects (A, λ, η, H), where (A, λ, η) is an object in A_{2,1,N} and H ⊂ A[p] is a finite flat subgroup scheme of rank p. It is known that the moduli scheme A_{2,1,N,(p)} has semi-stable reduction and the reduction A_{2,1,N,(p)} ⊗ \overline{𝔽}_{p} modulo p has two irreducible components. Let S_{2,1,N,(p)} (resp. S_{2,1,N}) denote the supersingular locus of the moduli space A_{2,1,N,(p)} ⊗ \overline{𝔽}_{p} (resp. A_{2,1,N} ⊗ \overline{𝔽}_{p}). Recall that an abelian variety A in characteristic p is called supersingular if it is isogenous to a product of supersingular elliptic curves over an algebraically closed field k; it is called superspecial if it is isomorphic to a product of supersingular elliptic curves over k. The supersingular locus S_{2,1,N} of the Siegel 3-fold is studied in Katsura and Oort [5]. We summarize the main results for S_{2,1,N} (the local results obtained earlier in Koblitz [7]):

The research is partially supported by NSC 95-2115-M-001-004.

Theorem 1.1.

(1) The scheme S_{2,1,N} is equi-dimensional and each irreducible component is isomorphic to P^{1}.

(2) The scheme S_{2,1,N} has

(1.1)  | Sp_{4}(ℤ/Nℤ)|\frac{(p^{2} − 1)}{5760}

irreducible components.

(3) The singular points of S_{2,1,N} are exactly the superspecial points and there are

(1.2)  | Sp_{4}(ℤ/Nℤ)|\frac{(p − 1)(p^{2} + 1)}{5760}

of them. Moreover, at each singular point there are p+1 irreducible components passing through and intersecting transversely.

Proof. See Koblitz [7, p.193] and Katsura-Oort [5, Section 2, Theorem 5.1, Theorem 5.3].

In this paper we extend their results to S_{2,1,N,(p)}. We show

Theorem 1.2.

(1) The scheme S_{2,1,N,(p)} is equi-dimensional and each irreducible component is isomorphic to P^{1}.

(2) The scheme S_{2,1,N,(p)} has

(1.3)  | Sp_{4}(ℤ/Nℤ)| · \frac{(−1)ζ(−1)ζ(−3)}{4} [(p^{2} − 1) + (p − 1)(p^{2} + 1)]

irreducible components, where ζ(s) is the Riemann zeta function.

(3) The scheme S_{2,1,N,(p)} has only ordinary double singular points and there are

(1.4)  | Sp_{4}(ℤ/Nℤ)| · \frac{(−1)ζ(−1)ζ(−3)}{4}(p − 1)(p^{2} + 1)(p + 1)

of them.

(4) The natural morphism S_{2,1,N,(p)} → S_{2,1,N} contracts

(1.5)  | Sp_{4}(ℤ/Nℤ)| · \frac{(−1)ζ(−1)ζ(−3)}{4}(p − 1)(p^{2} + 1)

projective lines onto the superspecial points of S_{2,1,N}.

Documenta Mathematica 11 (2006) 449–468

Remark 1.3. (1) By the basic fact that

ζ(−1) = \frac{−1}{12} and ζ(−3) = \frac{1}{120},

the number (1.5) (of the vertical components) equals the number (1.2) (of superspecial points), and the number (1.3) (sum of vertical and horizontal components) equals the sum of the numbers (1.1) (of irreducible components) and (1.2) (of superspecial points). Thus, the set of horizontal irreducible components of S_{2,1,N,(p)} is in bijection with the set of irreducible components of S_{2,1,N}

(2) Theorem 1.2 (4) says that S_{2,1,N,(p)} is a “desingularization” or a “blowup” of S_{2,1,N} at the singular points. Strictly speaking, the desingularization of S_{2,1,N} is its normalization, which is the (disjoint) union of horizontal components of S_{2,1,N,(p)}. The vertical components of S_{2,1,N,(p)} should be the exceptional divisors of the blowing up of a suitable ambient surface of S_{2,1,N} at superspecial points.

In the proof of Theorem 1.2 (Section 4) we see that

• the set of certain superspecial points (the set Λ in Subsection 4.1) in S_{2,p,N} (classifying degree-p^{2} polarized supersingular abelian surfaces) is in bijection with the set of irreducible components of S_{2,1,N}, and • the set of superspecial points in S_{2,1,N} is in bijection with the set of

irreducible components of S_{2,p,N}, furthermore

• the supersingular locus S_{2,1,N,(p)} provides the explicit link of this duality as a correspondence that performs simply through the “blowingups” and “blowing-downs”.

In the second part of this paper we attempt to generalize a similar picture to higher (even) dimensions.

Let g = 2D be an even positive integer. Let H be the moduli space over ℤ_{(p)}[ζ_{N}] which parametrizes equivalence classes of objects (φ : \underline{A}_{1} → \underline{A}_{2}), where

• \underline{A}_{1} = (A_{1}, λ_{1}, η_{1}) is an object in A_{g,1,N},

• \underline{A}_{2} = (A_{2}, λ_{2}, η_{2}) is an object in A_{g,p^{D},N}, and

• φ : A_{1} → A_{2} is an isogeny of degree p^{D} satisfying φ^{∗}λ_{2} = pλ_{1} and

φ_{∗}η_{1} = η_{2}.

The moduli space H with natural projections gives the following correspondence:

\xymatrix{ & H \ar[dl]_{_{pr_{1}}} \ar[dr]^{_{pr_{2}}} \\ A_{g,1,N} &  & A_{g,p^{D},N}.}

Let S be the supersingular locus of H ⊗ \overline{𝔽}_{p}, which is the reduced closed subscheme consisting of supersingular points (either A_{1} or A_{2} is supersingular, or equivalently both are so).

In the special case where g = 2, H is isomorphic to A_{2,1,N,(p)}, and S ≃ S_{2,1,N,(p)} under this isomorphism (See Subsection 4.5). As the second main result of this paper, we obtain

Theorem 1.4. Let C be the number of irreducible components of S_{g,1,N}. Then

C = | Sp_{2g}(ℤ/Nℤ)| · \frac{(−1)^{g(g+1)/2}}{2^{g}} \{∏_{i=1}^{g} ζ(1 − 2i)\} · L_{p} ,

where

L_{p} = \begin{cases}∏_{i=1}^{g} \{(p^{i} + (−1)^{i}\} & if g is odd; \\ ∏_{i=1}^{D}(p^{4i−2} − 1) & if g = 2D is even. \end{cases}

In the special case where g = 2, Theorem 1.4 recovers Theorem 1.1 (2). We give the idea of the proof. Let Λ_{g,1,N} denote the set of superspecial (geometric) points in A_{g,1,N} ⊗\overline{𝔽}_{p}. For g = 2D is even, let Λ_{g,p^{D},N}^{∗} denote the set of superspecial (geometric) points (A, λ, η) in A_{g,p^{D},N}⊗\overline{𝔽}_{p} satisfying ker λ = A[F], where F : A → A^{(p)} is the relative Frobenius morphism on A. These are finite sets and each member is defined over \overline{𝔽}_{p}. By a result of Li and Oort [8] (also see Section 5), we know

C = \begin{cases}|Λ_{g,1,N}| & if g is odd; \\ |Λ_{g,p^{D},N}^{∗}| & if g is even. \end{cases}

One can use the geometric mass formula due to Ekedahl [2] and some others (see Section 3) to compute |Λ_{g,1,N}|. Therefore, it remains to compute |Λ_{g,p^{D},N}^{∗}| when g is even. We restrict the correspondence S to the product Λ_{g,1,N} × Λ_{g,p^{D},N}^{∗} of superspecial points, and compute certain special points in S. This gives us relation between Λ_{g,p^{D},N}^{∗} and Λ_{g,1,N}. See Section 6 for details. Theorem 1.4 tells us how the number C = C(g, N, p) varies when p varies. For another application, one can use this result to compute the dimension of the space of Siegel cusp forms of certain level at p by the expected Jacquet-Langlands correspondence for symplectic groups. As far as the author knows, the latter for general g is not available yet in the literature. The paper is organized as follows. In Section 2, we recall the basic definitions and properties of the Siegel moduli spaces and supersingular abelian varieties. In Section 3, we state the mass formula for superspecial principally polarized abelian varieties due to Ekedahl (and some others). The proof of Theorem 1.2 is given in Section 4. In Section 5, we describe the results of Li and Oort on irreducible components of the supersingular locus. In Section 6, we introduce a correspondence and use this to evaluate the number of irreducible components of the supersingular locus.

Acknowledgments. The author is grateful to Katsura, Li and Oort for their inspiring papers on which the present work relies heavily. He also thanks C.-L. Chai for his encouragements on this subject for years, and thanks T. Ibukiyama for his interest of the present work. He is indebted to the referee for careful reading and helpful comments that improve the presentation of this paper.

2. Notation and preliminaries

2.1. Throughout this paper we fix a rational prime p and a prime-to-p positive integer N ≥ 3. Let d be a positive integer with (d, N) = 1. We choose a primitive N-th root of unity ζ_{N} in \overline{ℚ} ⊂ ℂ and an embedding \overline{ℚ} ↪ \overline{ℚ}_{p}. The element ζ_{N} gives rise to a trivialization ℤ/Nℤ ≃ μ_{N} over any ℤ_{(p)}[ζ_{N}]-scheme. For a polarized abelian variety (A, λ) of degree d^{2}, a full symplectic level-N structure with respect to the choice ζ_{N} is an isomorphism

η : (ℤ/Nℤ)^{2g} ≃ A[N]

such that the following diagram commutes

(ℤ/Nℤ)^{2g} × (ℤ/Nℤ)^{2g} ⟶^{(η,η)} A[N] × A[N]
⟨ , ⟩↓  ^{e_{λ}}↓
ℤ/Nℤ  ⟶^{ζ_{N}}  μ_{N},

where ⟨ , ⟩ is the standard non-degenerate alternating form on (ℤ/Nℤ)^{2g} and e_{λ} is the Weil pairing induced by the polarization λ. Let A_{g,d,N} denote the moduli space over ℤ_{(p)}[ζ_{N}] of g-dimensional polarized abelian varieties (A, λ, η) of degree d^{2} with a full symplectic level N structure with respect to ζ_{N}. Let S_{g,d,N} denote the supersingular locus of the reduction A_{g,d,N} ⊗ \overline{𝔽}_{p} modulo p, which is the closed reduced subscheme of A_{g,d,N} ⊗ \overline{𝔽}_{p} consisting of supersingular points in A_{g,d,N} ⊗ \overline{𝔽}_{p}. Let Λ_{g,d,N} denote the set of superspecial (geometric) points in S_{g,d,N}; this is a finite set and every member is defined over \overline{𝔽}_{p}.

For a scheme X of finite type over a field K, we denote by Π_{0}(X) the set of geometrically irreducible components of X. Let k be an algebraically closed field of characteristic p.

2.2. Over a ground field K of characteristic p, denote by α_{p} the finite group scheme of rank p that is the kernel of the Frobenius endomorphism from the additive group 𝔾_{a} to itself. One has

α_{p} = SpecK[X]/X^{p}, m(X) = X ⊗ 1 + 1 ⊗ X,

where m is the group law. By definition, an elliptic curve E over K is called supersingular if E[p](\overline{K}) = 0. An abelian variety A over K is called supersingular if it is isogenous to a product of supersingular elliptic curves over \overline{K}; A is called superspecial if it is isomorphic to a product of supersingular elliptic curves over \overline{K}. For any abelian variety A over K where K is perfect, the a-number of A is defined by

a(A) := dim_{K} Hom(α_{p}, A).

The following interesting results are well-known; see Subsection 1.6 of [8] for a detail discussion.

Theorem 2.1 (Oort). If a(A) = g, then A is superspecial.

Theorem 2.2 (Deligne, Ogus, Shioda). For g ≥ 2, there is only one gdimensional superspecial abelian variety up to isomorphism over k.

3. The mass formula

Let Λ_{g} denote the set of isomorphism classes of g-dimensional principally polarized superspecial abelian varieties over \overline{𝔽}_{p}. Write

M_{g} := ∑_{(A,λ)∈Λ_{g}} \frac{1}{| Aut(A, λ)|}

for the mass attached to Λ_{g}. The following mass formula is due to Ekedahl [2, p.159] and Hashimoto-Ibukiyama [3, Proposition 9].

Theorem 3.1. Notation as above. One has

(3.1)  M_{g} = \frac{(−1)^{g(g+1)/2}}{2^{g}} \{∏_{k=1}^{g} ζ(1 − 2k)\} · ∏_{k=1}^{g} \{(p^{k} + (−1)^{k}\} .

Similarly, we set

M_{g,1.N} := ∑_{(A,λ,η)∈Λ_{g,1,N}} \frac{1}{| Aut(A, λ, η)|}.

Lemma 3.2. We have M_{g,1,N} = |Λ_{g,1,N}| = | Sp_{2g}(ℤ/Nℤ)| · M_{g}.

Proof. The first equality follows from a basic fact that (A, λ, η) has no non-trivial automorphism. The proof of the second equality is elementary; see Subsection 4.6 of [11].

Corollary 3.3. One has

|Λ_{2,1,N}| = | Sp_{4}(ℤ/Nℤ)| · \frac{(−1)ζ(−1)ζ(−3)}{4}(p − 1)(p^{2} + 1).

4. Proof of Theorem 1.2

4.1. In this section we consider the case where g = 2. Let

Λ := {(A, λ, η) ∈ S_{2,p,N}; ker λ ≃ α_{p} × α_{p}}.

Note that every member \underline{A} of Λ is superspecial (because A ⊃ α_{p} × α_{p}), that is, Λ ⊂ Λ_{2,p,N}. For a point ξ in Λ, consider the space S_{ξ} which parametrizes the isogenies of degree p

φ : (A_{ξ}, λ_{ξ}, η_{ξ}) → \underline{A} = (A, λ, η)

which preserve the polarizations and level structures. Let

ψ_{ξ} : S_{ξ} → S_{2,1,N}

be the morphism which sends (φ : ξ → \underline{A}) to \underline{A}. Let V_{ξ} ⊂ S_{2,1,N} be the image of S_{ξ} under ψ_{ξ}. The following theorem is due to Katsura and Oort [5, Theorem 2.1 and Theorem 5.1]:

Theorem 4.1 (Katsura-Oort). Notation as above. The map ξ ↦ V_{ξ} gives rise to a bijection Λ ≃ Π_{0}(S_{2,1,N}) and one has

|Λ| = | Sp_{4}(ℤ/Nℤ)|(p^{2} − 1)/5760.

We will give a different way of evaluating |Λ| that is based on the geometric mass formula (see Corollary 4.6).

4.2. Dually we can consider the space S′_{ξ} for each ξ ∈ Λ that parametrizes the isogenies of degree p

φ′ : \underline{A} = (A, λ, η) → ξ = (A_{ξ}, λ_{ξ}, η_{ξ}),

with \underline{A} ∈ A_{2,1,N} ⊗ \overline{𝔽}_{p}, such that φ′_{∗}η = η_{ξ} and φ^{′∗}λ_{ξ} = p λ. Let

ψ′_{ξ} : S′_{ξ} → S_{2,1,N}

be the morphism which sends (φ′ : \underline{A} → ξ) to \underline{A}. Let V′_{ξ} ⊂ S_{2,1,N} be the image of S′_{ξ} under ψ′_{ξ}. For a degree p isogeny (φ : \underline{A}_{1} → \underline{A}_{2}) with \underline{A}_{2} in A_{2,1,N}, φ^{∗}λ_{2} = λ_{1} and φ_{∗}η_{1} = η_{2}, we define

(φ : \underline{A}_{1} → \underline{A}_{2})^{∗} = (φ′ : \underline{A}′_{2} → \underline{A}′_{1}),

where φ′ = φ^{t} and

\underline{A}′_{2} = (A_{2}^{t}, λ_{2}^{−1}, λ_{2} ∘ η_{2}),
\underline{A}′_{1} = (A_{1}^{t}, p λ_{1}^{−1}, λ_{1} ∘ η_{1}).

Note that φ′_{∗}η′_{2} = η′_{1} as we have the commutative diagram:

(ℤ/Nℤ)^{4} ⟶^{η_{2}} A_{2}[N] ⟶^{λ_{2}} A_{2}^{t}[N]
↓^{=}  ↑^{φ}  ↓^{φ^{t}}
(ℤ/Nℤ)^{4} ⟶^{η_{1}} A_{1}[N] ⟶^{λ_{1}} A_{1}^{t}[N].

If \underline{A}_{1} ∈ Λ, then \underline{A}′_{1} is also in Λ. Therefore, the map ξ ↦ V′_{ξ} also gives rise to a bijection Λ ≃ Π_{0}(S_{2,1,N}).

4.3. We use the classical contravariant Dieudonné theory. We refer the reader to Demazure [1] for a basic account of this theory. Let K be a perfect field of characteristic p, W := W (K) the ring of Witt vectors over K, B(K) the fraction field of W (K). Let σ be the Frobenius map on B(K). A quasipolarization on a Dieudonné module M here is a non-degenerate (meaning of non-zero discriminant) alternating pairing

⟨ , ⟩ : M × M → B(K),

such that ⟨F x, y⟩ = ⟨x, V y⟩^{σ} for x, y ∈ M and ⟨M^{t}, M^{t}⟩ ⊂ W. Here we regard the dual M^{t} of M as a Dieudonné submodule in M ⊗B(K) using the pairing. A quasi-polarization is called separable if M^{t} = M. Any polarized abelian variety (A, λ) over K naturally gives rise to a quasi-polarized Dieudonné module. The induced quasi-polarization is separable if and only if (p, deg λ) = 1. Recall (Subsection 2.1) that k denotes an algebraically closed field of characteristic p.

Lemma 4.2.

(1) Let M be a separably quasi-polarized superspecial Dieudonné module over k of rank 4. Then there exists a basis f_{1}, f_{2}, f_{3}, f_{4} for M over W := W (k) such that

F f_{1} = f_{3}, F f_{3} = pf_{1}, F f_{2} = f_{4}, F f_{4} = pf_{2}

and the non-zero pairings are

⟨f_{1}, f_{3}⟩ = −⟨f_{3}, f_{1}⟩ = β_{1}, ⟨f_{2}, f_{4}⟩ = −⟨f_{4}, f_{2}⟩ = β_{1},

where β_{1} ∈ W (𝔽_{p^{2}})^{×} with β_{1}^{σ} = −β_{1}.

(2) Let ξ be a point in Λ, and let M_{ξ} be the Dieudonné module of ξ. Then there is a W -basis e_{1}, e_{2}, e_{3}, e_{4} for M_{ξ} such that

F e_{1} = e_{3}, F e_{2} = e_{4}, F e_{3} = pe_{1}, F e_{4} = pe_{2},

and the non-zero pairings are

⟨e_{1}, e_{2}⟩ = −⟨e_{2}, e_{1}⟩ = \frac{1}{p}, ⟨e_{3}, e_{4}⟩ = −⟨e_{4}, e_{3}⟩ = 1.

Proof. (1) This is a special case of Proposition 6.1 of [8].

(2) By Proposition 6.1 of [8], (M_{ξ}, ⟨ , ⟩) either is indecomposable or decomposes into a product of two quasi-polarized supersingular Dieudonné modules of rank 2. In the indecomposable case, one can choose such a basis e_{i} for M_{ξ}. Hence it remains to show that (M_{ξ}, ⟨ , ⟩) is indecomposable. Let (A_{ξ}[p^{∞}], λ_{ξ}) be the associated polarized p-divisible group. Suppose it decomposes into (H_{1}, λ_{1}) × (H_{2}, λ_{2}). Then the kernel of λ is isomorphic to E[p] for a supersingular elliptic curve E. Since E[p] is a nontrivial extension of α_{p} by α_{p}, one gets contradiction. This completes the proof.

4.4. Let (A_{0}, λ_{0}) be a superspecial principally polarized abelian surface and (M_{0}, ⟨ , ⟩_{0}) be the associated Dieudonné module. Let φ′ : (A_{0}, λ_{0}) → (A, λ) be an isogeny of degree p with φ^{′∗}λ = p λ_{0}. Write (M, ⟨ , ⟩) for the Dieudonné module of (A, λ). Choose a basis f_{1}, f_{2}, f_{3}, f_{4} for M_{0} as in Lemma 4.2. We have the inclusions

(F, V)M_{0} ⊂ M ⊂ M_{0}.

Modulo (F, V)M_{0}, a module M corresponds a one-dimensional subspace \overline{M} in \overline{M}_{0} := M_{0}/(F, V)M_{0}. As \overline{M}_{0} = k < f_{1}, f_{2} >, \overline{M} is of the form

\overline{M} = k < af_{1} + bf_{2} >, [a : b] ∈ P^{1}(k).

The following result is due to Moret-Bailly [9, p.138-9]. We include a proof for the reader’s convenience.

Lemma 4.3. Notation as above, ker λ ≃ α_{p}×α_{p} if and only if the corresponding point [a : b] satisfies a^{p+1} + b^{p+1} = 0. Consequently, there are p + 1 isogenies φ′ so that ker λ ≃ α_{p} × α_{p}.

Proof. As φ^{′∗}λ = pλ_{0}, we have ⟨ , ⟩ = \frac{1}{p}⟨ , ⟩_{0}. The Dieudonné module M(ker λ) of the subgroup ker λ is equal to M/M^{t}. Hence the condition ker λ ≃ α_{p} × α_{p} is equivalent to that F and V vanish on M(ker λ) = M/M^{t}. Since ⟨ , ⟩ is a perfect pairing on F M_{0}, that is, (F M_{0})^{t} = F M_{0}, we have

pM_{0} ⊂ M^{t} ⊂ F M_{0} ⊂ M ⊂ M_{0}.

Changing the notation, put \overline{M}_{0} := M_{0}/pM_{0} and let

⟨ , ⟩_{0} : \overline{M}_{0} × \overline{M}_{0} → k.

be the induced perfect pairing. In \overline{M}_{0}, the subspace \overline{M^{t}} is equal to \overline{M}^{⊥}. Indeed,

\begin{aligned}M^{t} & = \{m ∈ M_{0}; ⟨m, x⟩_{0} ∈ pW ∀ x ∈ M\}, \\ \overline{M^{t}} & = \{m ∈ \overline{M}_{0}; ⟨m, x⟩_{0} = 0 ∀ x ∈ \overline{M}\} = \overline{M}^{⊥}. \end{aligned} ; (4.1)

From this we see that the condition ker λ ≃ α_{p}×α_{p} is equivalent to ⟨\overline{M}, F \overline{M}⟩ = ⟨\overline{M}, V \overline{M}⟩ = 0. Since \overline{F M_{0}} = k < f_{3}, f_{4} >, one has \overline{M} = k < f′_{1}, f_{3}, f_{4} > where f′_{1} = af_{1} + bf_{2}. The condition ⟨\overline{M}, F \overline{M}⟩ = ⟨\overline{M}, V \overline{M}⟩ = 0, same as ⟨f′_{1}, F f′_{1}⟩ = ⟨f′_{1}, V f′_{1}⟩ = 0, gives the equation a^{p+1} + b^{p+1} = 0. This completes the proof.

Conversely, fix a polarized superspecial abelian surface (A, λ) such that ker λ ≃ α_{p} × α_{p}. Then there are p^{2} + 1 degree-p isogenies φ′ : (A_{0}, λ_{0}) → (A, λ) such that A_{0} is superspecial and φ^{′∗}λ = p λ_{0}. Indeed, each isogeny φ′ always has the property φ^{′∗}λ = p λ_{0} for a principal polarization λ_{0}, and there are |P^{1}(𝔽_{p^{2}})| isogenies with A_{0} superspecial.

4.5. We denote by A′_{2,1,N,(p)} the moduli space which parametrizes equivalence classes of isogenies (φ′ : \underline{A}_{0} → \underline{A}_{1}) of degree p, where \underline{A}_{1} is an object in A_{2,p,N} and \underline{A}_{0} is an object in A_{2,1,N}, such that φ^{′∗}λ_{1} = p λ_{0} and φ′_{∗}η_{0} = η_{1}. There is a natural isomorphism from A_{2,1,N,(p)} to A′_{2,1,N,(p)}. Given an object (A, λ, η, H) in A_{2,1,N,(p)}, let \underline{A}_{0} := \underline{A}, A_{1} := A/H and φ′ : A_{0} → A_{1} be the natural projection. The polarization p λ_{0} descends to one, denoted by λ_{1}, on A_{1}. Put η_{1} := φ′_{∗}η_{0} and \underline{A}_{1} = (A_{1}, λ_{1}, η_{1}). Then (φ′ : \underline{A}_{0} → \underline{A}_{1}) lies in A′_{2,1,N,(p)} and the morphism

q : A_{2,1,N,(p)} → A′_{2,1,N,(p)}, (A, λ, η, H) ↦ (φ′ : \underline{A}_{0} → \underline{A}_{1})

is an isomorphism. We denote by S′_{2,1,N,(p)} the supersingular locus of A′_{2,1,N,(p)} ⊗ \overline{𝔽}_{p}. Thus we have S′_{2,1,N,(p)} ≃ S_{2,1,N,(p)}. It is clear that S′_{ξ} ⊂ S′_{2,1,N,(p)} for each ξ ∈ Λ, and S′_{ξ} ∩ S′_{ξ′} = ∅ if ξ ≠ ξ′. For each γ ∈ Λ_{2,1,N}, let S′′_{γ} be the subspace of S′_{2,1,N,(p)} that consists of objects (φ′ : \underline{A}_{0} → \underline{A}_{1}) with \underline{A}_{0} = \underline{A}_{γ}. One also has S′′_{γ} ∩ S′′_{γ′} = ∅ if γ ≠ γ′.

Lemma 4.4. (1) Let (M_{0}, ⟨ , ⟩_{0}) be a separably quasi-polarized supersingular Dieudonné module of rank 4 and suppose a(M_{0}) = 1. Let M_{1} := (F, V)M_{0} and N be the unique Dieudonné module containing M_{0} with N/M_{0} = k. Let ⟨ , ⟩_{1} := \frac{1}{p}⟨ , ⟩_{0} be the quasi-polarization for M_{1}. Then one has a(N) = a(M_{1}) = 2, V N = M_{1}, and M_{1}/M_{1}^{t} ≃ k ⊕ k as Dieudonné modules.

(2) Let (M_{1}, ⟨ , ⟩_{1}) be a quasi-polarized supersingular Dieudonné module of rank 4. Suppose that M_{1}/M_{1}^{t} is of length 2, that is, the quasi-polarization has degree p^{2}.

(i) If a(M_{1}) = 1, then letting M_{2} := (F, V)M_{1}, one has that a(M_{2}) = 2 and ⟨ , ⟩_{1} is a separable quasi-polarization on M_{2}.

(ii) Suppose (M_{1}, ⟨ , ⟩_{1}) decomposes as the product of two quasi-polarized Dieudonné submodules of rank 2. Then there are a unique Dieudonné submodule M_{2} of M_{1} with M_{1}/M_{2} = k and a unique Dieudonné module M_{0} containing M_{1} with M_{0}/M_{1} = k so that ⟨ , ⟩_{1} (resp. p⟨ , ⟩_{1}) is a separable quasi-polarization on M_{2} (resp. M_{0}).

(iii) Suppose M_{1}/M_{1}^{t} ≃ k ⊕ k as Dieudonné modules. Let M_{2} ⊂ M_{1} be any Dieudonné submodule with M_{1}/M_{2} = k, and M_{0} ⊃ M_{1} be any Dieudonné overmodule with M_{0}/M_{1} = k. Then ⟨ , ⟩_{1} (resp. p⟨ , ⟩_{1}) is a separable quasi-polarization on M_{2} (resp. M_{0}).

This is well-known; the proof is elementary and omitted.

Proposition 4.5. Notation as above.

(1) One has

S′_{2,1,N,(p)} = (∐_{ξ∈Λ} S′_{ξ}) ∪ ( ∐_{γ∈Λ_{2,1,N}} S′′_{γ}) .

(2) The scheme S′_{2,1,N,(p)} has ordinary double singular points and

(S′_{2,1,N,(p)})^{sing} = (∐_{ξ∈Λ} S′_{ξ}) ∩ ( ∐_{γ∈Λ_{2,1,N}} S′′_{γ}) .

Moreover, one has

|(S′_{2,1,N,(p)})^{sing}| = |Λ_{2,1,N}|(p + 1) = |Λ|(p^{2} + 1).

Proof. (1) Let (φ′ : \underline{A}_{0} → \underline{A}_{1}) be a point of S′_{2,1,N,(p)}. If a(A_{0}) = 1, then ker φ′ is the unique α-subgroup of A_{0}[p] and thus \underline{A}_{1} ∈ Λ. Hence this point lies in S′_{ξ} for some ξ. Suppose that \underline{A}_{1} is not in Λ, then there is a unique lifting (φ′_{1} : \underline{A}′_{0} → \underline{A}_{1}) in S′_{2,1,N,(p)} and the source \underline{A}′_{0} is superspecial. Hence \underline{A}_{0} = \underline{A}′_{0} is superspecial and the point (φ′ : \underline{A}_{0} → \underline{A}_{1}) lies in S′′_{γ} for some γ.

(2) It is clear that the singularities only occur at the intersection of S′_{ξ}’s and S′′_{γ}’s, as S′_{ξ} and S′′_{γ} are smooth. Let x = (φ′ : \underline{A}_{γ} → \underline{A}_{ξ}) ∈ S′_{ξ} ∩ S′′_{γ}. We know that the projection pr_{0} : S′_{2,1,N,(p)} → S_{2,1,N} induces an isomorphism from S′_{ξ} to V′_{ξ}. Therefore, pr_{0} maps the one-dimensional subspace T_{x}(S′_{ξ}) of T_{x}(A′_{2,1,N,(p)}⊗ \overline{𝔽}_{p}) onto the one-dimensional subspace T_{pr_{0}(x)}(V′_{ξ}) of T_{pr_{0}(x)}(A_{2,1,N} ⊗\overline{𝔽}_{p}), where T_{x}(X) denotes the tangent space of a variety X at a point x. On the other hand, pr_{0} maps the subspace T_{x}(S′′_{γ}) to zero. This shows T_{x}(S′_{ξ}) ≠ T_{x}(S′′_{γ}) in T_{x}(A′_{2,1,N,(p)} ⊗ \overline{𝔽}_{p}); particularly S′_{2,1,N,(p)} has ordinary double singularity at x. Since every singular point lies in both S′_{ξ} and S′′_{γ} for some ξ, γ, by Subsection 4.4 each S′_{ξ} has p^{2} + 1 singular points and each S′′_{γ} has p + 1 singular points. We get

|(S′_{2,1,N,(p)})^{sing}| = |Λ_{2,1,N}|(p + 1), and |(S′_{2,1,N,(p)})^{sing}| = |Λ|(p^{2} + 1).

This completes the proof.

Corollary 4.6. We have

|(S′_{2,1,N,(p)})^{sing}| = | Sp_{4}(ℤ/Nℤ)| · \frac{(−1)ζ(−1)ζ(−3)}{4}(p − 1)(p^{2} + 1)(p + 1)

and

|Λ| = | Sp_{4}(ℤ/Nℤ)| · \frac{(−1)ζ(−1)ζ(−3)}{4}(p^{2} − 1).

Proof. This follows from Corollary 3.3 and (2) of Proposition 4.5.

Note that the evaluation of |Λ| here is different from that given in Katsura-Oort [5]. Their method does not rely on the mass formula but the computation is more complicated. Since S_{2,1,N,(p)} ≃ S′_{2,1,N,(p)} (Subsection 4.5), Theorem 1.2 follows from Proposition 4.5 and Corollary 4.6. As a byproduct, we obtain the description of the supersingular locus S_{2,p,N}.

Theorem 4.7.

(1) The scheme S_{2,p,N} is equi-dimensional and each irreducible component is isomorphic to P^{1}.

(2) The scheme S_{2,p,N} has |Λ_{2,1,N}| irreducible components.

(3) The singular locus of S_{2,p,N} consists of superspecial points (A, λ, η) with ker λ ≃ α_{p} ×α_{p}, and thus |S_{2,p,N}^{sing}| = |Λ|. Moreover, at each singular point there are p^{2} +1 irreducible components passing through and intersecting transversely.

(4) The natural morphism pr_{1} : S_{2,1,N,(p)} → S_{2,p,N} contracts |Λ| projective lines onto the singular locus of S_{2,p,N}.

5. The class numbers H_{n}(p, 1) and H_{n}(1, p)

In this section we describe the arithmetic part of the results in Li and Oort [8]. Our references are Ibukiyama-Katsura-Oort [4, Section 2] and Li-Oort [8, Section 4]. Let B be the definite quaternion algebra over ℚ with discriminant p, and O be a maximal order of B. Let V = B^{⊕n}, regarded as a left B-module of row vectors, and let ψ(x, y) = ∑_{i=1}^{n} x_{i}ȳ_{i} be the standard hermitian form on V, where y_{i} ↦ ȳ_{i} is the canonical involution on B. Let G be the group of ψ-similitudes over ℚ; its group of ℚ-points is

G(ℚ) := {h ∈ M_{n}(B) | hh̄^{t} = rI_{n} for some r ∈ ℚ^{×} }.

Two O-lattices L and L′ in B^{⊕n} are called globally equivalent (denoted by L ∼ L′) if L′ = Lh for some h ∈ G(ℚ). For a finite place v of ℚ, we write B_{v} := B ⊗ ℚ_{v}, O_{v} := O ⊗ ℤ_{v} and L_{v} := L ⊗ ℤ_{v}. Two O-lattices L and L′ in B^{⊕n} are called locally equivalent at v (denoted by L_{v} ∼ L′_{v}) if L′_{v} = L_{v}h_{v} for some h_{v} ∈ G(ℚ_{v}). A genus of O-lattices is a set of (global) O-lattices in B^{⊕n} which are equivalent to each other locally at every finite place v. Let

N_{p} = O_{p}^{⊕n} · (\begin{matrix}I_{r} & 0 \\ 0 & πI_{n−r} \end{matrix}) · ξ ⊂ B_{p}^{⊕n},

where r is the integer [n/2], π is a uniformizer in O_{p}, and ξ is an element in GL_{n}(B_{p}) such that

ξξ̄^{t} = anti-diag(1, 1, …, 1).

Definition 5.1. (1) Let L_{n}(p, 1) denote the set of global equivalence classes of O-lattices L in B^{⊕n} such that L_{v} ∼ O_{v}^{⊕n} at every finite place v. The genus L_{n}(p, 1) is called the principal genus, and let H_{n}(p, 1) := |L_{n}(p, 1)|.

(2) Let L_{n}(1, p) denote the set of global equivalence classes of O-lattices L in B^{⊕n} such that L_{p} ∼ N_{p} and L_{v} ∼ O_{v}^{⊕n} at every finite place v ≠ p. The genus L_{n}(p, 1) is called the non-principal genus, and let H_{n}(1, p) := |L_{n}(1, p)|.

Recall (Section 3) that Λ_{g} is the set of isomorphism classes of g-dimensional principally polarized superspecial abelian varieties over \overline{𝔽}_{p}. When g = 2D > 0 is even, we denote by Λ_{g,p^{D}}^{∗} the set of isomorphism classes of g-dimensional polarized superspecial abelian varieties (A, λ) of degree p^{2D} over \overline{𝔽}_{p} satisfying ker λ = A[F]. Let A_{g,1} be the coarse moduli scheme of g-dimensional principally polarized abelian varieties, and let S_{g,1} be the supersingular locus of A_{g,1} ⊗ \overline{𝔽}_{p}. Recall (Subsection 2.1) that Π_{0}(S_{g,1}) denotes the set of irreducible components of S_{g,1}.

Theorem 5.2 (Li-Oort). We have

|Π_{0}(S_{g,1})| = \begin{cases}|Λ_{g}| & if g is odd; \\ |Λ_{g,p^{D}}^{∗}| & if g = 2D is even. \end{cases}

The arithmetic part for Π_{0}(S_{g,1}) is given by the following

Proposition 5.3.

(1) For any positive integer g, one has |Λ_{g}| = H_{g}(p, 1).

(2) For any even positive integer g = 2D, one has |Λ_{g,p^{D}}^{∗}| = H_{g}(1, p).

Proof. (1) See [4, Theorem 2.10]. (2) See [8, Proposition 4.7].

6. Correspondence computation

6.1. Let M_{0} be a superspecial Dieudonné module over k of rank 2g, and call

M̃_{0} := {x ∈ M_{0}; F^{2}x = px},

the skeleton of M_{0} (cf. [8, 5.7]). We know that M̃_{0} is a Dieudonné module over 𝔽_{p^{2}} and M̃_{0} ⊗_{W (𝔽_{p^{2}} )} W (k) = M_{0}. The vector space M̃_{0}/V M̃_{0} defines an 𝔽_{p^{2}}-structure of the k-vector space M_{0}/V M_{0}. Let Gr(n, m) be the Grassmannian variety of n-dimensional subspaces in an m-dimensional vector space. Suppose M_{1} is a Dieudonné submodule of M_{0} such that

V M_{0} ⊂ M_{1} ⊂ M_{0}, dim_{k} M_{1}/V M_{0} = r,

for some integer 0 ≤ r ≤ g. As dim M_{0}/V M_{0} = g, the subspace \overline{M}_{1} := M_{1}/V M_{0} corresponds to a point in Gr(r, g)(k).

Lemma 6.1. Notation as above. Then M_{1} is superspecial (i.e. F^{2}M_{1} = pM_{1}) if and only if \overline{M}_{1} ∈ Gr(r, g)(𝔽_{p^{2}}).

Proof. If M_{1} is generated by V M̃_{0} and x_{1}, x_{2}, …, x_{r}, x_{i} ∈ M̃_{0} over W. Then M̃_{1} generates M_{1} and thus F^{2}M_{1} = pM_{1}. Therefore, M_{1} is superspecial. Conversely if M_{1} is superspecial, then we have

V M̃_{0} ⊂ M̃_{1} ⊂ M̃_{0}.

Therefore, M̃_{1} gives rise to an element in Gr(r, g)(𝔽_{p^{2}}).

6.2. Let L(n, 2n) ⊂ Gr(n, 2n) be the Lagrangian variety of maximal isotropic subspaces in a 2n-dimensional vector space with a non-degenerate alternating form. From now on g = 2D is an even positive integer. Recall (in Introduction and Section 5) that Λ_{g,p^{D},N}^{∗} denotes the set of superspecial (geometric) points (A, λ, η) in A_{g,p^{D},N} ⊗ \overline{𝔽}_{p} satisfying ker λ = A[F].

Lemma 6.2. Let (A_{2}, λ_{2}, η_{2}) ∈ Λ_{g,p^{D},N}^{∗} and (M_{2}, ⟨ , ⟩_{2}) be the associated Dieudonné module. There is a W -basis e_{1}, …, e_{2g} for M_{2} such that for 1 ≤ i ≤ g

F e_{i} = e_{g+i}, F e_{g+i} = pe_{i},

and the non-zero pairings are

⟨e_{i}, e_{D+i}⟩_{2} = −⟨e_{D+i}, e_{i}⟩_{2} = \frac{1}{p},

⟨e_{g+i}, e_{g+D+i}⟩_{2} = −⟨e_{g+D+i}, e_{g+i}⟩_{2} = 1,

for 1 ≤ i ≤ D.

Proof. Use the same argument of Lemma 4.2 (2).

6.3. Let H be the moduli space over ℤ_{(p)}[ζ_{N}] which parametrizes equivalence classes of objects (φ : \underline{A}_{1} → \underline{A}_{2}), where

• \underline{A}_{1} = (A_{1}, λ_{1}, η_{1}) is an object in A_{g,1,N},

• \underline{A}_{2} = (A_{2}, λ_{2}, η_{2}) is an object in A_{g,p^{D},N}, and

• φ : A_{1} → A_{2} is an isogeny of degree p^{D} satisfying φ^{∗}λ_{2} = pλ_{1} and

φ_{∗}η_{1} = η_{2}.

The moduli space H with two natural projections gives the following correspondence:

\xymatrix{ & H \ar[dl]_{_{pr_{1}}} \ar[dr]^{_{pr_{2}}} \\ A_{g,1,N} &  & A_{g,p^{D},N}.}

Let S be the supersingular locus of H ⊗ \overline{𝔽}_{p}, which is the reduced closed subscheme consisting of supersingular points (either A_{1} or A_{2} is supersingular, or equivalently both are supersingular). Restricting the natural projections on S, we have the following correspondence

\xymatrix{ & S \ar[dl]_{_{pr_{1}}} \ar[dr]^{_{pr_{2}}} \\ S_{g,1,N} &  & S_{g,p^{D},N}.}

Suppose that \underline{A}_{2} ∈ Λ_{g,p^{D},N}^{∗}. Let (φ : \underline{A}_{1} → \underline{A}_{2}) ∈ S(k) be a point in the pre-image pr_{2}^{−1}(\underline{A}_{2}), and let (M_{1}, ⟨ , ⟩_{1}) be the Dieudonné module associated to \underline{A}_{1}. We have

M_{1} ⊂ M_{2}, p⟨ , ⟩_{2} = ⟨ , ⟩_{1}, F M_{2} = M_{2}^{t}.

Since \underline{A}_{2} is superspecial and ⟨ , ⟩_{1} is a perfect pairing on M_{1}, we get

F M_{2} = V M_{2} = M_{2}^{t}, M_{2}^{t} ⊂ M_{1}^{t} = M_{1}.

Therefore, we have

F M_{2} = V M_{2} ⊂ M_{1} ⊂ M_{2}, dim_{k} M_{1}/V M_{2} = D.

Put ⟨ , ⟩ := p⟨ , ⟩_{2}. The pairing

⟨ , ⟩ : M_{2} × M_{2} → W

induces a pairing

⟨ , ⟩ : M_{2}/V M_{2} × M_{2}/V M_{2} → k

which is perfect (by Lemma 6.2). Furthermore, M_{1}/V M_{2} is a maximal isotropic subspace for the pairing ⟨ , ⟩. This is because ⟨ , ⟩_{1} is a perfect pairing on M_{1} and dim M_{1}/V M_{2} = D is the maximal dimension of isotropic subspaces. We conclude that the point (φ : \underline{A}_{1} → \underline{A}_{2}) lies in pr_{2}^{−1}(\underline{A}_{2}) if and only if V M_{2} ⊂ M_{1} ⊂ M_{2} and M_{1}/V M_{2} is a maximal isotropic subspace of the symplectic space (M_{2}/V M_{2}, ⟨ , ⟩). By Lemma 6.1, we have proved

Proposition 6.3. Let \underline{A}_{2} be a point in Λ_{g,p^{D},N}^{∗}.

(1) The pre-image pr_{2}^{−1}(\underline{A}_{2}) is naturally isomorphic to the projective variety L(D, 2D) over k.

(2) The set pr_{1}^{−1}(Λ_{g,1,N}) ∩ pr_{2}^{−1}(\underline{A}_{2}) is in bijection with L(D, 2D)(𝔽_{p^{2}}), where the W (𝔽_{p^{2}})-structure of M_{2} is given by the skeleton M̃_{2}.

6.4. We compute pr_{1}^{−1}(\underline{A}_{1}) ∩ pr_{2}^{−1}(Λ_{g,p^{D},N}^{∗}) for a point \underline{A}_{1} in Λ_{g,1,N}. Let T be the closed subscheme of S consisting of the points (φ : \underline{A}_{1} → \underline{A}_{2}) such that ker λ_{2} = A_{2}[F]. We compute the closed subvariety pr_{1}^{−1}(\underline{A}_{1}) ∩ T first. Let \underline{A}_{1} ∈ Λ_{g,1,N}, and let (φ : \underline{A}_{1} → \underline{A}_{2}) ∈ S(k) be a point in the pre-image pr_{1}^{−1}(\underline{A}_{1}). Let (M_{1}, ⟨ , ⟩_{1}) and (M_{2}, ⟨ , ⟩_{2}) be the Dieudonné modules associated to \underline{A}_{1} and \underline{A}_{2}, respectively. One has

M_{1}^{t} = M_{1} ⊃ M_{2}^{t} = F M_{2},

and thus has

F M_{2} ⊂ M_{1} ⊂ M_{2} ⊂ p^{−1}V M_{1}.

Since M_{1} is superspecial, p^{−1}V M_{1} = p^{−1}F M_{1}. Put M_{0} := p^{−1}V M_{1} and ⟨ , ⟩ := p⟨ , ⟩_{2}. We have

pM_{0} ⊂ M_{2}^{t} = F M_{2} ⊂ M_{1} = V M_{0} ⊂ M_{2} ⊂ M_{0}

and that ⟨ , ⟩ is a perfect pairing on M_{0}. By Proposition 6.1 of [8] (cf. Lemma 4.2 (1)), there is a W -basis f_{1} …, f_{2g} for M_{0} such that for 1 ≤ i ≤ g

F f_{i} = f_{g+i}, F f_{g+i} = pf_{i},

and the non-zero pairings are

⟨f_{i}, f_{g+i}⟩ = −⟨f_{g+i}, f_{i}⟩ = β_{1}, ∀ 1 ≤ i ≤ g

where β_{1} ∈ W (𝔽_{p^{2}})^{×} with β_{1}^{σ} = −β_{1}. In the vector space \overline{M}_{0} := M_{0}/pM_{0}, \overline{M}_{2} is a vector subspace over k of dimension g + D with

\overline{M}_{2} ⊃ \overline{V M}_{0} = k < f_{g+1}, …, f_{2g} > and ⟨\overline{M}_{2}, F \overline{M}_{2}⟩ = 0.

We can write

\overline{M}_{2} = k < v_{1}, …, v_{D} > +\overline{V M}_{0}, v_{i} = ∑_{r=1}^{g} a_{ir}f_{r}.

One computes

⟨v_{i}, F v_{j}⟩ = ⟨∑_{r=1}^{g} a_{ir}f_{r}, F (∑_{q=1}^{g} a_{jq}f_{q})⟩ = ⟨∑_{r=1}^{g} a_{ir}f_{r}, ∑_{q=1}^{g} a_{jq}^{p}f_{g+q}⟩ = β_{1} ∑_{r=1}^{g} a_{ir}a_{jr}^{p}.

This computation leads us to the following definition.

6.5. Let V := 𝔽_{p^{2}}^{2n}. For any field K ⊃ 𝔽_{p^{2}}, we put V_{K} := V ⊗_{𝔽_{p^{2}}} K and define a pairing on V_{K}

⟨ , ⟩′ : V_{K} × V_{K} → K, ⟨(a_{i}), (b_{i})⟩′ := ∑_{i=1}^{2n} a_{i} b_{i}^{p}.

Let X(n, 2n) ⊂ Gr(n, 2n) be the subvariety over 𝔽_{p^{2}} which parametrizes ndimensional (maximal) isotropic subspaces in V with respect to the pairing ⟨ , ⟩′. With the computation in Subsection 6.4 and Lemma 6.1, we have proved Proposition 6.4. Let \underline{A}_{1} be a point in Λ_{g,1,N} and g = 2D.

(1) The intersection pr_{1}^{−1}(\underline{A}_{1}) ∩ T is naturally isomorphic to the projective variety X(n, 2n) over k.

(2) The set pr_{1}^{−1}(\underline{A}_{1}) ∩ pr_{2}^{−1}(Λ_{g,p^{D},N}^{∗}) is in bijection with X(D, 2D)(𝔽_{p^{2}}).

When g = 2, Proposition 6.4 (2) is a result of Moret-Bailly (Lemma 4.3).

6.6. To compute X(n, 2n)(𝔽_{p^{2}}), we show that it is the set of rational points of a homogeneous space under the quasi-split group U(n, n). Let V = 𝔽_{p^{2}}^{2n} and let x ↦ x̄ be the involution of 𝔽_{p^{2}} over 𝔽_{p}. Let ψ((x_{i}), (y_{i})) = ∑_{i} x_{i}ȳ_{i} be the standard hermitian form on V. For any field K ⊃ 𝔽_{p}, we put V_{K} := V ⊗_{𝔽_{p}} K and extend ψ to a from

ψ : V_{K} ⊗ V_{K} → 𝔽_{p^{2}} ⊗ K

by K-linearity. Let U(n, n) be the group of automorphisms of V that preserve the hermitian form ψ. Let LU(n, 2n)(K) be the space of n-dimensional (maximal) isotropic K-subspaces in V_{K} with respect to ψ. We know that LU(n, 2n) is a projective scheme over 𝔽_{p} of finite type, and this is a homogeneous space under U(n, n). It follows from the definition that

LU(n, 2n)(𝔽_{p}) = X(n, 2n)(𝔽_{p^{2}}).

However, the space LU(n, 2n) is not isomorphic to the space X(n, 2n) over k. Let Λ̃ be the subset of S consisting of elements (φ : \underline{A}_{1} → \underline{A}_{2}) such that \underline{A}_{2} ∈ Λ_{g,p^{D},N}^{∗} and \underline{A}_{1} ∈ Λ_{g,1,N}. We have natural projections

\xymatrix{ & Λ̃ \ar[dl]_{_{pr_{1}}} \ar[dr]^{_{pr_{2}}} \\ Λ_{g,1,N} &  & Λ_{g,p^{D},N}^{∗}.}

By Propositions 6.3 and 6.4, and Subsection 6.6, we have proved

Proposition 6.5. Notation as above, one has

|Λ̃| = |L(D, 2D)(𝔽_{p^{2}})| · |Λ_{g,p^{D},N}^{∗}| = |LU(D, 2D)(𝔽_{p})| · |Λ_{g,1,N}|.

Theorem 6.6. We have

|Λ_{g,p^{D},N}^{∗}| = | Sp_{2g}(ℤ/Nℤ)| · \frac{(−1)^{g(g+1)/2}}{2^{g}} \{∏_{i=1}^{g} ζ(1 − 2i)\} · ∏_{i=1}^{D}(p^{4i−2} − 1).

Proof. We compute in Section 7 that

|L(D, 2D)(𝔽_{p^{2}})| = ∏_{i=1}^{D}(p^{2i} + 1),

|LU(D, 2D)(𝔽_{p})| = ∏_{i=1}^{D}(p^{2i−1} + 1).

Using Proposition 6.5, Theorem 3.1 and Lemma 3.2, we get the value of |Λ_{g,p^{D},N}^{∗}|.

6.7. Proof of Theorem 1.4. By a theorem of Li and Oort (Theorem 5.2), we know

|Π_{0}(S_{g,1,N})| = \begin{cases}|Λ_{g,1,N}| & if g is odd; \\ |Λ_{g,p^{D},N}^{∗}| & if g = 2D is even. \end{cases}

Note that the result of Li and Oort is formulated for the coarse moduli space S_{g,1}. However, it is clear that adding the level-N structure yields a modification as above. Theorem 1.4 then follows from Theorem 3.1, Lemma 3.2 and Theorem 6.6.

7. L(n, 2n)(𝔽_{q}) and LU(n, 2n)(𝔽_{q})

Let L(n, 2n) be the Lagrangian variety of maximal isotropic subspaces in a 2n-dimensional vector space V_{0} with a non-degenerate alternating form ψ_{0}.

Lemma 7.1. |L(n, 2n)(𝔽_{q})| = ∏_{i=1}^{n}(q^{i} + 1).

Proof. Let e_{1}, …, e_{2n} be the standard symplectic basis for V_{0}. The group Sp_{2n}(𝔽_{q}) acts transitively on the space L(n, 2n)(𝔽_{q}). For h ∈ Sp_{2n}(𝔽_{q}), the map h ↦ {he_{1}, … he_{2n}} induces a bijection between Sp_{2n}(𝔽_{q}) and the set B(n) of ordered symplectic bases {v_{1}, …, v_{2n}} for V_{0}. The first vector v_{1} has q^{2n} − 1 choices. The first companion vector v_{n+1} has (q^{2n} − q^{2n−1})/(q − 1) choices as it does not lie in the hyperplane v_{1}^{⊥} and we require ψ_{0}(v_{1}, v_{n+1}) = 1. The remaining ordered symplectic basis can be chosen from the complement 𝔽_{q} < v_{1}, v_{n+1} >^{⊥}. Therefore, we have proved the recursive formula

| Sp_{2n}(𝔽_{q})| = (q^{2n} − 1)q^{2n−1}| Sp_{2n−2}(𝔽_{q})|.

From this, we get

| Sp_{2n}(𝔽_{q})| = q^{n^{2}} ∏_{i=1}^{n}(q^{2i} − 1).

Let P be the stabilizer of the standard maximal isotropic subspace 𝔽_{q} < e_{1}, …, e_{n} >. It is easy to see that

P = \{(\begin{matrix}A & B \\ 0 & D \end{matrix}); AD^{t} = I_{n}, BA^{t} = AB^{t}\} .

The matrix BA^{t} is symmetric and the space of n × n symmetric matrices has dimension (n^{2} + n)/2. This yields

|P | = q^{\frac{n^{2}+n}{2}} | GL_{n}(𝔽_{q})| = q^{n^{2}} ∏_{i=1}^{n}(q^{i} − 1)

as one has

| GL_{n}(𝔽_{q})| = q^{\frac{n^{2}−n}{2}} ∏_{i=1}^{n}(q^{i} − 1).

Since L(n, 2n)(𝔽_{q}) ≃ Sp_{2n}(𝔽_{q})/P, we get |L(n, 2n)(𝔽_{q})| = ∏_{i=1}^{n}(q^{i} + 1).

7.1. Let V = 𝔽_{q^{2}}^{2n} and let x ↦ x̄ = x^{q} be the involution of 𝔽_{q^{2}} over 𝔽_{q}. Let ψ((x_{i}), (y_{i})) = ∑_{i} x_{i}ȳ_{i} be the standard hermitian form on V. For any field K ⊃ 𝔽_{q}, we put V_{K} := V ⊗_{𝔽_{p}} K and extend ψ to a from

ψ : V_{K} ⊗ V_{K} → 𝔽_{q^{2}} ⊗_{𝔽_{q}} K

by K-linearity. Let U(n, n) be the group of automorphisms of V that preserve the hermitian form ψ. Let LU(n, 2n)(K) be the set of n-dimensional (maximal) isotropic Ksubspaces in V_{K} with respect to ψ. We know that LU(n, 2n) is a homogeneous space under U(n, n). Let

I_{m} := \{\underline{a} = (a_{1}, …, a_{m}) ∈ 𝔽_{q^{2}}^{m} ; Q(\underline{a}) = 0\},

where Q(\underline{a}) = a_{1}^{q+1} + ⋯ + a_{m}^{q+1}.

Lemma 7.2. We have |I_{m}| = q^{2m−1} + (−1)^{m}q^{m} + (−1)^{m−1}q^{m−1}.

Proof. For m > 1, consider the projection p : I_{m} → 𝔽_{q^{2}}^{m−1} which sends (a_{1}, …, a_{m}) to (a_{1}, …, a_{m−1}). Let I_{m−1}^{c} be the complement of I_{m−1} in 𝔽_{q^{2}}^{m−1}. If x ∈ I_{m−1}, then the pre-image p^{−1}(x) consists of one element. If x ∈ I_{m−1}^{c}, then the pre-image p^{−1}(x) consists of solutions of the equation a_{m}^{q+1} = −Q(x) ∈ 𝔽_{q}^{×} and thus p^{−1}(x) has q+1 elements. Therefore, |I_{m}| = |I_{m−1}|+(q+1)|I_{m−1}^{c}|. From this we get the recursive formula

|I_{m}| = (q + 1)q^{2(m−1)} − q|I_{m−1}|.

We show the lemma by induction. When m = 1, |I_{m}| = 1 and the statement holds. Suppose the statement holds for m = k, i.e. |I_{k}| = q^{2k−1} + (−1)^{k}q^{k} + (−1)^{k−1}q^{k−1}. When m = k + 1,

\begin{aligned}|I_{k+1}| & = (q + 1)q^{2k} − q[q^{2k−1} + (−1)^{k}q^{k} + (−1)^{k−1}q^{k−1}] \\ & = q^{2k+1} + (−1)^{k+1}q^{k+1} + (−1)^{k}q^{k}. \end{aligned}

This completes the proof.

Proposition 7.3. |LU(n, 2n)(𝔽_{q})| = ∏_{i=1}^{n}(q^{2i−1} + 1).

Proof. We can choose a new basis e_{1}, …, e_{2n} for V such that the non-zero pairings are

ψ(e_{i}, e_{n+i}) = ψ(e_{n+i}, e_{i}) = 1, ∀ 1 ≤ i ≤ n.

The representing matrix for ψ with respect to {e_{1}, …, e_{2n}} is

J = (\begin{matrix}0 & I_{n} \\ I_{n} & 0 \end{matrix}) .

Let P be the stabilizer of the standard maximal isotropic subspace 𝔽_{q^{2}} < e_{1}, …, e_{n} >. It is easy to see that

P = \{(\begin{matrix}A & B \\ 0 & D \end{matrix}); AD^{∗} = I_{n}, BA^{∗} + AB^{∗} = 0\} .

The matrix BA^{∗} is skew-symmetric hermitian. The space of n × n skewsymmetric hermitian matrices has dimension n^{2} over 𝔽_{q}. Indeed, the diagonal consists of entries in the kernel of the trace; this gives dimension n. The upper triangular has (n^{2} − n)/2 entries in 𝔽_{q^{2}}; this gives dimension n^{2} − n. Hence,

|P | = q^{n^{2}}| GL_{n}(𝔽_{q^{2}})| = q^{2n^{2}−n} ∏_{i=1}^{n}(q^{2i} − 1).

We compute |U(n, n)(𝔽_{q})|. For h ∈ U(n, n)(𝔽_{q}), the map

h ↦ {he_{1}, …, he_{2n}}

gives a bijection between U(n, n)(𝔽_{q}) and the set B(n) of ordered bases {v_{1}, …, v_{2n}} for which the representing matrix of ψ is J. The first vector v_{1} has

|I_{2n}| − 1 = q^{4n−1} + q^{2n} − q^{2n−1} − 1 = (q^{2n} − 1)(q^{2n−1} + 1).

choices (Lemma 7.2). For the choices of the companion vector v_{n+1} with ψ(v_{n+1}, v_{n+1}) = 0 and ψ(v_{1}, v_{n+1}) = 1, consider the set

Y := {v ∈ V; ψ(v_{1}, v) = 1}.

Clearly, |Y | = q^{4n−2}. The additive group 𝔽_{q^{2}} acts on Y by a · v = v + av_{1} for a ∈ 𝔽_{q^{2}}, v ∈ Y. It follows from

ψ(v + av_{1}, v + av_{1}) = ψ(v, v) + ā + a

that every orbit O(v) contains an isotropic vector v_{0} and any isotropic vector in O(v) has the form v_{0} + av_{1} with ā + a = 0. Hence, the vector v_{n+1} has

\frac{| Y | q}{q^{2}} = q^{4n−3}

choices. In conclusion, we have proved the recursive formula

|U(n, n)(𝔽_{q})| = (q^{2n} − 1)(q^{2n−1} + 1)q^{4n−3}|U(n − 1, n − 1)(𝔽_{q})|.

It follows that

|U(n, n)(𝔽_{q})| = q^{2n^{2}−n} ∏_{i=1}^{n}(q^{2i} − 1)(q^{2i−1} + 1).

Since LU(n, 2n)(𝔽_{q}) ≃ U(n, n)(𝔽_{q})/P, we get |LU(n, 2n)(𝔽_{q})| = ∏_{i=1}^{n}(q^{2i−1} + 1).

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Chia-Fu Yu

Institute of Mathematics Academia Sinica 128 Academia Rd. Sec. 2 Nankang Taipei Taiwan and NCTS (Taipei Office) chiafu@math.sinica.edu.tw