On Superspecial Abelian Surfaces over Finite Fields

Jiangwei Xue,Tse-Chung Yang, and Chia-Fu Yu

Received: April 20, 2016 Revised: September 4, 2016

Communicated by Takeshi Saito

Abstract. In this paper we establish a new lattice description for superspecial abelian varieties over a finite field 𝔽_{q} of q = p^{a} elements. Our description depends on the parity of the exponent a of q. When q is an odd power of the prime p, we give an explicit formula for the number of superspecial abelian surfaces over 𝔽_{q}.

2010 Mathematics Subject Classification: 11R52, 11G10

Keywords and Phrases: supersingular abelian surfaces, class number formula, Galois cohomology.

1. Introduction

Throughout this paper p denotes a prime number, and q = p^{a} a power of p with an exponent a ∈ ℕ, the set of strictly positive integers. The goal of this paper is to calculate explicitly the number of superspecial abelian surfaces over a finite field 𝔽_{q}. This can be regarded as a natural extension of works of the authors [22, 23] and the last named author [26] contributed to the study of supersingular abelian varieties over finite fields. Recall that an abelian variety over a field k of characteristic p is said to be supersingular if it is isogenous to a product of supersingular elliptic curves over an algebraic closure \overline{k} of k; it is said to be superspecial if it is isomorphic to a product of supersingular elliptic curves over \overline{k}. As any supersingular abelian variety is isogenous to a superspecial abelian variety, it is very common to study supersingular abelian varieties through investigating the classification of superspecial abelian varieties. For any integer d ≥ 1, let Sp_{d}(𝔽_{q}) denote the set of isomorphism classes of ddimensional superspecial abelian varieties over the finite field 𝔽_{q} of q elements. The case where d = 1 concerns the classification of supersingular elliptic curves over finite fields. The theory of elliptic curves over finite fields has been studied by Deuring since 1940’s and becomes well known. There are explicit descriptions for each isogeny class; see Waterhouse [21, Section 4]. However, the authors could not find an explicit formula for | Sp_{1}(𝔽_{q})| in the literature. For the sake of completeness we include a formula for | Sp_{1}(𝔽_{q})|, based on the exposition of Deuring’s results by Waterhouse [21]. The goal of the present paper is then to find an explicit formula for the number | Sp_{d}(𝔽_{q})| in the case where d = 2. Before stating our main results, we describe a basic method for counting Sp_{d}(𝔽_{q}). For simplicity, assume that 𝔽_{q} = 𝔽_{p} is the prime finite field for the moment. One can divide the finite set Sp_{d}(𝔽_{p}) into finitely many subsets according to the isogeny classes of members. Therefore, it suffices to classify all d-dimensional supersingular isogeny classes and to count the number of superspecial members in each supersingular isogeny class. The Honda-Tate theorem allows us to describe isogeny classes over 𝔽_{q} in terms of multiple Weil q-numbers (which are simply finite nonnegative integral formal sums of Weil q-numbers up to conjugate; see Section 4.1). If π is a supersingular multiple Weil q-number, we denote by [X_{π}] the corresponding supersingular isogeny class (here X_{π} is an abelian variety in this class), H(π) the number of isomorphism classes of abelian varieties in [X_{π}] and H_{sp}(π) the number of isomorphism classes of superspecial abelian varieties in [X_{π}]. Then we have

(1.1)  | Sp_{d}(𝔽_{p})| = ∑_{π} H_{sp}(π),

where π runs through all supersingular multiple Weil p-numbers with dim X_{π} = d. We classify all possible isogeny classes of π’s occurring in the sum (see Sections 2–3). The problem then is to compute each term H_{sp}(π). One should distinguish the cases according to whether the endomorphism algebra End^{0}(X_{π}) = End(X_{π}) ⊗ ℚ of X_{π} satisfies the Eichler condition [19, Section III.4, p.81] or not. We now focus on the case where d = 2. Consider the case where π is the Weil p-number \sqrt{p}. Correspondingly, X_{π} is a supersingular abelian surface. It is known (see Tate [17]) that the endomorphism algebra End^{0}(X_{π}) of X_{π} is isomorphic to the totally definite quaternion algebra algebra D = D_{∞_{1},∞_{2}} over the quadratic real field F = ℚ(\sqrt{p}) ramified exactly at the two real places {∞_{1}, ∞_{2}} of F. In this case all abelian surfaces in the isogeny class [X\sqrt{p}] are superspecial, i.e. H(\sqrt{p}) = H_{sp}(\sqrt{p}). When p = 2 or p ≡ 3 (mod 4), Waterhouse proved that the number H(\sqrt{p}) is equal to the class number h(D) of D. The current authors analyzed the remaining case in [22, Section 6] and showed that when p ≡ 1 (mod 4), the number H(\sqrt{p}) is equal to the sum of h(D) and the class numbers of two other proper ℤ[\sqrt{p}]-orders in D of index 8 and 16, respectively (the descriptions of these orders are made concrete by results of [25]). These class numbers are computed systematically in [22], which produces the explicit formulas for H(\sqrt{p}) given in Theorem 1.1 below. In what follows we write K_{m,j} for the number field ℚ(\sqrt{m} ,\sqrt{−j}) for any square-free integers m > 1 and j ≥ 1. If m ≡ 1 (mod 4), then we define

(1.2)  ϖ_{m} := 3[O_{ℚ(\sqrt{m})}^{×} : ℤ[\sqrt{m}]^{×}]^{−1},

where O_{ℚ(\sqrt{m})} denotes the ring of integers of ℚ(\sqrt{m}). By similar arguments as those in [23, Lemma 4.1 and Section 4.2], we have ϖ_{m} ∈ {1, 3}, and ϖ_{m} = 3 if m ≡ 1 (mod 8). The class number of a number field K is denoted by h(K). When K = ℚ(\sqrt{m}), we write h(\sqrt{m}) for h(ℚ(\sqrt{m})) instead.

Theorem 1.1. Let H(\sqrt{p}) be the number of 𝔽_{p}-isomorphism classes of abelian varieties in the simple isogeny class corresponding to the Weil p-number π = \sqrt{p}, and let F = ℚ(\sqrt{p}). Then

(1) H(\sqrt{p}) = 1, 2, 3 for p = 2, 3, 5, respectively.

(2) For p > 5 and p ≡ 3 (mod 4), we have (1.3)

H(\sqrt{p}) = \frac{1}{2}h(F)ζ_{F} (−1)+(\frac{3}{8} + \frac{5}{8} (2 − (\frac{2}{p})))h(K_{p,1})+ \frac{1}{4}h(K_{p,2})+ \frac{1}{3}h(K_{p,3}),

where ζ_{F} (s) is the Dedekind zeta function of F.

(3) For p > 5 and p ≡ 1 (mod 4), we have (1.4)

H(\sqrt{p}) = \{8ζ_{F} (−1)h(F) + h(K_{p,1}) + \frac{4}{3}h(K_{p,3}) for p ≡ 1 (mod 8); \begin{matrix}\frac{1}{2}(15ϖ_{p} + 1)ζ_{F} (−1)h(F) + \frac{1}{4}(3ϖ_{p} + 1)h(K_{p,1}) + \frac{4}{3}h(K_{p,3}) \\ for p ≡ 5 (mod 8). \end{matrix}

The computation in Theorem 1.1 is based on the generalized Eichler class formula [22, Theorem 1.4] that the authors developed. Compared with the classical Eichler class number formula [19, Corollary V.2.5] which treats only the Eichler orders, this generalized formula allows us to compute the class number of an arbitrary ℤ-order in a totally definite quaternion over a totally real field F. This ℤ-order does not necessarily contains the maximal order O_{F} of F. For a quadratic real field F, the special zeta value ζ_{F} (−1) can be calculated by Siegel’s formula [28, Table 2, p. 70]

(1.5)  ζ_{F} (−1) = \frac{1}{60} ∑_{\begin{matrix}b^{2}+4ac=𝔡_{F} \\ a,c>0 \end{matrix}} a,

where 𝔡_{F} is the discriminant of F/ℚ, b ∈ ℤ and a, c ∈ ℕ. The first main result of this paper gives the following explicit formula for |Sp_{2}(𝔽_{p})|, the number of isomorphism classes of superspecial abelian surfaces over 𝔽_{p}. To obtain this formula, we calculate all terms H_{sp}(π) with π ≠ ±\sqrt{p} in (1.1), and then sum them up together with H(\sqrt{p}). The computation of H_{sp}(π) uses a lattice description for superspecial abelian varieties; see Section 5 for details. Similar to Theorem 1.1, special attentions have to be paid to the cases with small primes p.

Theorem 1.2. We have | Sp_{2}(𝔽_{p})| = H(\sqrt{p}) + Δ(p), where the formula for H(\sqrt{p}) is stated in Theorem 1.1 and Δ(p) is the number described as follows.

(1) Δ(p) = 15, 20, 9 for p = 2, 3, 5, respectively.

(2) For p > 5 and p ≡ 1 (mod 4), we have

(1.6)  Δ(p) = (ϖ_{p} + 1)h(K_{p,3}) + h(K_{2p,1}) + h(K_{3p,3}) + h(\sqrt{−p}).

(3) For p > 5 and p ≡ 3 (mod 4), we have

(1.7) Δ(p) = h(K_{p,3}) + h(K_{2p,1}) + (ϖ_{3p} + 1)h(K_{3p,3}) + (4 − (\frac{2}{p})) h(\sqrt{−p}).

A key ingredient of our computation for Sp_{2}(𝔽_{p}) is Proposition 5.1, which works only for the prime finite fields. Centeleghe and Stix [4] provide a categorical description of Proposition 5.1 (also compare [26, Theorem 3,1]). However, their results are also limited to the prime finite fields. When the base field 𝔽_{q} is no longer the prime finite field, direct calculations via the counting method described earlier for Sp_{d}(𝔽_{q}) (even when d = 2) become more complicated. Our second main result extends the computations of Sp_{2}(𝔽_{p}) to Sp_{2}(𝔽_{q}) for more general finite fields 𝔽_{q} via Galois cohomology. Observe that if d > 1, then there is only one isomorphism class of d-dimensional superspecial abelian varieties over \overline{𝔽}_{p} (see [12, Section 1.6, p. 13] or Theorem 6.6). Suppose X_{0} is any d-dimensional superspecial abelian variety over 𝔽_{p}. Then there is a bijection of finite pointed sets

(1.8)  Sp_{d}(𝔽_{p}) ≃ H^{1}(Γ_{𝔽_{p}}, G), d > 1,

where Γ_{𝔽_{p}} = Gal(\overline{𝔽}_{p}/𝔽_{p}) is the absolute Galois group of 𝔽_{p}, and G = Aut(X_{0} ⊗ \overline{𝔽}_{p}). Thus, computing the Galois cohomology would lead to a second proof of Theorem 1.2. However, the complexity of the final formula as in Theorem 1.2 suggests that the computation of this Galois cohomology is likely on the same level of difficulty as the counting method via (1.1). Nevertheless, the true advantages of connecting to Galois cohomology are two folds.

(a) It naturally relates Sp_{d}(𝔽_{q}) and Sp_{d}(𝔽_{q′}) in the sense of Theorem 1.3 when the exponents in q = p^{a} and q′ = p^{a′} have the same parity.

(b) It gives rise to a lattice description of Sp_{d}(𝔽_{q}) when q = p^{a} is an even power of p; see Proposition 6.11.

Theorem 1.3. Let q and q′ be powers of p with same exponent parity and d ≥ 1 an integer. Then there is a natural bijection Sp_{d}(𝔽_{q}) ≃ Sp_{d}(𝔽_{q′}) preserving isogeny classes. In particular, the same formulas in Theorem 1.2 hold for | Sp_{2}(𝔽_{q})| since | Sp_{d}(𝔽_{q})| = | Sp_{d}(𝔽_{p})| when q is an odd power of p.

The bijection for the case d = 1 is handled separately in Section 4 (see Remark 4.5). For d ≥ 2, the bijection is established in Theorem 6.7. Along the way, we prove in Section 6.2 the following general result connecting isogeny classes of abelian varieties over 𝔽_{q} with cohomology classes.

Proposition 1.4. Let [X_{0}] be the 𝔽_{q}-isogeny class of an arbitrary abelian variety X_{0} over 𝔽_{q}, and G_{ℚ} = End^{0}(\overline{X}_{0})^{×} where \overline{X}_{0} = X_{0} ⊗_{𝔽_{q}} \overline{𝔽}_{q}. We write E^{0}(\overline{𝔽}_{q}/𝔽_{q}, [X_{0}]) for the set of 𝔽_{q}-isogeny classes of abelian varieties [X] such that \overline{X} is isogenous to \overline{X}_{0} over \overline{𝔽}_{q}. Then there is a canonical bijection of pointed sets

E^{0}(\overline{𝔽}_{q}/𝔽_{q}, [X_{0}]) ⟶^{∼} H^{1}(Γ_{𝔽_{q}}, G_{ℚ})

sending [X_{0}] to the trivial cohomology class.

Theorem 1.3 together with Proposition 5.1 give a new lattice description in Corollary 6.9 for Sp_{d}(𝔽_{q}) when q is an odd power of p. When q is an even power of p, a lattice description of Sp_{d}(𝔽_{q}) completely different from the odd case is given in Proposition 6.11, which paves the way to explicit formulas of |Sp_{2}(𝔽_{q})|. The detailed formulas and computations will be presented in a separated paper. The paper is organized as follows. In Section 2, we parameterize simple isogeny classes of supersingular abelian varieties over 𝔽_{q} using Weil q-numbers. Their dimensions are calculated in Section 3. In Section 4 we treat the dimension 1 case and calculate the number of isomorphism classes of supersingular elliptic curves over finite fields. The dimension 2 case is then treated in Section 5, except we work exclusively over the prime field 𝔽_{p}, and some arithmetic calculations are postponed to Section 7. Section 6 studies the parity property via Galois cohomology, thus providing means to extend results of Section 5 to all 𝔽_{p^{a}} with a odd. The aforementioned lattices descriptions are obtained in this process.

2. Parameterization of supersingular isogeny classes

2.1. Let q = p^{a} be a power of a prime number p. In this section we parameterize simple isogeny classes of supersingular abelian varieties over 𝔽_{q}. Let \overline{ℚ} ⊂ ℂ be the algebraic closure of ℚ in ℂ. If two algebraic numbers α, β ∈ \overline{ℚ} are conjugate over ℚ, then we write α ∼ β. Recall that an algebraic integer π ∈ \overline{ℚ} is said to be a Weil q-number if |ι(π)| = q^{1/2} for any embedding ι : ℚ(π) ↪ ℂ. By the Honda-Tate theory, the simple isogeny classes of abelian varieties over 𝔽_{q} are in bijection with the conjugacy classes of Weil q-numbers. A Weil q-number is said to be supersingular if the corresponding isogeny class consists of supersingular abelian varieties. Let W_{q}^{ss} denote the set of conjugacy classes of supersingular Weil q-numbers. We will find a unique representative for each conjugacy class in W_{q}^{ss}. Let π be a supersingular Weil q-number. It is known (the Manin-Oort Theorem, cf. [27, Theorem 2.9]) that π = \sqrt{q} ζ for a root of unity ζ. Let K := ℚ(π) and L := ℚ(\sqrt{q} , ζ). Note that both L and K are abelian extensions over ℚ. For any n ∈ ℕ (the set of positive integers), write ζ_{n} := e^{2πi/n} ∈ \overline{ℚ}.

Lemma 2.1. Any supersingular Weil q-number π is conjugate to \sqrt{q} ζ_{n} or −\sqrt{q} ζ_{n} with n ≢ 2 (mod 4).

Proof. Let π = \sqrt{q} ζ_{m}^{ν} for some positive integers ν and m with (ν, m) = 1. Choose an element σ ∈ Gal(L/ℚ) such that σ(ζ_{m}^{ν}) = ζ_{m}, Then σ(π) = ±\sqrt{q} ζ_{m}.

If m ≢ 2 (mod 4), then we are done. Suppose that m = 2k for an odd integer k = 1 − 2u. Clearly (k, u) = 1. Since ζ_{2k} = ζ_{2k}^{k+2u} = −ζ_{2k}^{2u} = −ζ_{k}^{u}, we have

±\sqrt{q} ζ_{2k} = ∓\sqrt{q} ζ_{k}^{u} ∼ ϵ\sqrt{q} ζ_{k}, for some ϵ ∈ \{±1\}

by the previous argument. □

By Lemma 2.1, there is a unique subset W of \{±\sqrt{q} ζ_{n}; n ≢ 2 (mod 4)\} that contains \{\sqrt{q} ζ_{n}; n ≢ 2 (mod 4)\} and represents W_{q}^{ss}. We often identify W with W_{q}^{ss}. To determine the set W_{q}^{ss}, we need to characterize when \sqrt{q} ζ_{n} and −\sqrt{q} ζ_{n} are conjugate. As usual, the Galois group G_{n} := Gal(ℚ(ζ_{n})/ℚ) is naturally identified with (ℤ/nℤ)^{×} by mapping any r ∈ (ℤ/nℤ)^{×} to the element σ_{r} ∈ G_{n} with σ_{r}(ζ_{n}) = ζ_{n}^{r}.

2.2. Let us first assume that a is even, i.e., \sqrt{q} ∈ ℚ. Then \sqrt{q} ζ_{n} ∼ −\sqrt{q} ζ_{n} if and only if there is an element σ_{r} ∈ G_{n} such that σ_{r}(ζ_{n}) = −ζ_{n}. It is easy to see that

(2.1)  ζ_{n}^{r} = −ζ_{n} ⟺ 2|n and r = \frac{n}{2} + 1,

and if 4|n, then (r, n) = 1. As n ≢ 2 (mod 4), this gives

(2.2)  \sqrt{q} ζ_{n} ∼ −\sqrt{q} ζ_{n} ⟺ 4|n.

Thus,

(2.3)  W_{q}^{ss} ≃ \{±\sqrt{q} ζ_{n} ; 2 ∤ n \} ∪ \{\sqrt{q} ζ_{n} ; 4|n \}.

Alternatively, since \sqrt{q} ∈ ℚ, we have \sqrt{q} ζ_{n}^{ν} ∼ \sqrt{q} ζ_{n} for any ν ∈ ℕ with (ν, n) = 1. It follows that

(2.4)  W_{q}^{ss} ≃ \{\sqrt{q} ζ_{n} ; n ∈ ℕ\}.

The two descriptions (2.3) and (2.4) match, because when n is odd, −ζ_{n} is a primitive 2n-th root of unity and hence −\sqrt{q} ζ_{n} is conjugate to \sqrt{q} ζ_{2n}.

2.3. We now assume that a is odd. Let 𝔡_{p} be the discriminant of ℚ(\sqrt{p}). In other words, 𝔡_{p} = p if p ≡ 1 (mod 4), otherwise 𝔡_{p} = 4p. By [7, Chapter V, Theorem 48], \sqrt{p} ∈ ℚ(ζ_{n}) if and only if 𝔡_{p} | n. Suppose this is the case. Let

(2.5) χ : G_{n} = (ℤ/nℤ)^{×} → Gal(ℚ(\sqrt{p})/ℚ) = \{±1\}, σ_{r}(\sqrt{p}) = χ(r)\sqrt{p}

be the associated quadratic character. Clearly, χ factors through G_{𝔡_{p}} = Gal(ℚ(ζ_{𝔡_{p}})/ℚ).

Lemma 2.2. Let n be a positive integer with n ≢ 2 (mod 4) and q = p^{a} an odd power of p.

(i) If \sqrt{p} ∉ ℚ(ζ_{n}), then \sqrt{q} ζ_{n} ∼ −\sqrt{q} ζ_{n}. (ii) Suppose that \sqrt{p} ∈ ℚ(ζ_{n}), i.e., n is divisible by 𝔡_{p}. Then

(2.6)  \sqrt{q} ζ_{n} ∼ −\sqrt{q} ζ_{n} ⟺ 4|n and χ(n/2 + 1) = 1.

Proof. (i) As \sqrt{p} ∉ ℚ(ζ_{n}), there is an element σ ∈ Gal(L/ℚ) such that σ(ζ_{n}) = ζ_{n} and σ(\sqrt{p}) = −\sqrt{p}. Then σ(\sqrt{q} ζ_{n}) = −\sqrt{q} ζ_{n}. (ii) First, \sqrt{q} ζ_{n} ∼ −\sqrt{q} ζ_{n} if and only if there is an element σ_{r} ∈ G_{n} such that σ_{r}(\sqrt{q} ζ_{n}) = χ(r)\sqrt{q} ζ_{n}^{r} = −\sqrt{q} ζ_{n}. If χ(r) = −1, then ζ_{n}^{r} = ζ_{n} and σ_{r} = 1, which is impossible. If χ(r) = 1, then ζ_{n}^{r} = −ζ_{n} and hence 4|n and r = n/2 + 1 by (2.1). This concludes our assertion (2.6). □

Proposition 2.3. Let n and q be as in Lemma 2.2.

(a) Suppose that p = 2. Then

(2.7)  \sqrt{q} ζ_{n} ∼ −\sqrt{q} ζ_{n} ⟺ 8 ∤ n or 16|n.

(b) Suppose that p ≡ 1 (mod 4). Then

(2.8)  \sqrt{q} ζ_{n} ∼ −\sqrt{q} ζ_{n} ⟺ p ∤ n or 4p|n.

(c) Suppose that p ≡ 3 (mod 4). Then

(2.9)  \sqrt{q} ζ_{n} ∼ −\sqrt{q} ζ_{n} ⟺ 4p ∤ n or 8p|n.

Proof. (a) By Lemma 2.2, we have \sqrt{q} ζ_{n} ∼ −\sqrt{q} ζ_{n} if and only if either 8 ∤ n, or both 8|n and χ(n/2 + 1) = 1. Suppose 8|n. Note that ℚ(ζ_{8}) = ℚ(\sqrt{2} ,\sqrt{−1} ) and \sqrt{2} = ζ_{8} + ζ_{8}^{−1}. It follows that

(2.10)  χ(r) = \begin{cases}1 & if r ≡ 1, 7 (mod 8); \\ −1 & if r ≡ 3, 5 (mod 8). \end{cases}

If 8||n, then r = n/2 + 1 ≡ 5 (mod 8) and χ(r) = −1. If 16|n, then r = n/2 + 1 ≡ 1 (mod 8) and χ(r) = 1. Thus, \sqrt{q} ζ_{n} ∼ −\sqrt{q} ζ_{n} ⟺ 8 ∤ n or 16|n.

(b) By Lemma 2.2, we have \sqrt{q} ζ_{n} ∼ −\sqrt{q} ζ_{n} if and only if one of the following two conditions holds: (i) p ∤ n; (ii) 4p|n and χ(n/2 + 1) = 1. If 4p |n, then χ(n/2 + 1) = 1 since n/2 + 1 ≡ 1 (mod p). Thus, \sqrt{q} ζ_{n} ∼ −\sqrt{q} ζ_{n} ⟺ p ∤ n or 4p|n.

(c) By Lemma 2.2, we have \sqrt{q} ζ_{n} ∼ −\sqrt{q} ζ_{n} if and only if one of the following two conditions holds: (i) 4p ∤ n; (ii) 4p |n and χ(n/2 + 1) = 1. Suppose that 4p|n and write G_{4p} = G_{4} × G_{p}. Since r = n/2 + 1 ≡ 1 (mod p), the image of σ_{r} in G_{p} is trivial. In particular, it fixes \sqrt{−p} ∈ ℚ(ζ_{p}). As \sqrt{−p} ·\sqrt{−1} = −\sqrt{p}, one has χ(r) = 1 if and only if r ≡ 1 (mod 4). Write n = 4pk for some integer k. Then r = 2pk + 1 ≡ 1 (mod 4) if and only if k ≡ 0 (mod 2). Therefore, we get \sqrt{q} ζ_{n} ∼ −\sqrt{q} ζ_{n} ⟺ 4p ∤ n or 8p|n. □

As typical examples, we have (a) \sqrt{2} ζ_{8} ≁ −\sqrt{2} ζ_{8} and \sqrt{2} ζ_{16} ∼ −\sqrt{2} ζ_{16}, (b) \sqrt{5} ζ_{5} ≁ −\sqrt{5} ζ_{5} and \sqrt{5} ζ_{20} ∼ −\sqrt{5} ζ_{20}, and (c) \sqrt{3} ζ_{12} ≁ −\sqrt{3} ζ_{12} and \sqrt{3} ζ_{24} ∼ −\sqrt{3} ζ_{24}.

Corollary 2.4. Suppose that q is an odd power of p and n ≢ 2 (mod 4).

(1) If p ≡ 1 (mod 4), then

W_{q}^{ss} = \{\sqrt{q} ζ_{n}; n ≢ 2 (mod 4) \} ∪ \{−\sqrt{q} ζ_{n}; 2 ∤ n and p|n \}.

(2) If p ≡ 3 (mod 4) or p = 2, then

W_{q}^{ss} = \{\sqrt{q} ζ_{n}; n ≢ 2 (mod 4) \} ∪ \{−\sqrt{q} ζ_{n}; 4p | n and 8p ∤ n \}.

Proof. (1) By Proposition 2.3, \sqrt{q} ζ_{n} ≁ −\sqrt{q} ζ_{n} if and only if p|n and 4p ∤ n, i.e. p|n and 2 ∤ n. (2) We have \sqrt{q} ζ_{n} ≁ −\sqrt{q} ζ_{n} if and only if 4p|n and 8p ∤ n. □

Definition 2.5. Let 𝔡_{q} be the smallest positive integer such that ℚ(\sqrt{q}) ⊂ ℚ(ζ_{𝔡_{q}}). More specifically, 𝔡_{q} = 𝔡_{p} if q is an odd power of p, otherwise 𝔡_{q} = 1. We say a positive integer n is critical at q if 𝔡_{q}|n and 2𝔡_{q} ∤ n.

It is clear from the definition that for a fixed n ∈ ℕ, the condition that n is critical at q = p^{a} depends only on p and the parity of a.

Proposition 2.6. Let n ≢ 2 (mod 4) be a positive integer and q = p^{a} a power of a prime number p. Then \sqrt{q} ζ_{n} ∼ −\sqrt{q} ζ_{n} if and only if n is not critical at q.

Proof. The proposition reduces to either (2.2) or Proposition 2.3 according to whether a is even or odd respectively. □

Corollary 2.7. We have

W_{q}^{ss} = \{\sqrt{q} ζ_{n}; n ≢ 2 (mod 4) \} ∪\{−\sqrt{q} ζ_{n}; n ≢ 2 (mod 4) and n is critical at q \}.

3. Dimension of supersingular abelian varieties

3.1. Let q = p^{a} be a power of a prime number p, and π a supersingular Weil q-number as in the previous section. Replacing π by a suitable conjugate, we may assume that π = ±\sqrt{q} ζ_{n} for a positive integer n with n ≢ 2 (mod 4). Let X_{π} be a simple abelian variety over 𝔽_{q} in the isogeny class corresponding to π. Its endomorphism algebra E = E_{π} := End^{0}(X_{π}) is a central division algebra over K := ℚ(π), unique up to isomorphism depending only on π and not on the choice of X_{π}. The field K is either a totally real field or a CM field [18, Section 1]. The goal of this section is to determine the dimension d(π) of X_{π}. For each d ∈ ℕ, define

(3.1)  W_{q}^{ss}(d) := {π ∈ W_{q}^{ss} | d(π) = d}.

According to the Honda-Tate theory (ibid.), one has

d(π) = \frac{1}{2}[K : ℚ]\sqrt{[E : K]} = \frac{1}{2} deg_{ℚ}(E).

(For a semisimple algebra over a field F, its F -degree is the degree of any of its maximal commutative semi-simple F -subalgebras.) Moreover, the invariants of E at a place v of K is given by

inv_{v}(E) = \begin{cases}1/2 & if v is real; \\ [K_{v} : ℚ_{p}]v(π)/v(q) & if v|p; \\ 0 & otherwise. \end{cases}

Here K_{v} is the completion of K at the place v. Observe that d(π) = d(−π). As v(π)/v(q) = 1/2 for all v|p, every invariant inv_{v}(E) is a 2-torsion. It follows from the Albert-Brauer-Hasse-Noether theorem that E is either a quaternion Kalgebra or the field K itself (henceforth labeled as case (Q) or (F) respectively).

3.2. Totally real case. The case where K is a totally real field is well known.

(a) If a is even, then K = ℚ and E is the quaternion algebra over ℚ ramified exactly at {p, ∞}. One has π = ±p^{a/2} (two isogeny classes) and d(π) = 1.

(b) If a is odd, then K = ℚ(\sqrt{p}) and E is the quaternion algebra over K ramified exactly at the two real places {∞_{1}, ∞_{2}} of K. One has π = q^{1/2} (one isogeny class) and d(π) = 2.

3.3. CM case. Consider the case where K is a CM field, i.e., n > 2. Put L := ℚ(\sqrt{q} , ζ_{n}) ⊇ K. As K and L are abelian extensions of ℚ, the degree [K_{v} : ℚ_{p}] is even for one v|p if and only if it is so for all v|p. Thus, we have the following two possibilities:

(F) [K_{v} : ℚ_{p}] is even for all v|p.

(Q) [K_{v} : ℚ_{p}] is odd for all v|p. As K is CM, Condition (F) holds if and only if all invariants of E vanish. In this case E = K and d(π) = [K : ℚ]/2.

3.4. The case where a is even. Suppose that n > 2. One has K = ℚ(ζ_{n}) and [K : ℚ] = φ(n). Thus,

(3.2)  d(π) = \begin{cases}φ(n)/2 & if (F) holds; \\ φ(n) & if (Q) holds. \end{cases}

The ramification index of any ramified prime p in ℚ(ζ_{n}) is even, so if p | n, then (F) holds. When p ∤ n, Condition (F) holds if and only if the order of p ∈ (ℤ/nℤ)^{×} is even. In particular, if [K : ℚ] is a power of 2, then Condition (Q) holds if and only if K_{v} = ℚ_{p}, or equivalently p ≡ 1 (mod n). We have the following list, which enables us to list concretely all π with small values of d(π).

n ≢ 2 (mod 4) 3 4 5 7 8 9 11 12 15 16 20 21 24 rest
d(π), (Q) holds 2 2 4 6 4 6 10 4 8 8 8 12 8 > 8
d(π), (F) holds 1 1 2 3 2 3 5 2 4 4 4 6 4 > 4

Proposition 3.1. Let π = ±\sqrt{q} ζ_{n} be a supersingular Weil q-number with n ≥ 1 and n ≢ 2 (mod 4). Suppose that q = p^{a} is an even power of p.

(1) We have d(π) = 1 if and only if n = 1, or n = 3, 4 and p ≢ 1 (mod n).

(2) We have d(π) = 2 if and only if (a) n = 3, 4 and p ≡ 1 (mod n), or

(b) n = 5, 8, 12 and p ≢ 1 (mod n).

(3) We have d(π) = 3 if and only if n = 7 and p ≢ 1, 2, 4 (mod 7), or n = 9 and p ≢ 1, 4, 7 (mod 9).

(4) We have d(π) = 4 if and only if (a) n = 5, 8, 12 and p ≡ 1 (mod n), or (b) n = 15, 16, 20, 24 and p ≢ 1 (mod n).

3.5. The case where a is odd. Suppose that n > 1 and n ≢ 2 (mod 4). Put

(3.3)  m := \begin{cases}n/2 & if n is even, \\ n & if n is odd, \end{cases} and K := ℚ(\sqrt{p} ζ_{n}).

We have the following towers of number fields.

(3.4)  [figure: L = ℚ(\sqrt{p} , ζ_{n}); ℚ(\sqrt{p} , ζ_{m}); K = ℚ(\sqrt{p} ζ_{n}); ℚ(ζ_{n}); E = ℚ(ζ_{m})]

Note that the prime p is ramified in K with even ramification index, and hence Condition (F) always holds. Therefore,

(3.5)  E = K and d(π) = \frac{1}{2} [K : ℚ].

Lemma 3.2. Let K and E be as in (3.4). We have K = E if and only if n is critical at q.

Proof. Clearly [K : E] = 1 or 2. If π ∼ −π, then π ↦ −π induces a nontrivial automorphism of K with fixed field E. Thus, π ∼ −π if and only if [K : E] = 2. By Proposition 2.6, [K : E] = 1 if and only if n is critical at q. Note that the lemma also holds when a is even with K = ℚ(\sqrt{q} ζ_{n}) = ℚ(ζ_{n}). □

Lemma 3.3. Suppose that a is odd and n > 1 with 4 ∤ n. Then

(3.6)  d(π) = \frac{1}{2}[K : ℚ] = \begin{cases}φ(n)/2 & if p |n and p ≡ 1 (mod 4); \\ φ(n) & otherwise. \end{cases}

Proof. Since n is odd one has E = ℚ(ζ_{n}) and [E : ℚ] = φ(n). We have 𝔡_{q} = p or 4p according as p ≡ 1 (mod 4) or not. It is easy to see that n is critical at q if and only if p ≡ 1 (mod 4) and p|n. The assertion then follows from Lemma 3.2 and (3.5). □

Lemma 3.4. Suppose that a is odd and n = 4k with k ∈ ℕ. Then

d(π) = \frac{1}{2}[K : ℚ] = \begin{cases}φ(n)/4 & if p ≢ 1 (mod 4), 4p | n and 8p ∤ n; \\ φ(n)/2 & otherwise. \end{cases}

Proof. Since 4|n we have [E : ℚ] = φ(n)/2. By Lemma 3.2 we have [K : ℚ] = δ_{n}φ(n)/2, where δ_{n} = 1 or 2 depending on whether n is critical at q or not. The lemma follows once we note that n = 4k is never critical when p ≡ 1 (mod 4). □

The following are tables of d(π) for π = \sqrt{q} ζ_{n} with 4 ∤ n and 4|n, respectively. The symbol (∗) denotes the primes satisfying the conditions p |n and p ≡ 1 (4), and (∗∗) denotes the primes satisfying the three conditions p ≢ 1 (mod 4), 4p | n and 8p ∤ n. For the sake of completeness, the case n = 1 is included and also marked with a ♮ to make a distinction.

n odd 1^{♮} 3  5  7 9 11  13  15  rest
φ(n) 1 2  4  6 6 10  12  8  > 8
(∗) ∅ ∅ p = 5 ∅ ∅ ∅  p = 13  p = 5
d(π) 2 2 2 (p = 5) 6 6 10 6 (p = 13) 4 (p = 5) > 4
4 (p ≠ 5)  12 (p ≠ 13) 8 (p ≠ 5)
n = 4k 4  8  12  16  20  24  28
φ(n) 2  4  4  8  8  8  12
(∗∗)  ∅  2  3  ∅  ∅  2  7
d(π) 1 1 (p = 2) 1 (p = 3)  4  4 2 (p = 2) 3 (p = 7)
2 (p ≠ 2) 2 (p ≠ 3)  4 (p ≠ 2) 6 (p ≠ 7)
n = 4k 32  36  40  44  48  56  60
φ(n) 16  12  16  20  16  24  16
(∗∗)  ∅  p = 3  p = 2  p = 11  ∅  p = 2  p = 3
d(π) 8 3 (p = 3) 4 (p = 2) 5 (p = 11) 8 6 (p = 2) 4 (p = 3)
6 (p ≠ 3) 8 (p ≠ 2) 10 (p ≠ 11)  12 (p ≠ 2) 8 (p ≠ 3)

It is easy to see that when 4|n and either n = 52 or n > 60, the value φ(n) > 16 and hence d(\sqrt{q} ζ_{n}) > 4.

Proposition 3.5. Suppose that q = p^{a} is an odd power of p.

(1) W_{q}^{ss}(1) consists of

\sqrt{q} ζ_{4}, ±\sqrt{q} ζ_{8} (p = 2), ±\sqrt{q} ζ_{12} (p = 3).

(2) W_{q}^{ss}(2) consists of

\sqrt{q} , \sqrt{q} ζ_{3}, ±\sqrt{q} ζ_{5} (p = 5), \sqrt{q} ζ_{8} (p ≠ 2), \sqrt{q} ζ_{12} (p ≠ 3), ±\sqrt{q} ζ_{24} (p = 2).

(3) W_{q}^{ss}(3) consists of ±\sqrt{q} ζ_{28} if p = 7, or ±\sqrt{q} ζ_{36} if p = 3.

(4) W_{q}^{ss}(4) consists of

\sqrt{q} ζ_{5} (p ≠ 5), ±\sqrt{q} ζ_{15} (p = 5), \sqrt{q} ζ_{16},

\sqrt{q} ζ_{20}, \sqrt{q} ζ_{24} (p ≠ 2), ±\sqrt{q} ζ_{40} (p = 2), ±\sqrt{q} ζ_{60} (p = 3).

4. Supersingular elliptic curves over finite fields

4.1. Isogeny classes over finite fields. Let Isog_{q} denote the set of isogeny classes of abelian varieties over 𝔽_{q}, where q = p^{a} is a power of the prime number p. Let ℤW_{q} be the free abelian group (written multiplicatively) generated by the set W_{q} of conjugacy classes of Weil q-numbers. A nontrivial element π ∈ ℤW_{q} can be put in the form π_{1}^{m_{1}} × ⋯ × π_{r}^{m_{r}} for some r ∈ ℕ, where each π_{i} ∈ W_{q}, π_{i} ≁ π_{j} if i ≠ j, and m_{i} ≠ 0 for all 1 ≤ i ≤ r. Such an element is called a multiple Weil q-number if m_{i} > 0 for all i, and the set of all these elements is denoted by MW_{q}. Put X_{π} := ∏_{i} X_{π_{i}}^{m_{i}}, where X_{π_{i}} is the simple abelian variety (up to isogeny) over 𝔽_{q} corresponding to π_{i}. The Honda-Tate theorem naturally extends to a bijection MW_{q} ≃ Isog_{q} which sends each π ∈ MW_{q} to the isogeny class [X_{π}] ∈ Isog_{q} of X_{π}. For each π ∈ MW_{q}, we define its dimension as

d(π) := dim X_{π} = ∑_{i=1}^{r} m_{i}d(π_{i}).

Let Isog(π) = Isog(X_{π}) denote the set of 𝔽_{q}-isomorphism classes of abelian varieties isogenous to X_{π} over 𝔽_{q}, and denote H(π) := |Isog(π)|. Let MW_{q}^{ss} ⊂ MW_{q} be the subset of supersingular multiple Weil q-numbers, i.e. those π ∈ MW_{q} whose corresponding abelian varieties X_{π} are supersingular. For any integer d ≥ 1, let MW_{q}(d) (resp. MW_{q}^{ss}(d)) denote the subset consisting of all elements π in MW_{q} (resp. in MW_{q}^{ss}) of dimension d. Let S_{d}(𝔽_{q}) (resp. Sp_{d}(𝔽_{q})) be the set of isomorphism classes of d-dimensional supersingular (resp. superspecial) abelian varieties over 𝔽_{q}. When π ∈ MW_{q}^{ss}, we let Sp(π) ⊂ Isog(π) be the subset consisting of superspecial isomorphism classes and denote H_{sp}(π) := |Sp(π)|. Thus,

(4.1)  |S_{d}(𝔽_{q})| = ∑_{π∈MW_{q}^{ss}(d)} H(π), |Sp_{d}(𝔽_{q})| = ∑_{π∈MW_{q}^{ss}(d)} H_{sp}(π).

4.2. Supersingular elliptic curves. We compute the number |S_{1}(𝔽_{q})| of isomorphism classes of supersingular elliptic curves over 𝔽_{q}, where q = p^{a} as before. The method is based almost entirely on the results of Waterhouse [21], except certain details need to be cleared up (compare with [21, Theorem 4.5]).

Proposition 4.1. Let π be the Frobenius endomorphism of an elliptic curve E_{0} over 𝔽_{q}, and K := ℚ(π). Assume that π ∉ ℚ so that K is an imaginary quadratic field. Equivalently, the central K-algebra End^{0}(E_{0}) of the elliptic curve E_{0} is assumed to be commutative and thus necessarily an imaginary quadratic field.

(1) Any endomorphism ring R = End(E) of an elliptic curve E in the isogeny class [E_{0}] of E_{0} contains π and is maximal at p, that is, R ⊗ ℤ_{p} is the maximal order in K ⊗ ℚ_{p}. Conversely, any order R of K satisfying these two properties occurs as an endomorphism ring of an elliptic curve in this isogeny class.

(2) Suppose that R ⊂ K is a quadratic order as in (1). Then the Picard group Pic(R) of R acts freely on the set [E_{0}]_{R} ⊂ [E_{0}] of isomorphism classes of elliptic curves in [E_{0}] with endomorphism ring R. Moreover, the number N of orbits is 2 if p is inert in K and a is even, and N = 1 otherwise.

Proof. Statement (1) is [21, Theorem 4.2]. We give a proof of the second part of Statement (2) since it differs from [21, Theorem 4.5] in some cases. We assert that the statement of [21, Theorem 5.1] for principal abelian varieties is directly applicable to this situation. Namely, the number of orbits here is also given by N = ∏_{v|p} N_{v}, where v runs through the set of all places of K over p, and each N_{v} is the number described as follows. Let e_{v} and f_{v} be the ramification index and residue degree of v, respectively, and set g_{v} = gcd(f_{v}, a) and m_{v} := g_{v} ord_{v}(π)/a. Note that m_{v} is an integer since End^{0}(E_{0}) is commutative and thus f_{v} ord_{v}(π)/a ∈ ℕ. Then N_{v} is the number of all g_{v}-tuples (n_{1}, …, n_{g_{v}}) of integers satisfying 0 ≤ n_{j} ≤ e_{v} and ∑_{j=1}^{g_{v}} n_{j} = m_{v}. In the present situation End^{0}(E_{0}) = K is commutative and R is maximal at p. As in the proof of [21, Theorem 5.1], to find the number of orbits for the action of Pic(R) on [E_{0}]_{R}, one needs to classify the Tate-modules T_{ℓ}E at all primes ℓ ≠ p and the Dieudonné modules at the prime p of E ∈ [E_{0}]_{R}. The number of orbits is then the product of the number of isomorphism classes of the above modules at each prime. The Tate-module T_{ℓ}E of each E ∈ [E_{0}]_{R} at a prime ℓ ≠ p is naturally an R_{ℓ}- module with R_{ℓ} = R ⊗_{ℤ} ℤ_{ℓ}. Since R[1/p] is a quadratic order, any fractional R[1/p]-ideal I whose order ring equals R[1/p] must be locally free over R[1/p]. Particularly, there is only one isomorphism class of the prime-to-p Tate modules of E for all E ∈ [E_{0}]_{R}. Thus, N is equal to the number of isomorphism classes of Dieudonné modules occurring in the isogeny class [E_{0}], which is equal to ∏_{v} N_{v} as given in the proof of [21, Theorem 5.1]. Now it is easy to compute the number N of orbits. Notice N_{v} ≠ 1 only when g_{v} > 1. For our case with [K : ℚ] = 2 this occurs only when p is inert in K and a is even. In this case there is only one place v over p, g_{v} = 2 and e_{v} = 1. Then N = N_{v} is the number of pairs (n_{1}, n_{2}) with 0 ≤ n_{1}, n_{2} ≤ 1 and n_{1} + n_{2} = 1, which is 2. □

Remark 4.2. In [21, Theorem 5.1] the assumption that the endomorphism ring R = End(A) is the maximal order can be replaced by the weaker assumption that R is both Gorenstein and maximal at p. Indeed, any proper R-lattice of rank one over a Gorenstein order R is locally free [5, Theorem 37.16 p. 789], so the same proof of [21, Theorem 5.1] applies.

Remark 4.3. Suppose that a is even and p is inert in the imaginary quadratic field K = ℚ(π) so that N = 2. By the classification of Waterhouse ([21, Lemma, p.537], see also Proposition 3.1), this occurs only for supersingular Weil q-numbers π where

(4.2) π ∼ ±p^{a/2}ζ_{3}, p ≡ 2 (mod 3) or π ∼ p^{a/2}ζ_{4}, p ≡ 3 (mod 4).

Then by part (1) of Proposition 4.1, End(E) = O_{K} for any elliptic curve E in the isogeny class corresponding to π. Since h(O_{K}) = 1, part (2) of Proposition 4.1 implies that a complete set of representatives of Sp(π) consists a pair of elliptic curves of the form {E, E^{(p)}}, where E^{(p)} := E ⊗_{𝔽_{q},σ_{p}} 𝔽_{q}, and σ_{p} ∈ Gal(𝔽_{q}/𝔽_{p}) is the Frobenius automorphism of 𝔽_{q}/𝔽_{p}. These two elliptic curves are distinguished by the actions of O_{K} on the respective 1dimensional Lie-algebras Lie(E) and Lie(E^{(p)}) over 𝔽_{q}, which are given by distinct embeddings O_{K}/(p) ≃ 𝔽_{p^{2}} ↪ 𝔽_{q}. This establishes a natural bijection Sp(π) ≃ Hom(O_{K}/(p), 𝔽_{q}) for every π in (4.2).

We return to the calculation of | Sp_{1}(𝔽_{q})| by the counting method. The isogeny classes of supersingular elliptic curves over 𝔽_{q} are completely listed by the following Weil numbers

W_{q}^{ss}(1) = \{\sqrt{q} ζ_{4}, ±\sqrt{q} ζ_{8} (p = 2), ±\sqrt{q} ζ_{12} (p = 3) \}, for a odd;
W_{q}^{ss}(1) = \{±\sqrt{q} , ±\sqrt{q} ζ_{3} (p ≢ 1 (3)), \sqrt{q} ζ_{4} (p ≢ 1 (4)) \}, for a even.

(4.3)

For each Weil q-number π ∈ W_{q}^{ss}(1), let R_{0} be the smallest quadratic order in K = ℚ(π) which contains π and is maximal at p. It is easy to see that R_{0} is the maximal order except when π = \sqrt{q} ζ_{4}, p ≡ 3 (mod 4) and a is odd. In the latter case R_{0} = ℤ[\sqrt{−p}] and we have by Proposition 4.1 that

(4.4)  H(\sqrt{q} ζ_{4}) = \begin{cases}h(O_{K}) & for p = 2 or p ≡ 1 (mod 4); \\ h(R_{0}) + h(O_{K}) & for p ≡ 3 (mod 4). \end{cases}

For the other cases, the order R_{0} is maximal and we have

(4.5)  H(π) = N · h(O_{K})

where N = 2 if p is inert in K and a is even, and N = 1 otherwise. Recall that for a square free m ∈ ℤ, the class number of ℚ(\sqrt{m}) is denoted by h(\sqrt{m}). Suppose first that a is odd. For p = 2, we have

(4.6)  | Sp_{1}(𝔽_{q})| = H(\sqrt{q} ζ_{4}) + 2H(\sqrt{q} ζ_{8}) = h(\sqrt{−2} ) + 2h(\sqrt{−1} ) = 3.

For p = 3, we have

\begin{aligned}| Sp_{1}(𝔽_{q})| & = H(\sqrt{q} ζ_{4}) + 2H(\sqrt{q} ζ_{12}) \\ & = h(ℤ[\sqrt{−3} ]) + h(\sqrt{−3} ) + 2h(\sqrt{−3} ) = 4. \end{aligned} ; (4.7)

For p > 3, we have by [26, Theorem 1.1] that

| Sp_{1}(𝔽_{q})| = H(\sqrt{q} ζ_{4})

= \begin{cases}h(\sqrt{−p}) & for p ≡ 1 (mod 4); \\ 2h(\sqrt{−p}) & for p ≡ 7 (mod 8) (2 splits in ℚ(\sqrt{−p})) ; \\ 4h(\sqrt{−p}) & for p ≡ 3 (mod 8) (2 is inert in ℚ(\sqrt{−p})). \end{cases} (4.8)

Since (\frac{2}{p}) = 1 for p ≡ 1, 7 (mod 8) and (\frac{2}{p}) = −1 for p ≡ 3, 5 (mod 8), we can rewrite (4.8) as

(4.9)  | Sp_{1}(𝔽_{q})| = \begin{cases}h(\sqrt{−p}) & for p ≡ 1 (mod 4); \\ (3 − (\frac{2}{p})) h(\sqrt{−p}) & for p ≡ 3 (mod 4). \end{cases}

Suppose now that a is even. By (4.3), we have

(4.10)  | Sp_{1}(𝔽_{q})| = 2H(\sqrt{q}) + 2δ_{3}(p)H(\sqrt{q} ζ_{3}) + δ_{4}(p)H(\sqrt{q} ζ_{4}),

where δ_{m}(p) = 1, 0 according as p ≢ 1 (mod m) or not for m = 3, 4. It is well known that H(\sqrt{q}) is equal to the class number h(B_{p,∞}) of the quaternion ℚ-algebra B_{p,∞} ramified only at p and ∞. Thus,

(4.11)  H(\sqrt{q}) = \frac{p − 1}{12} + \frac{1}{3} (1 − (\frac{−3}{p})) + \frac{1}{4} (1 − (\frac{−4}{p})) .

By Proposition 4.1, we have

(4.12)  δ_{3}(p)H(\sqrt{q} ζ_{3}) = \begin{cases}1 & for p = 3; \\ 2 & for p ≡ 2 (mod 3); \\ 0 & for p ≡ 1 (mod 3); \end{cases}

and get δ_{3}(p)H(\sqrt{q} ζ_{3}) = 1 − (\frac{−3}{p}). Similarly, we have δ_{4}(p)H(\sqrt{q} ζ_{4}) = 1 −

(\frac{−4}{p}). Using (4.10) and (4.11), we get

\begin{aligned}| Sp_{1}(𝔽_{q})| & = \frac{p − 1}{6} + \frac{2}{3} (1 − (\frac{−3}{p})) + \frac{1}{2} (1 − (\frac{−4}{p})) \\ & + 2 (1 − (\frac{−3}{p})) + (1 − (\frac{−4}{p})) \\ & = \frac{p − 1}{6} + \frac{8}{3} (1 − (\frac{−3}{p})) + \frac{3}{2} (1 − (\frac{−4}{p})) . \end{aligned} ; (4.13)

From (4.6), (4.7), (4.9) and (4.13), we obtain an explicit formula for the number | Sp_{1}(𝔽_{q})| of supersingular elliptic curves over 𝔽_{q}.

Proposition 4.4. Suppose q = p^{a} is a power of the prime number p. (1) If a is odd, then

(4.14) | Sp_{1}(𝔽_{q})| = \begin{cases}3, 4 & for p = 2, 3, respectively; \\ h(\sqrt{−p}) & for p ≡ 1 (mod 4); \\ (3 − (\frac{2}{p})) h(\sqrt{−p}) & for p ≡ 3 (mod 4) and p > 3. \end{cases}

(2) If a is even, then

\begin{aligned}& | Sp_{1}(𝔽_{q})| = \frac{p − 1}{6} + \frac{8}{3} (1 − (\frac{−3}{p})) + \frac{3}{2} (1 − (\frac{−4}{p})) . \\ (4.15) \end{aligned}

Remark 4.5. From the formulas above we observe a phenomenon that the number | Sp_{1}(𝔽_{q})| depends only on the parity of the exponent a of q = p^{a}. We have already seen in Section 2 that the classification of supersingular isogeny classes depends only on the parity of a. More explicitly, if the exponents a and a′ of q and q′ respectively have the same parity, then a bijective correspondence between supersingular isogeny classes over 𝔽_{q} and those over 𝔽_{q′} can be given by matching π ∈ W_{q}^{ss}(1) with π′ = (−p)^{(a′−a)/2}π (see Remark 6.8). The parity phenomenon of | Sp_{1}(𝔽_{q})| arises because there is a bijection Sp(π) ≃ Sp(π′) for all pairs (π, π′) as above. Indeed, if π and π′ are of the form in (4.2), then a canonical bijection Sp(π) ≃ Sp(π′) is given by identifying both with Hom(O_{K}/(p), 𝔽_{q}) as in Remark 4.3. For the remaining cases, first suppose that K = ℚ(π) = ℚ(π′) is imaginary quadratic. Then the endomorphism rings occurring for both isogeny classes are the same by Proposition 4.1. We partition Sp(π) into ∐_{R} Sp(π, R), where R runs over all possible endomorphism rings, and Sp(π, R) ⊆ Sp(π) consists of those members with endomorphism ring R. Every Sp(π, R) is a principal homogeneous space of Pic(R). Thus a Pic(R)-equivariant bijection between Sp(π, R) and Sp(π′, R) is established whenever a base point is chosen respectively in each of them. Lastly, suppose that ℚ(π) = ℚ(π′) = ℚ. Then π^{a′} = (π′)^{a} = p^{aa′/2}. So we have canonical bijections Sp(π) ≃ Sp(π^{a′}) ≃ Sp(π′) by extending both base fields to 𝔽_{p^{aa′}} ([21, Remark, p. 542]). Equivalently, the bijection Sp(π) ≃ Sp(π′) can be obtained by matching the j-invariants.

5. Superspecial abelian surfaces over 𝔽_{p}

In this section we assume that the ground field is the prime field 𝔽_{p}; abelian varieties and their morphisms are all defined over 𝔽_{p} unless otherwise stated.

5.1. Supersingular abelian varieties over 𝔽_{p}. We describe a result which allows us to count supersingular and superspecial abelian varieties over 𝔽_{p}, based on a result of Waterhouse [21, Theorem 6.1 (3)] (see also [26, Theorem 3.1] for an extension to non-simple isogenies). Let X_{0} be a fixed supersingular abelian variety over 𝔽_{p} and let π = π_{1}^{m_{1}} × ⋯ × π_{r}^{m_{r}} be a multiple Weil p-number corresponding to the isogeny class [X_{0}]. One has X_{0} ∼ ∏_{i=1}^{r} X_{i}^{m_{i}}, where each X_{i} with 1 ≤ i ≤ r is a simple abelian variety with Frobenius endomorphism π_{i}. The endomorphism algebra E = End^{0}(X_{0}) of X_{0} is equal to ∏_{i=1}^{r} Mat_{m_{i}}(End^{0}(X_{i})). Let π_{0} ∈ End(X_{0}) be the Frobenius endomorphism. The ℚ-subalgebra K = ℚ(π_{0}) ⊂ E generated by π_{0} is semi-simple and coincides with the center of E. One has K = ∏_{i} K_{i} and π_{0} = (π_{1}, …, π_{r}), where K_{i} = ℚ(π_{i}). Let R := ℤ[π_{0}, pπ_{0}^{−1}] ⊂ K and R_{sp} := R[π_{0}^{2}/p] ⊂ K. Clearly π_{0}^{2}/p is an integral element of finite multiplicative order, and p/π_{0} = π_{0} · (π_{0}^{2}/p)^{−1}, so R_{sp} = ℤ[π_{0}, π_{0}^{2}/p] ⊆ O_{K}, where O_{K} = ∏_{i} O_{K_{i}} is the maximal order K. Observe that the Tate module T_{ℓ}(X_{0}) (for any prime ℓ ≠ p), as a ℤ_{ℓ}[Gal(\overline{𝔽}_{p}/𝔽_{p})]-module, is nothing but an R_{ℓ}-module, and the (covariant) Dieudonné module M(X_{0}) is simply an R_{p}-module, where R_{ℓ} = R ⊗ ℤ_{ℓ} and R_{p} = R ⊗ ℤ_{p}.

Proposition 5.1. Let π = π_{1}^{m_{1}} × … π_{r}^{m_{r}}, and K, R and R_{sp} be as above. Assume that K has no real place, that is, none of π_{i} is conjugate to \sqrt{p}, and set V := ∏_{i=1}^{r} K_{i}^{m_{i}}.

(1) There is a natural bijection between the set Isog(π) and the set of isomorphism classes of R-lattices in V.

(2) Under the above map the subset Sp(π) is in bijection with the set of isomorphism classes of R_{sp}-lattices in V.

Proof. Set Λ := ∏_{i=1}^{r} O_{K_{i}}^{m_{i}} ⊂ V, and view V and Λ as a K-module and an R-lattice, respectively. We choose an identification V ⊗_{ℚ} ℚ_{ℓ} = T_{ℓ}(X_{0}) ⊗ ℚ_{ℓ} for primes ℓ ≠ p and V ⊗_{ℚ} ℚ_{p} = M(X_{0}) ⊗ ℚ_{p} such that Λ_{ℓ} = T_{ℓ}(X_{0}) for almost all primes ℓ. Under this identification, any R-lattice Λ′ in V gives rise to a unique quasi-isogeny φ : X → X_{0} such that φ_{∗}(T_{ℓ}(X)) = Λ′ ⊗ ℤ_{ℓ} for ℓ ≠ p and φ_{∗}(M(X)) = Λ′ ⊗ ℤ_{p}. Two lattices Λ_{1} and Λ_{2} are isomorphic as R-modules if and only if there is an element g ∈ GL_{K}(V) such that Λ_{2} = gΛ_{1}. Two quasi-isogenies are isomorphic if and only if they differ by an element in E^{×}. Our assumption ensures that GL_{K}(V) ≃ E^{×}. Then the above correspondence induces the desired bijection (also see [26, Theorem 3.1] for a detailed proof). Note that the abelian variety X in [X_{0}] as above is superspecial if and only if π_{0}^{2}M(X) = pM(X), or equivalently, M(X) is a (R_{sp})_{p}-lattice in M(X_{0}) ⊗ ℚ_{p}. That is, X is superspecial if and only if the corresponding R-module is R_{sp}- stable. The statement (2) then follows from (1). □

Remark 5.2. Let π = π_{1}^{e_{1}} be a multiple supersingular Weil p-number with π_{1} = ±\sqrt{p} ζ_{n} and n critical at p. Then by Lemma 3.2, K = ℚ(π_{1}) = ℚ(ζ_{m}) and O_{K} = ℤ[ζ_{m}], where m is defined in (3.3). Since R_{sp} = R[π_{1}^{2}/p] ∋ ζ_{m}, it follows that R_{sp} coincides with the maximal order O_{K} in this case.

5.2. Proof of the main theorem. By Section 3, we list the sets W_{p}^{ss}(1) and W_{p}^{ss}(2) of supersingular Weil p-numbers of dimension 1 or 2 as follows:

W_{2}^{ss}(1) = \{\sqrt{2} ζ_{4}, ±\sqrt{2} ζ_{8}\},

W_{3}^{ss}(1) = \{\sqrt{3} ζ_{4}, ±\sqrt{3} ζ_{12}\}, (5.1)

W_{p}^{ss}(1) = \{\sqrt{p} ζ_{4}\}, p ≥ 5;

and

W_{2}^{ss}(2) = \{\sqrt{2} ,\sqrt{2} ζ_{3},\sqrt{2} ζ_{12}, ±\sqrt{2} ζ_{24}\},

W_{3}^{ss}(2) = \{\sqrt{3} ,\sqrt{3} ζ_{3},\sqrt{3} ζ_{8}\}, (5.2)

W_{5}^{ss}(2) = \{\sqrt{5} ,\sqrt{5} ζ_{3},\sqrt{5} ζ_{8},\sqrt{5} ζ_{12}, ±\sqrt{5} ζ_{5}\},
W_{p}^{ss}(2) = \{\sqrt{p} ,\sqrt{p} ζ_{3},\sqrt{p} ζ_{8},\sqrt{p} ζ_{12}\}, p ≥ 7.

Consider the case π ∈ W_{p}^{ss}(2) or π = π_{1} × π_{2} with π_{1}, π_{2} ∈ W_{p}^{ss}(1). By (4.1) we have

(5.3)  |Sp_{2}(𝔽_{p})| = ∑_{π∈W_{p}^{ss}(2)} H_{sp}(π) + ∑_{π_{1},π_{2}∈W_{p}^{ss}(1)} H_{sp}(π_{1} × π_{2}).

The number H_{sp}(\sqrt{p}) = H(\sqrt{p}) has been calculated in [22], so this case will be excluded from our discussion. We refer to [5, Section 37] for the definition of a Bass order. Note that when π = π_{1} × π_{1}, R_{sp} is an order in the quadratic field ℚ(π_{1}), and such orders are well known to be Bass. It will be shown in Section 7.2 that R_{sp} is a Bass order for all π considered (i.e. π ∈ MW_{p}^{ss}(2)). Thus, when the K-module V is free of rank one (i.e. in the case where π ≠ π_{1} × π_{1}), Proposition 5.1 gives

(5.4)  H_{sp}(π) = ∑_{R_{sp}⊂B⊂O_{K}} h(B).

In the case when V is free of higher rank (in fact, rank 2 when π = π_{1} × π_{1}), one can use the results of Borevič and Faddeev on lattices over orders of cyclic index to compute H_{sp}(π) (cf. [5, Section 37, p. 789]). In the following, the notation B_{π,j} (or B_{j} for short) with j ∈ ℕ, will stand for an order B of K with R_{sp} ⊂ B ⊂ O_{K} and [O_{K} : B] = j. The dependence of K, R_{sp} and B_{j} on the choice of the Weil p-number π should be understood though it is omitted from the notation. For any two square-free integers d > 1 and j ≥ 1, we write K_{d,j} for the CM field ℚ(\sqrt{d} ,\sqrt{−j}). For a finite collection of algebraic numbers α_{1}, …, α_{n}, the notation h(α_{1}, …, α_{n}) denotes the class number of the number field ℚ(α_{1}, …, α_{n}). Particularly, h(\sqrt{d} ,\sqrt{−j}) and h(K_{d,j}) have the same meaning.

Case π = π_{1} × π_{1}. For π_{1} = ±\sqrt{2} ζ_{8}, one has K = ℚ(\sqrt{−1} ), R_{sp} = R = O_{K}, and H_{sp}(π) = H(π) = 1. For π_{1} = ±\sqrt{3} ζ_{12}, one has K = ℚ(\sqrt{−3} ), R_{sp} = R = O_{K}, and H_{sp}(π) = H(π) = 1. For π_{1} = \sqrt{−p}, we have K = ℚ(\sqrt{−p}), R_{sp} = R and [O_{K} : R_{sp}] = 2 or 1 depending on p ≡ 3 (mod 4) or not. In this case we have H_{sp}(π) = 1, 3 for p = 2, 3, respectively, and

(5.5)  H_{sp}(π) = \begin{cases}h(\sqrt{−p}) & for p ≡ 1 (mod 4); \\ (4 − (\frac{2}{p})) h(\sqrt{−p}) & for p ≡ 3 (mod 4) and p > 3; \end{cases}

see [26, Theorem 1.1]. Therefore, the contribution of the self-product cases is given by (5.6)

∑_{π_{1}∈W_{p}^{ss}(1)} H_{sp}(π_{1}×π_{1})=\begin{cases}3, 5 & for p = 2, 3, respectively; \\ h(\sqrt{−p}) & for p ≡ 1 (mod 4); \\ (4 − (\frac{2}{p})) h(\sqrt{−p}) & for p ≡ 3 (mod 4) and p > 3. \end{cases}

Case π = π_{1} × π_{2}, π_{1} ≠ π_{2}. This occurs only when p = 2 or 3. The following are class numbers of B with R_{sp} ⊂ B ⊂ O_{K} obtained in Section 7.3.

\begin{matrix}π = π_{1} × π_{2} & K & [O_{K} : R_{sp}] & R_{sp} ⊂ B ⊂ O_{K} & h(B) \\ \sqrt{2} ζ_{4} × ±\sqrt{2} ζ_{8} & ℚ(\sqrt{−2} ) × ℚ(\sqrt{−1} ) & 2 & R_{sp}, O_{K} & 1, 1 \\ \sqrt{2} ζ_{8} × −\sqrt{2} ζ_{8} & ℚ(\sqrt{−1} ) × ℚ(\sqrt{−1} ) & 8 & R_{sp}, B_{4}, B_{2}, O_{K} & 1, 1, 1, 1 \\ \sqrt{3} ζ_{4} × ±\sqrt{3} ζ_{12} & ℚ(\sqrt{−3} ) × ℚ(\sqrt{−3} ) & 6 & R_{sp}, B_{3}, B_{2}, O_{K} & 1, 1, 1, 1 \\ \sqrt{3} ζ_{12} × −\sqrt{3} ζ_{12} & ℚ(\sqrt{−3} ) × ℚ(\sqrt{−3} ) & 12 & R_{sp}, B_{4}, B_{3}, O_{K} & 1, 1, 1, 1 \end{matrix}

The orders B_{j} are listed here for the convenience of the reader:

B_{2} = ℤ[(1 + ζ_{4}, 0), (ζ_{4}, ζ_{4})]  for π = \sqrt{2} ζ_{8} × −\sqrt{2} ζ_{8};
B_{2} = ℤ[\sqrt{−3} ] × ℤ[ζ_{6}]  for π = \sqrt{3} ζ_{4} × ±\sqrt{3} ζ_{12};
B_{3} = ℤ[(\sqrt{−3} , 0), (ζ_{6}, ζ_{6})]  for π = \sqrt{3} ζ_{4} × ±\sqrt{3} ζ_{12} or \sqrt{3} ζ_{12} × −\sqrt{3} ζ_{12};
B_{4} = ℤ[(2, 0), (ζ_{2p}, ζ_{2p})]  for π = \sqrt{p} ζ_{4p} × −\sqrt{p} ζ_{4p} and p = 2, 3.

The contribution of other non-simple cases is

(5.7)  ∑_{π_{1}≠π_{2}} H_{sp}(π_{1} × π_{2}) = \begin{cases}2 × 2 + 4 = 8 & for p = 2; \\ 2 × 4 + 4 = 12 & for p = 3. \end{cases}

Case π ∈ W_{p}^{ss}(2). We have π ∈ \{±\sqrt{2} ζ_{24}, ±\sqrt{5} ζ_{5},\sqrt{p} ζ_{8} (p ≠ 2),\sqrt{p} ζ_{3},\sqrt{p} ζ_{12} (p ≠ 3)\}. For π = ±\sqrt{p} ζ_{n} with (p, n) = (5, 5) or (2, 24), we have R_{sp} = O_{K} by Remark 5.2 since n is critical at p. For π = \sqrt{p} ζ_{8} with p ≠ 2, we have K = ℚ(\sqrt{p} ζ_{8}) = ℚ(\sqrt{−1} ,\sqrt{2p}) and R_{sp} = ℤ[(\sqrt{2p} +\sqrt{−2p})/2,\sqrt{−1} ], which is the maximal order in K by Exercise 42(b) of [13, Chapter 2]. Therefore,

(5.8) H_{sp}(±\sqrt{2} ζ_{24}) = H_{sp}(±\sqrt{5} ζ_{5}) = 1, h(\sqrt{p} ζ_{8}) = h(\sqrt{2p} ,\sqrt{−1} ), p ≠ 2.

For π = \sqrt{p} ζ_{3}, we have K = ℚ(\sqrt{p} ,\sqrt{−3} ) and R_{sp} = ℤ[\sqrt{p} , ζ_{3}]. The suborders B ⊆ O_{K} containing ℤ[\sqrt{p}] with the property [B^{×} : ℤ[\sqrt{p}]^{×}] > 1 are classified in [23]. We list the suporders of R_{sp} in O_{K} and their class numbers in the following table.

\begin{matrix}π = \sqrt{p} ζ_{3} & [O_{K} : R_{sp}] & R_{sp} ⊂ B ⊂ O_{K} & h(B) \\ p = 2 & 1 & O_{K} & 1 \\ p = 3 & 3 & R_{sp}, O_{K} & 1, 1 \\ p ≡ 3 (mod 4), p ≠ 3 & 1 & O_{K} & h(K) \\ p ≡ 1 (mod 4) & 4 & R_{sp}, O_{K} & ϖ_{p} h(K), h(K) \end{matrix}

Thus,

(5.9) H_{sp}(\sqrt{p} ζ_{3}) = \begin{cases}1, 2 & for p = 2, 3, respectively; \\ (ϖ_{p} + 1)h(\sqrt{p} ,\sqrt{−3} ) & for p ≡ 1 (mod 4); \\ h(\sqrt{p} ,\sqrt{−3} ) & for p ≡ 3 (mod 4) and p > 3. \end{cases}

For π = \sqrt{p} ζ_{12} (p ≠ 3), we have K = ℚ(\sqrt{−p} ,\sqrt{−3} ) and R_{sp} = ℤ[\sqrt{p} ζ_{12}, ζ_{6}] = ℤ[\sqrt{−p} , ζ_{6}]. We have the following results from Section 7.4.

\begin{matrix}π = \sqrt{p} ζ_{12} (p ≠ 3) & [O_{K} : R_{sp}] & R_{sp} ⊂ B ⊂ O_{K} & h(B) \\ p = 2 & 1 & O_{K} & 1 \\ p ≡ 1 (mod 4) & 1 & O_{K} & h(K) \\ p ≡ 3 (mod 4) & 4 & R_{sp}, O_{K} & ϖ_{3p} h(K), h(K) \end{matrix}

Thus, (5.10)

H_{sp}(\sqrt{p} ζ_{12}) = \begin{cases}1 & for p = 2; \\ h(\sqrt{−p} ,\sqrt{−3} ) & for p ≡ 1 (mod 4); \\ (ϖ_{3p} + 1)h(\sqrt{−p} ,\sqrt{−3} ) & for p ≡ 3 (mod 4) (p ≠ 3). \end{cases}

The following are the class numbers of the fields K = ℚ(\sqrt{p} ζ_{n}) for n ∈ \{3, 8, 12\} and p ∈ {2, 3, 5}. They are checked using the Magma algebra system [2].

\begin{matrix}h(K) & p = 2 & p = 3 & p = 5 \\ ℚ(\sqrt{p} ζ_{3}) = ℚ(\sqrt{p} ,\sqrt{−3} ) & 1 & 1 & 1 \\ ℚ(\sqrt{p} ζ_{8}) = ℚ(\sqrt{2p} ,\sqrt{−3} ) & 1 & 2 & 2 \\ ℚ(\sqrt{p} ζ_{12}) = ℚ(\sqrt{−p} ,\sqrt{−3} ) & 1 & 1 & 2 \end{matrix}

We collect the contribution of simple cases. For p = 2, we have

(5.11)  H_{sp}(\sqrt{2} ζ_{3}) + H_{sp}(\sqrt{2} ζ_{12}) + 2H_{sp}(\sqrt{2} ζ_{24}) = 1 + 1 + 2 = 4.

For p = 3, we have

(5.12)  H_{sp}(\sqrt{3} ζ_{3}) + H_{sp}(\sqrt{3} ζ_{8}) = 1 + 2 = 3.

For p = 5, we have

(5.13) H_{sp}(\sqrt{5} ζ_{3})+H_{sp}(\sqrt{3} ζ_{8})+H_{sp}(\sqrt{5} ζ_{12})+2H_{sp}(\sqrt{5} ζ_{5}) = 1+2+2+2 = 7.

For p ≥ 7, we have

∑_{π≠\sqrt{p} ∈W_{p}^{ss}(2)} H_{sp}(π) = H_{sp}(\sqrt{p} ζ_{3}) + H_{sp}(\sqrt{p} ζ_{8}) + H_{sp}(\sqrt{p} ζ_{12})

= \begin{cases}(ϖ_{p} + 1)h(K_{p,3}) + h(K_{2p,1}) + h(K_{3p,3}), & for p ≡ 1 (mod 4); \\ h(K_{p,3}) + h(K_{2p,1}) + (ϖ_{3p} + 1)h(K_{3p,3}), & for p ≡ 3 (mod 4). \end{cases}

(5.14)

Let Δ(p) be the number of isomorphism classes of superspecial abelian surfaces whose Frobenius endomorphism not equal to ±\sqrt{p}. Then we have (5.15)

Δ(p) = ∑_{π∈W_{p}^{ss}(2),π≠\sqrt{p}} H_{sp}(π)+ ∑_{π_{1}×π_{2},π_{1}≠π_{2}} H_{sp}(π_{1}×π_{2})+ ∑_{π_{1}∈W_{p}^{ss}(1)} H_{sp}(π_{1}×π_{1}).

Collecting the results (5.6), (5.7), (5.11) (5.12), (5.13) and (5.14), we obtain the following result.

Theorem 5.3.

(1) The number Δ(p) is 15, 20, 9 for p = 2, 3, 5, respectively.

(2) For p > 5 and p ≡ 1 (mod 4), we have

(5.16)  Δ(p) = (ϖ_{p} + 1)h(K_{p,3}) + h(K_{2p,1}) + h(K_{3p,3}) + h(\sqrt{−p}),

where ϖ_{p} is defined in (1.2).

(3) For p > 5 and p ≡ 3 (mod 4), we have

(5.17) Δ(p) = h(K_{p,3})+h(K_{2p,1})+(ϖ_{3p} +1)h(K_{3p,3})+(4 − (\frac{2}{p})) h(\sqrt{−p}),

where ϖ_{3p} is defined in (1.2).

Theorem 1.2 then follows from Theorems 1.1 and 5.3.

Remark 5.4. Based on our computation we observe that the endomorphism ring of a superspecial abelian surface over 𝔽_{p} may be a non-maximal order, or even non-maximal at p. For example, when p = 3 and π = \sqrt{3} ζ_{3}, the order R_{sp}, which occurs as the endomorphism ring of a superspecial abelian surface [21, Theorem 6.1], has index 3 in the maximal order.

5.3. Asymptotic behavior of | Sp_{2}(𝔽_{p})|. We now determine the asymptotic behavior of the size of Sp_{2}(𝔽_{p}) as the prime p goes to infinity. For simplicity, let F = ℚ(\sqrt{p}). By Theorem 1.2, | Sp_{2}(𝔽_{p})| is expressed as a linear combination of ζ_{F} (−1)h(F), h(\sqrt{−p}), and class numbers of certain biquadratic CM fields. The term cζ_{F} (−1)h(\sqrt{p}) (for a suitable constant c) comes from the contribution of the isogeny class corresponding to the Weil p-number π = \sqrt{p}. More precisely, it arises from the mass part in the Eichler class number formula for the calculation of H(\sqrt{p}). We recall from Theorem 1.1 that the mass part for p > 5 is

(5.18)  Mass(p) = \begin{cases}\frac{1}{2}ζ_{F} (−1)h(F) & for p ≡ 3 (mod 4); \\ 8ζ_{F} (−1)h(F) & for p ≡ 1 (mod 8); \\ \frac{1}{2}(15ϖ_{p} + 1)ζ_{F} (−1)h(F) & for p ≡ 5 (mod 8). \end{cases}

In [22, Theorem 6.3.1] we showed that the mass part Mass(p) is the main term of H(\sqrt{p}). It is expected that Mass(p) is also the main term of | Sp_{2}(𝔽_{p})|. This is true and we have the following asymptotic formula for the size of Sp_{2}(𝔽_{p}).

Proposition 5.5. We have

lim_{p→∞} \frac{| Sp_{2}(𝔽_{p})|}{Mass(p)} = 1.

Proof. It is enough to show that lim_{p→∞} h(\sqrt{−p})/h(F)ζ_{F} (−1) = 0, and for all the biquadratic CM-fields K_{d,j} appearing in the formula of | Sp_{2}(𝔽_{p})|,

lim_{p→∞} h(K_{d,j})/h(F)ζ_{F} (−1) = 0.

The above limit has been verified for the pairs (d, j) with d = p and j = 1, 2, 3 in [22, Theorem 6.3.1], and it remains to consider the pairs (2p, 1) and (3p, 3). Recall that the discriminant of F is denoted by 𝔡_{F}, which is either p or 4p. Using the functional equation and the trivial inequality ζ_{F} (2) > 1, we have ζ_{F} (−1) > c_{1}(𝔡_{F})^{3/2} for a constant c_{1} > 0. For any CM-field K, let h^{−}(K) be the relative class number of K defined as h(K)/h(K^{+}), where K^{+} is the maximal totally real subfield of K. By [8, Lemma 4], when K ranges over a sequence of CM-fields with bounded degree and 𝔡_{K} → ∞, we have

(5.19)  lim_{𝔡_{K}→∞}(log h^{−}(K))/(log\sqrt{𝔡_{K}/𝔡_{K^{+}}} ) = 1.

In particular, applying this to the quadratic imaginary fields ℚ(\sqrt{−p}), we obtain that h(\sqrt{−p})/ζ_{F} (−1) → 0 as p → ∞. Assume (d, j) = (2p, 1) or (3p, 3). One calculates that 𝔡_{K_{d,j}}/𝔡_{K_{d,j}^{+}} ≤ 32p. Let ϵ_{d}

be the fundamental unit of the quadratic real field ℚ(\sqrt{d}). By Siegel’s theorem [9, Theorem 15.4, Chapter 12], the growth of h(K_{d,j}^{+}) = h(\sqrt{d}) satisfies the following formula

lim_{d→∞} \frac{log(h(\sqrt{d}) log ϵ_{d})}{log\sqrt{d}} = 1.

Documenta Mathematica 21 (2016) 1607–1643

Note that ϵ_{d} is bounded below by (1 +\sqrt{5} )/2 for all d. Recall that h(K_{d,j}) = h^{−}(K_{d,j})h(\sqrt{d}). Combining these bounds yields that h(K_{d,j})/ζ_{F} (−1) → 0 as p goes to infinity. □

6. Galois cohomology and Superspecial abelian varieties

6.1. Galois cohomology and conjugacy classes. We refer to [15, Section I.5] for the definition of nonabelian Galois cohomology. Let Γ_{𝔽_{q}} = Gal(\overline{𝔽}_{q}/𝔽_{q}) be the absolute Galois group of 𝔽_{q}, and G a group with discrete topology on which Γ_{𝔽_{q}} acts continuously. Let σ_{q} be the arithmetic Frobenius automorphism of \overline{𝔽}_{q}, which raises each element of \overline{𝔽}_{q} to its q-th power. The group Γ_{𝔽_{q}} is isomorphic to the profinite group ℤ̂ = \underleftarrow{lim}_{n∈ℕ} ℤ/nℤ with canonical generator σ_{q}. Each 1-cocycle (a_{σ})_{σ∈Γ_{𝔽q}} is uniquely determined by its value x = a_{σ_{q}} ∈ G at σ_{q}. An element of G is called a 1-cocycle element if it arises from a 1-cocycle in this way. We will identify the set of 1-cocycles Z^{1}(Γ_{𝔽_{q}}, G) with the subset of 1-cocycle elements of G. Two 1-cocycle elements x, y ∈ Z^{1}(Γ_{𝔽_{q}}, G) define the same cohomology class if and only if they are σ_{q}-conjugate (denoted by x ∼_{σ_{q}} y), i.e., there exists z ∈ G such that x = z^{−1}yσ_{q}(z). Write [x]_{σ_{q}} for the σ_{q}-conjugacy class of x ∈ G, and B(G) for the set of all σ_{q}-conjugacy classes of G. Then

H^{1}(Γ_{𝔽_{q}}, G) = {[x]_{σ_{q}} ∈ B(G) | x ∈ Z^{1}(Γ_{𝔽_{q}}, G)} ⊆ B(G).

If the action of Γ_{𝔽_{q}} on G is trivial, then B(G) is reduced to the set Cl(G) of (the usual) conjugacy classes of G. Define Cl_{0}(G) := {[x] ∈ Cl(G) | x is of finite order} ⊆ Cl(G).

Lemma 6.1. Assume that the action of Γ_{𝔽_{q}} on G factors through a finite quotient Gal(𝔽_{q^{N}} /𝔽_{q}). We have

Z^{1}(Γ_{𝔽_{q}}, G) = {x ∈ G | xσ_{q}(x) ⋯ σ_{q}^{N−1}(x) is of finite order }.

In particular, if the action of Γ_{𝔽_{q}} on G is trivial, then H^{1}(Γ_{𝔽_{q}}, G) = Cl_{0}(G).

Proof. This follows directly from Exercise 2 of [15, Section I.5.1]. □

6.2. Abelian varieties over finite fields and twisted forms. Let X_{0} be an abelian variety over 𝔽_{q} with Frobenius endomorphism π_{X_{0}} ∈ End_{𝔽_{q}}(X_{0}). Set \overline{X}_{0} = X_{0} ⊗ \overline{𝔽}_{q}, and G = Aut(\overline{X}_{0}). The Galois group Γ_{𝔽_{q}} acts on End(\overline{X}_{0}) as follows (see [24, Lemma 3.3])

(6.1)  σ_{q}(x) = π_{X_{0}}xπ_{X_{0}}^{−1}, ∀ x ∈ End(\overline{X}_{0}),

where the conjugation by π_{X_{0}} is taken inside End^{0}(\overline{X}_{0}). As End(\overline{X}_{0}) is a free ℤ-module of finite rank, the action of Γ_{𝔽_{q}} factors through a finite quotient Gal(𝔽_{q^{N}} /𝔽_{q}), and hence (π_{X_{0}})^{N} is central in End(\overline{X}_{0}). Recall that an (\overline{𝔽}_{q}/𝔽_{q})-form of X_{0} is an abelian varieties X over 𝔽_{q} such that \overline{X} := X ⊗ \overline{𝔽}_{q} is \overline{𝔽}_{q}-isomorphic to \overline{X}_{0}. Let E(\overline{𝔽}_{q}/𝔽_{q}, X_{0}) be the set of 𝔽_{q}- isomorphism classes of (\overline{𝔽}_{q}/𝔽_{q})-forms of X_{0}. By [15, Section III.1.3], there is a canonical bijection of pointed sets

(6.2)  θ : E(\overline{𝔽}_{q}/𝔽_{q}, X_{0}) ⟶^{∼} H^{1}(Γ_{𝔽_{q}}, G),

sending the 𝔽_{q}-isomorphism class of X_{0} to the trivial class. The map θ is induced from mapping each \overline{𝔽}_{q}-isomorphism f : \overline{X}_{0} → \overline{X} to the 1-cocycle element x = f^{−1}σ_{q}(f) ∈ G. The injectivity of θ follows purely from cohomological formalism, and the surjectivity is a consequence of Weil’s Galois descent. An isomorphism f of abelian varieties as above induces an isomorphism

(6.3)  α_{f} : End(\overline{X}) ≃ End(\overline{X}_{0}), y ↦ f^{−1}yf.

The Frobenius endomorphisms π_{X_{0}} and π_{X} are related by the following commutative diagram (see [24, (3.2)]):

(6.4)  \begin{matrix}\overline{X}_{0} & ⟶^{f} & \overline{X} \\ ^{π_{X_{0}}} ↓ & & ↓^{π_{X}} \\ \overline{X}_{0} & ⟶^{σ_{q}(f)} & \overline{X}. \end{matrix}

We compute

(6.5)  α_{f}(π_{X}) = f^{−1}π_{X}f = f^{−1}σ_{q}(f)π_{X_{0}} = xπ_{X_{0}}.

Note that for x, y, z ∈ G,

x = z^{−1}yσ_{q}(z) ⇔ xπ_{X_{0}} = z^{−1}(yπ_{X_{0}})z.

Hence there is a well-defined injective map

(6.6)  Π : B(G) ↪ End(\overline{X}_{0})/G, [x]_{σ_{q}} ↦ [xπ_{X_{0}}],

where End(\overline{X}_{0})/G denotes the set of orbits of End(\overline{X}_{0}) under the right action of G by conjugation. In a sense, the image of H^{1}(Γ_{𝔽_{q}}, G) under Π consists of the conjugacy classes of Frobenius endomorphisms of members of E(\overline{𝔽}_{q}/𝔽_{q}, X_{0}). We can also work in the category of abelian varieties up to isogeny and study the (\overline{𝔽}_{q}/𝔽_{q})-forms of the isogeny class [X_{0}]. Thus we pass from isomorphisms of abelian varieties to quasi-isogenies, and endomorphism rings to endomorphism algebras, etc. Let E^{0}(\overline{𝔽}_{q}/𝔽_{q}, [X_{0}]) be the set of 𝔽_{q}-isogeny classes of abelian varieties [X] such that \overline{X} is isogenous to \overline{X}_{0} over \overline{𝔽}_{q}, and G_{ℚ} = End^{0}(\overline{X}_{0})^{×}. Many previous constructions can be carried over. In particular, both (6.4) and (6.5) hold true for any quasi-isogeny f : \overline{X}_{0} → \overline{X}, and one obtains a 1-cocycle element x = f^{−1}σ_{q}(f) ∈ G_{ℚ} as before. This gives a canonical injective map

(6.7)  θ : E^{0}(\overline{𝔽}_{q}/𝔽_{q}, [X_{0}]) ↪ H^{1}(Γ_{𝔽_{q}}, G_{ℚ}),

which fits into a commutative diagram

(6.8)  \begin{matrix}E(\overline{𝔽}_{q}/𝔽_{q}, X_{0}) & ⟶_{θ}^{≅} & H^{1}(Γ_{𝔽_{q}}, G) \\ ↓ & & ↓ \\ E^{0}(\overline{𝔽}_{q}/𝔽_{q}, [X_{0}]) & ⟶^{θ} & H^{1}(Γ_{𝔽_{q}}, G_{ℚ}). \end{matrix}

The left vertical map sends the 𝔽_{q}-isomorphism class of X to its 𝔽_{q}-isogeny class [X], and the right vertical map is induced from the inclusion of Γ_{𝔽_{q}}-groups G ⊂ G_{ℚ}. Thus (6.8) endows a geometric meaning for this cohomological map. We complete the picture by showing that the map θ in (6.7) is surjective and thus a bijection of pointed sets as stated in Proposition 1.4. Recall that the action of Γ_{𝔽_{q}} on End^{0}(\overline{X}) factors through Gal(𝔽_{q^{N}} /𝔽_{q}) for a fixed N ∈ ℕ. Without lose of generality, assume that X_{0} is 𝔽_{q^{N}} -isotypical, i.e., X_{0} ⊗ 𝔽_{q^{N}} is isogenous to (Y_{N})^{d}, where Y_{N} is an absolutely simple abelian variety over 𝔽_{q^{N}} with End(Y_{N}) = End(\overline{Y}_{N}). Equivalently, we assume that the multiple Weil q-number π_{0,1}^{t_{1}} × ⋯ × π_{0,u}^{t_{u}} ∈ MW_{q} corresponding to the 𝔽_{q}-isogeny class [X_{0}] satisfies that π_{0,1}^{N} = π_{0,2}^{N} = ⋯ = π_{0,u}^{N} after suitable conjugation, and ℚ((π_{X_{0}})^{N}) ⊂ End^{0}(X_{0}) is a field which coincides with ℚ((π_{X_{0}})^{sN}) for all s ∈ ℕ. Then End^{0}(\overline{X}_{0}) = Mat_{d}(End^{0}(\overline{Y}_{N})), and End^{0}(\overline{Y}_{N}) is a central division algebra over ℚ((π_{X_{0}})^{N}). For simplicity, let D = End^{0}(\overline{Y}_{N}) and K_{0} = ℚ((π_{X_{0}})^{N}). Then G_{ℚ} = End^{0}(\overline{X}_{0})^{×} = GL_{d}(D).

Lemma 6.2. There is a bijection between H^{1}(Γ_{𝔽_{q}}, G_{ℚ}) and the following subset of Cl(G_{ℚ}):

(6.9)  𝒞 (π_{X_{0}}) = \{[\underline{π}] ∈ Cl(G_{ℚ}) | ∃M ∈ ℕ : \underline{π}^{NM} = π_{X_{0}}^{NM}\}.

Proof. Since π_{X_{0}} ∈ G_{ℚ}, the map Π in (6.6) defines a bijection

Π : B(G_{ℚ}) ⟶^{∼} Cl(G_{ℚ}), [x]_{σ_{q}} ↦ [xπ_{X_{0}}].

Let π_{x} = xπ_{X_{0}} for each x ∈ G_{ℚ}. Then

xσ_{q}(x) ⋯ σ_{q}^{N−1}(x) = (xπ_{X_{0}})^{N}(π_{X_{0}})^{−N} = (π_{x})^{N}(π_{X_{0}})^{−N}.

By Lemma 6.1, x ∈ Z^{1}(Γ_{𝔽_{q}}, G_{ℚ}) if and only if (π_{x})^{N} = (π_{X_{0}})^{N}ξ for some ξ ∈ G_{ℚ} of finite order, or equivalently, π_{x}^{NM} = (π_{X_{0}})^{NM} for some M ∈ ℕ. □

Any \underline{π} ∈ G_{ℚ} with [\underline{π}] ∈ 𝒞 (π_{X_{0}}) is semisimple, as \underline{π}^{NM} = (π_{X_{0}})^{NM} lies in the center of the simple ℚ-algebra End^{0}(\overline{X}_{0}). The minimal polynomial of \underline{π} factorizes as a product of distinct irreducible polynomials over ℚ:

(6.10)  P (t) = ∏_{i=1}^{r} P_{i}(t) ∈ ℚ[t].

For all \underline{π}′ in the conjugacy class [\underline{π}], the ℚ-subalgebra K\underline{π}′ := ℚ(\underline{π}′) ⊂ End^{0}(\overline{X}_{0}) is canonically isomorphic to K := ℚ[t]/(P (t)) via the map \underline{π}′ ↦ t. Since π_{X_{0}}^{NM} = \underline{π}^{NM}, the field K_{0} = ℚ(π_{X_{0}}^{NM}) can be identified with the ℚsubalgebra of K generated by t^{NM}, thus providing a K_{0}-algebra structure on K. By (6.10), K factorizes as a products of number fields

(6.11)  K = K_{1} × ⋯ × K_{r}, with K_{i} = ℚ[t]/(P_{i}(t)) ⊇ K_{0}.

By an abuse of notation, we regard π_{X_{0}}^{N} as a Weil q^{N}-number via a embedding K_{0} ↪ \overline{ℚ}. Then for each 1 ≤ i ≤ r, the roots of P_{i}(t) in \overline{ℚ} is a conjugacy class of Weil q-numbers such that one of its representative π_{i} satisfies π_{i}^{NM} = π_{X_{0}}^{NM}.

Therefore, given \underline{π} ∈ 𝒞 (π_{X_{0}}), we find r Weil q-numbers representing distinct conjugacy classes (6.12)

{π_{1}, ⋯, π_{r}} with π_{i}^{NM} = π_{X_{0}}^{NM} for some M ∈ ℕ and all 1 ≤ i ≤ r.

Next, we fix P (t) ∈ ℚ[t] as above, and produce a discrete invariant for every conjugacy class [\underline{π}] ∈ 𝒞 (π_{X_{0}}) with minimal polynomial P (t). Let V = D^{d} be the right vector space over D of column vectors. Then End_{D}(V) = Mat_{d}(D) acts on V from the left by the usual matrix multiplication. We have a canonical K_{0}-algebra embedding K ↪ End_{D}(V) sending K to K_{π}. Thus \underline{π} endows a faithful (K, D)-bimodule structure on V, denoted by V\underline{π}. By (6.11), there is a decomposition of V into right D-subspaces:

(6.13)  V = ⨁_{i=1}^{r} V_{i}, d_{i} = dim_{D} V_{i}.

The action of K_{i} on V_{i} gives rise to a K_{0}-embedding K_{i} ↪ End_{D}(V_{i}) = Mat_{d_{i}}(D). We study each of the embeddings individually first.

Lemma 6.3. Let π ∈ W_{q} be a Weil q-number such that π^{NM} = π_{X_{0}}^{NM} for some integer M ∈ ℕ, and X_{π} a simple abelian variety over 𝔽_{q} in the isogeny class corresponding to π. Let e = e(π) be the smallest integer such that there is an K_{0}-embedding ℚ(π) ↪ Mat_{e}(D). Then \overline{X}_{π} = X_{π} ⊗ \overline{𝔽}_{q} is isogenous to (\overline{Y}_{N})^{e}, and End^{0}(X_{π}) is isomorphic to the centralizer C_{π} of ℚ(π) in Mat_{e}(D).

Proof. Since π^{NM} = π_{X_{0}}^{NM}, there exists an isogeny \overline{X}_{π} → (\overline{Y}_{N})^{e} for some e ∈ ℕ, which gives an identification of End^{0}(\overline{X}_{π}) with Mat_{e}(D) = End^{0}((\overline{Y}_{N})^{e}) in the same way as (6.3). Thus we obtain a K_{0}-embedding ℚ(π) ↪ Mat_{e}(D), and End^{0}(X_{π}) is recovered as the Γ_{𝔽_{q}}-invariants of End^{0}(\overline{X}_{π}), or equivalently, the centralizer C_{π} of ℚ(π) in Mat_{e}(D) by (6.1). On the other hand, C_{π} is also the endomorphism algebra of the (ℚ(π), D)-bimodule D^{e}. Now the minimality of e follows from the fact that C_{π} = End^{0}(X_{π}) is a division algebra. □

Given e′ ∈ ℕ, a K_{0}-embedding of ℚ(π) into the simple algebra Mat_{e′}(D) exists if and only if e(π) divides e′. Therefore, every d_{i} in (6.13) is of the form m_{i}e(π_{i}) for some positive integer m_{i} ∈ ℕ subjecting to the condition

(6.14)  m_{1}e(π_{1}) + ⋯ + m_{r}e(π_{r}) = d.

We shall call the r-tuple \underline{m} = (m_{1}, …, m_{r}) the type of the (K, D)-bimodule V\underline{π} or simply the type of \underline{π}.

Lemma 6.4. There are natural bijections between the following sets:

(1) the set of conjugacy classes [\underline{π}] ∈ 𝒞 (π_{X_{0}}) with minimal polynomial P (t);

(2) the set of G_{ℚ}-conjugacy classes of K_{0}-embedding K ↪ End_{D}(V);

(3) the set of isomorphism classes of faithful (K, D)-bimodule structures on V;

(4) the set of r-tuples \underline{m} = (m_{1}, …, m_{r}) ∈ ℕ^{r} satisfying (6.14).

Proof. The bijection between between (1) and (2) is established by the map sending each K_{0}-embedding ϕ : K = ℚ[t]/(P (t)) ↪ End_{D}(V) to π = ϕ(t). Every faithful (K, D)-bimodule structure on V is given by a K_{0}-embedding ϕ : K ↪ End_{D}(V). Two such embeddings define isomorphic structures if and only if they are conjugate by an element of G_{ℚ}. Hence (2) is bijective to (3). The proof that (2) is bijective to (4) is similar to that of [16, Proposition 3.2] and is omitted. □

Proposition 6.5. Each cohomology class [x]_{σ_{q}} ∈ H^{1}(Γ_{𝔽_{q}}, G_{ℚ}) determines a unique conjugacy class of multiple Weil q-number π_{1}^{m_{1}} ×⋯×π_{r}^{m_{r}} ∈ MW_{q} such that

• π_{i}^{NM} = π_{X_{0}}^{NM} for some M ∈ ℕ and all 1 ≤ i ≤ r;
• \underline{m} = (m_{1}, …, m_{r}) ∈ ℕ^{r} satisfies (6.14).

In particular, the map θ in (6.7) is a bijection of pointed sets.

Proof. Given [x]_{σ_{q}} ∈ H^{1}(Γ_{𝔽_{q}}, G_{ℚ}), we produce the desired multiple Weil qnumber by combing the type \underline{m} = (m_{1}, ⋯, m_{r}) of [π_{x}] ∈ 𝒞 (π_{X_{0}}) and the set {π_{1}, ⋯, π_{r}} determined by [π_{x}] as in (6.12). Let X = ∏_{i=1}^{r}(X_{π_{i}})^{m_{i}} be an abelian variety over 𝔽_{q} corresponding to π_{1}^{m_{1}} × ⋯ × π_{r}^{m_{r}}. Then \overline{X} is isogenous to \overline{X}_{0} by Lemma 6.3 and (6.14). Identify End^{0}(\overline{X}) with End^{0}(\overline{X}_{0}) via an isogeny f : \overline{X}_{0} → \overline{X} as in (6.3). The conjugacy class of α_{f}(π_{X}) ∈ G_{ℚ} is independent of the choice of f. By the construction, α_{f}(π_{X}) is a semisimple element with the same minimal polynomial and type as π_{x} = xπ_{X_{0}}. It follows from Lemma 6.4 that they must lie in the same conjugacy class of G_{ℚ}. We conclude that θ is surjective by Lemma 6.2. □

6.3. Superspecial abelian varieties and the parity property. We apply the previous construction to the study of superspecial abelian varieties over finite fields. Let E_{0} be a supersingular elliptic curve over the prime finite field 𝔽_{p} whose Frobenius endomorphism π_{0} satisfying π_{0}^{2} + p = 0 (Recall that \sqrt{−p} ∈ W_{p}^{ss}(1) for all p by Proposition 3.5). Let O := End(E_{0} ⊗ \overline{𝔽}_{p}) be the endomorphism ring of E_{0} ⊗ \overline{𝔽}_{p}; this is a maximal order in the unique quaternion ℚ-algebra D = B_{p,∞} ramified exactly at {p, ∞}. Take X_{0} = E_{0}^{d} and \overline{X}_{0} := X_{0} ⊗ \overline{𝔽}_{p} for d ≥ 1. Then End(\overline{X}_{0}) = Mat_{d}(O). In what follows we denote by

G := Aut(\overline{X}_{0}) = GL_{d}(O)

the automorphism group of \overline{X}_{0}. Consider O as a subring of Mat_{d}(O) by the diagonal embedding and view π_{0} as an element in Mat_{d}(O). Then the action of Γ_{𝔽_{p}} on G = GL_{d}(O) is given by

(6.15)  σ_{p}(x) = π_{0}xπ_{0}^{−1}, x ∈ G.

We will also write G_{ℚ} for the group GL_{d}(D). Recall that Sp_{d}(𝔽_{q}) denotes the set of isomorphism classes of d-dimensional superspecial abelian varieties over 𝔽_{q}. For the classification of superspecial abelian varieties over the algebraic closure \overline{𝔽}_{q} of 𝔽_{q}, we have the following result, due to Deligne, Shioda and Ogus (cf. [12, Section 1.6, p. 13]).

Theorem 6.6. For any integer d ≥ 2, there is only one isomorphism class of d-dimensional superspecial abelian varieties over any algebraically closed field of characteristic p > 0.

According to this theorem, any d-dimensional superspecial abelian variety over 𝔽_{q} is an (\overline{𝔽}_{q}/𝔽_{q})-form of X_{0} ⊗ 𝔽_{q}. Thus we obtain a natural bijection by (6.2)

(6.16)  H^{1}(Γ_{𝔽_{q}}, G) ≃ Sp_{d}(𝔽_{q}), d > 1,

which sends the trivial class to the isomorphism class of X_{0} ⊗ 𝔽_{q}. The set Sp_{d}(𝔽_{q}) is partitioned into isogeny classes:

(6.17)  Sp_{d}(𝔽_{q}) = ∐_{π∈MW_{q}^{ss}(d)} Sp(π).

Theorem 6.7. Let q = p^{a} and q′ = p^{a′} be powers of the prime number p such that a ≡ a′ (mod 2). For any integer d ≥ 1, there are natural bijections

H^{1}(Γ_{𝔽_{q}}, G) ≃ H^{1}(Γ_{𝔽_{q′}} , G), (6.18)

Sp_{d}(𝔽_{q}) ≃ Sp_{d}(𝔽_{q′}). (6.19)

Proof. If d = 1, then (6.19) has been proven in Section 4.2; see Proposition 4.4 and Remark 4.5. If d > 1, then (6.19) follows from (6.16) and (6.18). Therefore, it remains to prove (6.18). Since π_{0}^{2} is a central element, the element σ_{p}^{2} acts trivially on G by (6.15). Thus σ_{q}(x) = σ_{q′}(x) for all x ∈ G. This together with the canonical isomorphism Γ_{𝔽_{q}} ≃ Γ_{𝔽_{q′}} (sending σ_{q} ↦ σ_{q′}) yields a natural bijection H^{1}(Γ_{𝔽_{q}}, G) ≃ H^{1}(Γ_{𝔽_{q′}} , G). The theorem is proved. □

Remark 6.8. By the same token, we have a natural bijection

(6.20)  H^{1}(Γ_{𝔽_{q}}, G_{ℚ}) ≃ H^{1}(Γ_{𝔽_{q′}} , G_{ℚ}).

Thus by Proposition 6.5, there is also a natural bijection between the isogeny classes of supersingular abelian varieties over 𝔽_{q} and those over 𝔽_{q′}. This can be made explicit in terms of multiple Weil numbers. The Frobenius endomorphism of X_{0} ⊗ 𝔽_{q} is π_{0}^{a}. Hence the Frobenius endomorphisms of the isogeny class corresponding to a cohomology class [x]_{σ_{q}} ∈ H^{1}(Γ_{𝔽_{q}}, G_{ℚ}) is given by the conjugacy class [xπ_{0}^{a}] by (6.5). Without lose of generality, assume that a−a′ = 2s ≥ 0. If π = π_{1}^{m_{1}} ×⋯×π_{r}^{m_{r}} is a multiple Weil q-number determined by [x]_{σ_{q}}, then the corresponding multiple Weil q′-number is π̃ = π_{1}^{m_{1}} ×⋯×π_{r}^{m_{r}}, with π̃_{i} = (−p)^{−s}π_{i} for all 1 ≤ i ≤ r. By the commutative diagram (6.8), the bijection (6.19) preserves isogeny classes in the sense that there is a natural bijection

(6.21)  Sp(π) ≃ Sp(π̃)  ∀ π ∈ MW_{q}^{ss}(d).

Corollary 6.9. Let q = p^{2s+1} be an odd power of p. Let Y_{0} be a fixed supersingular abelian variety over 𝔽_{q} and π = π_{1}^{m_{1}} × ⋯ × π_{r}^{m_{r}} the corresponding multiple Weil q-number. Let V and K be as in Proposition 5.1, and set R_{sp} := ℤ[π̃_{0}, pπ_{0}^{−1}] ⊂ K, where π̃_{0} = (−p)^{−s}(π_{1}, …, π_{r}). Assume that K has no real place. Then there is a natural bijection between the set Sp(π) of isomorphism classes of superspecial abelian varieties in the isogeny class [Y_{0}] and the set of isomorphism classes of R_{sp}-lattices in V.

Proof. This follows from Proposition 5.1 and Theorem 6.7. □

The above theorem provides an approach for computing the size of Sp_{d}(𝔽_{q}) explicitly in the odd exponent case, subject to the condition that K has no totally real factors. For the rest of this section we shall describe H^{1}(Γ_{𝔽_{q}}, GL_{d}(O)) (and hence Sp_{d}(𝔽_{q})) when q is an even power of p.

6.4. A description of H^{1}(Γ_{𝔽_{q}}, GL_{d}(O)) with even exponent. In what follows we assume that q = p^{a} is an even power of p and X_{0} = E_{0}^{d} ⊗ 𝔽_{q} with d ≥ 2. The Frobenius endomorphism π_{X_{0}} = (−p)^{a/2} lies in the center of End(\overline{X}_{0}) = Mat_{d}(O). Hence Γ_{𝔽_{q}} acts trivially on the group G := GL_{d}(O) by (6.15). Then it follows from Lemma 6.1 that H^{1}(Γ_{𝔽_{q}}, G) can be identified with the set Cl_{0}(G) of conjugacy classes of elements in G of finite order. We shall give a lattice description for Cl_{0}(G) and hence for Sp_{d}(𝔽_{q}) by the previous correspondence. See Proposition 6.11 for details. Suppose x ∈ G is an element of finite order, which is necessarily semi-simple. The minimal polynomial of x over ℚ has the form

(6.22)  P\underline{n}(t) = Φ_{n_{1}}(t)Φ_{n_{2}}(t) ⋯ Φ_{n_{r}}(t), 1 ≤ n_{1} < n_{2} < ⋯ < n_{r}

for some r-tuple \underline{n} = (n_{1}, …, n_{r}) ∈ ℕ^{r}, where Φ_{m}(t) ∈ ℤ[t] denotes the m-th cyclotomic polynomial. We define

K\underline{n} := \frac{ℚ[t]}{∏_{i=1}^{r} Φ_{n_{i}}(t)} and A\underline{n} := \frac{ℤ[t]}{∏_{i=1}^{r} Φ_{n_{i}}(t)}.

The ℚ-subalgebras of End^{0}(\overline{X}_{0}) = Mat_{d}(D) generated by x and π_{x} = xπ_{X_{0}} coincide and are isomorphic to K\underline{n}. Moreover, the subring ℤ[x] ⊂ Mat_{d}(O) is canonically isomorphic to A\underline{n}. We denote by C(\underline{n}) ⊂ Cl_{0}(G) the set of conjugacy classes of G with minimal polynomial P\underline{n}(t). By Proposition 6.5, each conjugacy class [x] ∈ C(\underline{n}) determines a (conjugacy class of) supersingular multiple Weil q-number π_{1}^{m_{1}} × ⋯ × π_{r}^{m_{r}}, where π_{i} = (−p)^{a/2}ζ_{n_{i}}, and \underline{m} = (m_{1}, ⋯, m_{r}) is the type of the faithful (K\underline{n}, D)-bimodule structure on V = D^{d} equipped by π_{x} ∈ Mat_{d}(D). Since ℚ(π_{x}) = ℚ(x) ≅ K\underline{n}, the (K\underline{n}, D)-bimodule structure on V is also equipped directly by x ∈ Mat_{d}(D). Thus \underline{m} is also called the type of [x], as it depends only on the conjugacy class. Recall that a (K\underline{n}, D)-bimodule V is said to be type \underline{m} if the decomposition into D-subspaces V = ⊕_{i=1}^{r}V_{i} induced from the decomposition K\underline{n} = ∏_{i=1}^{r} ℚ(ζ_{n_{i}}) satisfies that dim_{D} V_{i} = m_{i}e(π_{i}) for all 1 ≤ i ≤ r, where e(π_{i}) is defined in Lemma 6.3. Since dim E_{0} = 1, we have e(π_{i}) = d(π_{i}), the dimension of the Weil number π_{i}. Note that d(π_{i}) depends only on the integer n_{i} as q = p^{a} is fixed, so we write d(n_{i}) for it instead. Equation (6.14) becomes

(6.23)  m_{1}d(n_{1}) + ⋯ + m_{r}d(n_{r}) = d.

A pair of r-tuples (\underline{n}, \underline{m}) ∈ ℕ^{r} × ℕ^{r} with 1 ≤ n_{1} < ⋯ < n_{r} is said to be d-admissible if the condition (6.23) is satisfied. Let C(\underline{n}, \underline{m}) ⊂ C(\underline{n}) denote the subset of conjugacy classes of type \underline{m}. An element x ∈ G or its conjugacy [x] ∈ Cl(G) is said to be type (\underline{n}, \underline{m}) if [x] ∈ C(\underline{n}, \underline{m}).

Lemma 6.10. Fix a faithful (K\underline{n}, D)-bimodule V = D^{d} of type \underline{m}. There is a natural bijection between the set C(\underline{n}, \underline{m}) and the set of isomorphism classes of (A\underline{n}, O)-lattices in V.

Proof. Let M_{0} := O^{d} ⊂ V be the standard lattice in V. Every element x ∈ G of type (\underline{n}, \underline{m}) gives rise to an (A\underline{n}, O)-bimodule structure on M_{0}. Two elements x, x′ determine isomorphism bimodule structures if and only if they are conjugate in G. Therefore, the set C(\underline{n}, \underline{m}) is in bijection with the set of isomorphism classes of (A\underline{n}, O)-lattices in V that are O-isomorphic to M_{0}. Since d ≥ 2, every O-lattice in V is isomorphic to M_{0}. This follows from a theorem of Eichler [6] that the class number of Mat_{d}(O) is 1 for d ≥ 2 (see also [10, Theorem 2.1]). □

Proposition 6.11. Let Cl_{0}(G) be the set of conjugacy classes of G = GL_{d}(O) of finite order with d ≥ 2. Then

(6.24)  Cl_{0}(G) = ∐_{(\underline{n},\underline{m})} C(\underline{n}, \underline{m}),

where (\underline{n}, \underline{m}) runs through all d-admissible types. For each fixed (\underline{n}, \underline{m}), there are natural bijections between the following sets:

(1) C(\underline{n}, \underline{m}), the set of conjugacy classes of type (\underline{n}, \underline{m});

(2) Sp(π), where π = π_{1}^{m_{1}} × ⋯ × π_{r}^{m_{r}} and π_{i} = (−p)^{a/2}ζ_{n_{i}};

(3) the set of isomorphism classes of (A\underline{n}, O)-lattices in the (K\underline{n}, D)-

bimodule V of type \underline{m}.

Proof. The bijection between (1) and (2) is established by combining (6.16) and Proposition 6.5. The bijection between (1) and (3) follows from Lemma 6.10.

□

7. Arithmetic results

In this section, we prove the arithmetic results used in Section 5 concerning the order R_{sp}. In the light of (5.4), our goals are two fold: (1) show that R_{sp} is Bass for every supersingular multiple Weil p-number π ∈ MW_{p}^{ss}(2) of dimension 2 distinct from ±\sqrt{p}; (2) classify all suporders of R_{sp} (i.e., orders in K containing R_{sp}) and calculate their class numbers when π is not of the form π_{1} × π_{1} with π_{1} ∈ W_{p}^{ss}(1) (The case π = π_{1} × π_{1} has already been treated in Section 5.2).

7.1. Orders in products of number fields. Let K = ∏_{i=1}^{r} K_{i} be a product of number fields, and S be an order contained in the maximal order O_{K} = ∏_{i=1}^{r} O_{K_{i}}. We write pr_{i} : K → K_{i} for the projection map onto the i-th factor. By a theorem of Borevič and Faddeev [1] (see [5, Section 37, p. 789] or [11, Theorem 2.1]), S is Bass if and only if O_{K}/S is cyclic as an S-module. This leads to the following simple criterion when r = 2.

Lemma 7.1. A suborder S ⊆ O_{K_{1}} × O_{K_{2}} that projects surjectively onto both factors O_{K_{1}} and O_{K_{2}} is Bass.

Proof. Each O_{K_{i}} is equipped with an S-module structure via the projection map pr_{i} : S → O_{K_{i}}. Since pr_{2}(S) = O_{K_{2}}, the natural inclusion O_{K_{1}} ↪ O_{K_{1}} × O_{K_{2}} defined by x ↦ (x, 0) induces an isomorphism of S-modules

(7.1)  O_{K_{1}}/(O_{K_{1}} ∩ S) ⟶^{≃} (O_{K_{1}} × O_{K_{2}})/S.

The left hand side is a cyclic S-module because pr_{1}(S) = O_{K_{1}}. □

We return to the general case with r ≥ 1. Let 𝔞 be an O_{K}-lattice (i.e., a fractional O_{K}-ideal that contains a ℚ-basis of K) contained in S. There is a one-to-one correspondence between the orders B intermediate to S ⊆ O_{K} and the subrings of O_{K}/𝔞 containing S/𝔞. By [14, Theorem I.12.12], the class number h(B) can be calculated by

(7.2)  h(B) = \frac{h(O_{K})[(O_{K}/𝔞)^{×} : (B/𝔞)^{×}]}{[O_{K}^{×} : B^{×}]},

where h(O_{K}) = ∏_{i=1}^{r} h(O_{K_{i}}). A priori, [14, Theorem I.12.12] is only stated for the number field case with 𝔞 being the conductor of B, but the same proof applies in the current setting as well.

Lemma 7.2. Let 𝔞 ⊂ O_{K} be an O_{K}-lattice. If the natural map O_{K}^{×} → (O_{K}/𝔞)^{×} is surjective, then h(B) = h(O_{K}) for every suborder B ⊆ O_{K} containing 𝔞.

Proof. Let 𝔎 be the kernel of O_{K}^{×} → (O_{K}/𝔞)^{×}. Then 𝔎 ⊆ B^{×} and [O_{K}^{×} : B^{×}] = [O_{K}^{×}/𝔎 : B^{×}/𝔎]. We identify O_{K}^{×}/𝔎 with the image of O_{K}^{×} → (O_{K}/𝔞)^{×}, and similarly for B^{×}/𝔎. By [22, Lemma 2.7], B^{×} = O_{K}^{×} ∩ B. Hence

B^{×}/𝔎 = (O_{K}^{×}/𝔎) ∩ (B/𝔞).

When O_{K}^{×} maps surjectively onto (O_{K}/𝔞)^{×}, we have B^{×}/𝔎 = (O_{K}/𝔞)^{×} ∩ (B/𝔞) = (B/𝔞)^{×}. Therefore, h(B) = h(O_{K}) by (7.2). □

Lemma 7.3. Let S be a suborder of O_{K} = ∏_{i=1}^{r} O_{K_{i}}, and 𝔠_{1} be a nonzero ideal of O_{K_{1}} contained in pr_{1}(S). If x_{1} ∈ O_{K_{1}} is an element such that (x_{1}, 0, ⋯, 0) ∈ S, then (x_{1}𝔠_{1}, 0, ⋯, 0) is an ideal of O_{K} contained in S. Similar results hold for all 1 ≤ i ≤ r.

Proof. Clearly (x_{1}𝔠_{1}, 0, ⋯, 0) is an ideal of O_{K}. For any element y_{1} ∈ 𝔠_{1}, we may find y ∈ S such that pr_{1}(y) = y_{1}. Then (x_{1}y_{1}, 0, ⋯, 0) = (x_{1}, 0, ⋯, 0) · y ∈ S. □

7.2. The order R_{sp} is Bass when d(π) = 2. We recall the definition of R_{sp}. Suppose that π = π_{1}^{m_{1}} × ⋯ × π_{r}^{m_{r}} is a supersingular multiple Weil pnumber with m_{i} ∈ ℕ and π_{i} ≁ π_{j}. Let K = ∏_{i} K_{i} with K_{i} = ℚ(π_{i}), and π_{0} = (π_{1}, …, π_{r}) ∈ K. Then R_{sp} is defined to be the order ℤ[π_{0}, π_{0}^{2}/p] ⊆ O_{K}. Assume that π has dimension 2 and none of π_{i} is conjugate to \sqrt{p}. The case π = π_{1}^{2} with π_{1} ∈ W_{p}^{ss}(1) has already been studied in Section 5.2. It remains to treat the following two cases:

(1) π = π_{1} × π_{2} with both π_{1}, π_{2} ∈ W_{p}^{ss}(1) and π_{1} ≁ π_{2} (the nonisotypic product case);

(2) π = π_{1} ∈ W_{p}^{ss}(2) and π_{1} ≁ \sqrt{p} (the nonreal simple case). The first case occurs only when

(7.3) p = 2, 3, and π = \sqrt{p} ζ_{4} × (±\sqrt{p} ζ_{4p}), or \sqrt{p} ζ_{4p} × (−\sqrt{p} ζ_{4p}).

In the second case, the supersingular Weil p-numbers of dimension 2 distinct from ±\sqrt{p} are

(7.4) \sqrt{p} ζ_{3}, ±\sqrt{p} ζ_{5} (p = 5), \sqrt{p} ζ_{8} (p ≠ 2), \sqrt{p} ζ_{12} (p ≠ 3), ±\sqrt{p} ζ_{24} (p = 2).

Lemma 7.4. Assume p = 2 or 3. If π = \sqrt{p} ζ_{4} × (±\sqrt{p} ζ_{4p}), then

R_{sp} = ℤ[(\sqrt{−p} , 0), (0, 1 + ζ_{2p})] ⊂ ℚ(\sqrt{−p}) × ℚ(ζ_{2p}) = K.

If π = \sqrt{p} ζ_{4p} × (−\sqrt{p} ζ_{4p}), then

R_{sp} = ℤ[(2(1 + ζ_{2p}, 0), (ζ_{2p}, ζ_{2p})] ⊂ ℚ(ζ_{2p}) × ℚ(ζ_{2p}) = K.

Proof. Note that \sqrt{p} ζ_{4p} = 1 + ζ_{2p} when p = 2 or 3. If π = \sqrt{p} ζ_{4} × (±\sqrt{p} ζ_{4p}), then

\begin{aligned}R_{sp} & = ℤ[(\sqrt{−p} , ±\sqrt{p} ζ_{4p}), (−1, ζ_{2p})] = ℤ[(\sqrt{−p} , ±\sqrt{p} ζ_{4p}), (0, 1 + ζ_{2p})] \\ & = ℤ[(\sqrt{−p} , 0), (0, 1 + ζ_{2p})]. \end{aligned}

If π = \sqrt{p} ζ_{4p} × (−\sqrt{p} ζ_{4p}), then

\begin{aligned}R_{sp} & = ℤ[(\sqrt{p} ζ_{4p}, −\sqrt{p} ζ_{4p}), (ζ_{2p}, ζ_{2p})] = ℤ[(1 + ζ_{2p}, −(1 + ζ_{2p})), (ζ_{2p}, ζ_{2p})] \\ & = ℤ[(2(1 + ζ_{2p}), 0), (ζ_{2p}, ζ_{2p})]. \end{aligned} □

Proposition 7.5. The order R_{sp} is a Bass order for every supersingular multiple Weil p-number π ∈ MW_{p}^{ss}(2) distinct from ±\sqrt{p}.

Proof. We only need to consider the cases where π is not of the form π_{1}^{2} with π_{1} ∈ W_{p}^{ss}(1). Suppose that π = ±\sqrt{p} ζ_{n} ∈ W_{p}^{ss}(2) is one of the Weil p-numbers listed in (7.4), and m is defined as in (3.3). If n is critical at p, then R_{sp} equals to the maximal order ℤ[ζ_{m}] in K = ℚ(ζ_{m}) by Remark 5.2. Otherwise, [K : ℚ(ζ_{m})] = 2 and R_{sp} is a quadratic ℤ[ζ_{m}]-order, and such type of orders are Bass [11, Example 2.3]. If p = 2, 3 and π = \sqrt{p} ζ_{4p} × (−\sqrt{p} ζ_{4p}), or p = 2 and π = \sqrt{2} ζ_{4} × (±\sqrt{2} ζ_{8}), then R_{sp} projects surjectively onto both O_{K_{1}} and O_{K_{2}}, and hence R_{sp} is Bass by Lemma 7.1. Lastly, suppose that p = 3 and π = \sqrt{3} ζ_{4} × (±\sqrt{3} ζ_{12}). Then pr_{1}(R_{sp}) = ℤ[\sqrt{−3} ], a suborder of index 2 in O_{K_{1}} = ℤ[ζ_{6}], while pr_{2}(R_{sp}) = ℤ[ζ_{6}] = O_{K_{2}}.

So by (7.1), to show that R_{sp} is Bass, it is enough to prove that O_{K_{1}}/(O_{K_{1}} ∩ R_{sp}) is a cyclic R_{sp}-module. Note that O_{K_{1}} ⊂ O_{K_{1}} × O_{K_{2}} is generated by (1, 0) and (ζ_{6}, 0) over ℤ, and

R_{sp}(ζ_{6}, 0) ∋ (−1 +\sqrt{−3} , −1) · (ζ_{6}, 0) = (1, 0) + (\sqrt{−3} , 0)^{2} ≡ (1, 0)

(mod O_{K_{1}} ∩ R_{sp}).

Hence O_{K_{1}}/(O_{K_{1}} ∩ R_{sp}) is a cyclic R_{sp}-module generated by (ζ_{6}, 0). □

7.3. Suporders of R_{sp} and class numbers: the nonisotypic product case. Assume that p = 2 or 3 and π = π_{1} ×π_{2} is a supersingular multiple Weil p-number of dimension 2 listed in (7.3). Using Lemma 7.3 and Lemma 7.4, one may easily find an O_{K}-lattice 𝔞 contained in R_{sp} and compute the quotient rings O_{K}/𝔞 and R_{sp}/𝔞. We obtain the following table (For simplicity, we set i = ζ_{4} = \sqrt{−1} ).

\begin{matrix}π = π_{1} × π_{2} & 𝔞 ⊂ R_{sp} & O_{K}/𝔞 & R_{sp}/𝔞 \\ \sqrt{2} ζ_{4} × ±\sqrt{2} ζ_{8} & \sqrt{−2} O_{K_{1}} × (1 + i)O_{K_{2}} & (𝔽_{2})^{2} & 𝒟_{2} \\ \sqrt{2} ζ_{8} × −\sqrt{2} ζ_{8} & (2(1 + i)O_{K_{1}})^{2} & (ℤ[i]/(1 + i)^{3})^{2} & 𝒟_{8} \\ \sqrt{3} ζ_{4} × ±\sqrt{3} ζ_{12} & (2\sqrt{−3} )O_{K_{1}} ×\sqrt{−3} O_{K_{2}} & 𝔽_{4} × (𝔽_{3})^{2} & 𝔽_{2} × 𝒟_{3} \\ \sqrt{3} ζ_{12} × −\sqrt{3} ζ_{12} & (2\sqrt{−3} )O_{K_{1}} × (2\sqrt{−3} )O_{K_{2}} & (𝔽_{4} × 𝔽_{3})^{2} & 𝒟_{12} \end{matrix}

Here 𝒟_{2}, 𝒟_{8}, 𝒟_{3}, and 𝒟_{12} denote the diagonal in (𝔽_{2})^{2}, (ℤ[i]/(1 + i)^{3})^{2}, (𝔽_{3})^{2}, and (𝔽_{4} × 𝔽_{3})^{2} respectively. It is an exercise to show that O_{K}^{×} maps surjectively onto (O_{K}/𝔞)^{×} in all the above cases. By Lemma 7.2, h(B) = h(O_{K}) for every order B with R_{sp} ⊆ B ⊆ O_{K}. Note that h(O_{K}) = h(O_{K_{1}})h(O_{K_{2}}) = 1 since both ℤ[i] and ℤ[ζ_{6}] have class number 1. We obtain the following proposition.

Proposition 7.6. Assume that p = 2 or 3 and π = π_{1} × π_{2} is given in (7.3). Then any suporder B of R_{sp} has class number 1.

It remains to list all suporders B of R_{sp} for each π. We recall the convention in Section 5.2 that a suporder of R_{sp} with index j > 1 in O_{K} is denoted by B_{j}. Our calculation will show that for those π considered in this subsection, if such an order exists, then it is unique. So there is no ambiguity in this notation if π is clear from the context. We separate into cases. Case π = \sqrt{2} ζ_{4} × ±\sqrt{2} ζ_{8}. Since [O_{K} : R_{sp}] = [O_{K}/𝔞 : R_{sp}/𝔞] = 2, there are no other suporders of R_{sp} besides R_{sp} and O_{K}. Case π = \sqrt{3} ζ_{4} × ±\sqrt{3} ζ_{12}. We have [O_{K} : R_{sp}] = [𝔽_{4} × (𝔽_{3})^{2} : 𝔽_{2} × 𝒟_{3}] = 6. There are two rings properly intermediate to the inclusion 𝔽_{2}×𝒟_{3} ⊂ 𝔽_{4}×(𝔽_{3})^{2}, namely 𝔽_{4} ×𝒟_{3} and 𝔽_{2} ×(𝔽_{3})^{2}. Under the inclusion-preserving correspondence between suborders of O_{K} containing 𝔞 and subrings of O_{K}/𝔞, we have

B_{3} := ℤ[(\sqrt{−3} , 0), (ζ_{6}, ζ_{6})] = ℤ[(1 + ζ_{6}, 0), (0, 1 + ζ_{6})] ⟷ 𝔽_{4} × 𝒟_{3},
B_{2} := ℤ[\sqrt{−3} ] × ℤ[ζ_{6}] ⟷ 𝔽_{2} × (𝔽_{3})^{2}.

The remaining two cases are best seen in the light of the following lemma.

Lemma 7.7. Let R be a commutative ring, and 𝒟 be the diagonal of R^{2}. Every subring S of R^{2} containing 𝒟 decomposes uniquely as 𝒟 ⊕ (I_{S}, 0), where I_{S} is an ideal of R. In particular, there is an inclusion-preserving bijective correspondence between subrings of R^{2} containing 𝒟 and ideals of R.

Proof. Every subring of R^{2} containing 𝒟 is naturally an R-submodule of R^{2}. So the intersection (I_{S}, 0) := S ∩(R, 0) is again an R-submodule of R^{2}. Equivalently, I_{S} is an ideal of R. Clearly, we have S = 𝒟⊕(I_{S}, 0). Conversely, for any ideal I ⊆ R, the direct sum 𝒟 ⊕ (I, 0) is a subring of R^{2}. The correspondence is established. □

By Lemma 7.4, if π = \sqrt{p} ζ_{4p} × −\sqrt{p} ζ_{4p} with p = 2 or 3, then O_{K} = ℤ[ζ_{2p}]^{2}, and

R_{sp} = ℤ[(2(1 + ζ_{2p}), 0), (ζ_{2p}, ζ_{2p})] = 𝒟 ⊕ (2(1 + ζ_{2p})ℤ[ζ_{2p}], 0).

Case π = \sqrt{2} ζ_{8}×−\sqrt{2} ζ_{8}. We have 2(1+i)ℤ[i] = (1+i)^{3}ℤ[i]. So by Lemma 7.7, the suborders of O_{K} properly containing R_{sp} and distinct from O_{K} are

B_{4} := ℤ[(i, i), (2, 0)] ⟷ (1 + i)^{2}ℤ[i] = 2ℤ[i],
B_{2} := ℤ[(i, i), (1 + i, 0)] ⟷ (1 + i)ℤ[i].

Case π = \sqrt{3} ζ_{12} × −\sqrt{3} ζ_{12}. In this case, the ideal 2(1 + ζ_{6})ℤ[ζ_{6}] factors as the product of the prime ideals 2ℤ[ζ_{6}] and \sqrt{−3} ℤ[ζ_{6}]. The suborders of O_{K} properly containing R_{sp} and distinct from O_{K} are

B_{4} := ℤ[(ζ_{6}, ζ_{6}), (2, 0)] ⟷ 2ℤ[ζ_{6}],
B_{3} := ℤ[(ζ_{6}, ζ_{6}), (\sqrt{−3} , 0)] ⟷ \sqrt{−3} ℤ[ζ_{6}].

7.4. Suporders of R_{sp} and class numbers: the nonreal simple case. Assume that π is a supersingular Weil p-number of dimension 2 listed in (7.4). Only the case π = \sqrt{p} ζ_{12} with p ≠ 3 needs to be studied, as the rest have already been covered in Section 5.2. If π = \sqrt{p} ζ_{12}, we have K = ℚ(\sqrt{−p} ,\sqrt{−3} ), and R_{sp} = ℤ[\sqrt{−p} , ζ_{6}]. Since the discriminants of ℚ(\sqrt{−3} ) and ℚ(\sqrt{−p}) are coprime, O_{K} is the compositum of ℤ[ζ_{6}] and O_{ℚ(\sqrt{−p})}. If p = 2 or p ≡ 1 (mod 4), then O_{ℚ(\sqrt{−p})} = ℤ[\sqrt{−p}], and R_{sp} is the maximal order in K. We assume that p ≡ 3 (mod 4) and p ≠ 3 for the rest of this subsection. Note that 2O_{K} ⊆ R_{sp}, and R_{sp}/2O_{K} = ℤ[ζ_{6}]/(2) ≃ 𝔽_{4}, which embeds into O_{K}/2O_{K} ≃ 𝔽_{4} ⊕ 𝔽_{4} diagonally. It follows that R_{sp} and O_{K} are the only orders in O_{K} containing R_{sp}. By (7.2), h(R_{sp}) = 3h(O_{K})/[O_{K}^{×} : R_{sp}^{×}]. It remains to calculate the index [O_{K}^{×} : R_{sp}^{×}].

Lemma 7.8. Let p_{1} and p_{2} be distinct primes with p_{1} ≡ p_{2} ≡ 3 (mod 4), and ϵ be the fundamental unit of F = ℚ(\sqrt{p_{1}p_{2}} ). Then \sqrt{−ϵ} ∈ K = ℚ(\sqrt{−p_{1}} ,\sqrt{−p_{2}} ), and O_{K}^{×} = ⟨\sqrt{−ϵ} ⟩×μ_{K}, the direct product of the free abelian group generated by \sqrt{−ϵ} and the group μ_{K} of roots of unity in K. Moreover, if ϵ ∈ ℤ[\sqrt{p_{1}p_{2}} ], then \sqrt{−ϵ} lies in the ℤ-module ℤ\sqrt{−p_{1}} + ℤ\sqrt{−p_{2}} ⊂ O_{K}; otherwise \sqrt{−ϵ} ≡ (\sqrt{−p_{1}} +\sqrt{−p_{2}} )/2 (mod ℤ\sqrt{−p_{1}} + ℤ\sqrt{−p_{2}} ).

Proof. By Dirichlet’s Unit Theorem, the quotient group O_{K}^{×}/μ_{K} is a free abelian group of rank 1 containing O_{F}^{×}/{±1} ≅ ⟨−ϵ⟩ as a subgroup of finite index. In fact, we have [O_{K}^{×}/μ_{K} : O_{F}^{×}/{±1}] ≤ 2 by [20, Theorem 4.12] as K is a CM-field with maximal totally real subfield F. Since both p_{i} ≡ 3 (mod 4), it follows from [7, (V.1.7)] that the norm N_{F/ℚ}(ϵ) = +1. By [3, Lemma 3], p_{1}ϵ is a perfect square in F^{×}. Write p_{1}ϵ = (x+y\sqrt{p_{1}p_{2}} )^{2} with x, y ∈ ℚ. Then

\sqrt{−ϵ} = \sqrt{p_{1}ϵ} · \frac{−1}{\sqrt{−p_{1}}} = (x + y\sqrt{p_{1}p_{2}} ) · \frac{−1}{\sqrt{−p_{1}}} ∈ ℚ\sqrt{−p_{1}} + ℚ\sqrt{−p_{2}} ⊂ K.

In particular, [O_{K}^{×}/μ_{K} : O_{F}^{×}/{±1}] ≥ 2. It follows that [O_{K}^{×}/μ_{K} : O_{F}^{×}/{±1}] = 2, and O_{K}^{×}/μ_{K} ≅ ⟨\sqrt{−ϵ} ⟩. Hence O_{K}^{×} = ⟨\sqrt{−ϵ} ⟩ × μ_{K}. By our assumption on p_{i}, the prime 2 is unramified in O_{K}. One easily checks that the following statements are equivalent:

(1) ϵ ∈ ℤ[\sqrt{p_{1}p_{2}} ] = ℤ + 2O_{F};

(2) ϵ ≡ 1 (mod 2O_{F});
(3) \sqrt{−ϵ} ≡ 1 (mod 2O_{K});
(4) \sqrt{−ϵ} ∈ ℤ + 2O_{K}.

By Exercise 42(d) of [13, Chapter 2], a ℤ-basis of O_{K} is given by

\{1, \frac{1 +\sqrt{−p_{1}}}{2} , \frac{1 +\sqrt{−p_{2}}}{2} , \frac{(1 +\sqrt{−p_{1}} )(1 +\sqrt{−p_{2}} )}{4}\} .

It follows that

O_{K} ∩ (ℚ\sqrt{−p_{1}} + ℚ\sqrt{−p_{2}} ) = ℤ\sqrt{−p_{1}} + ℤ(\sqrt{−p_{1}} +\sqrt{−p_{2}} )/2;
(ℤ + 2O_{K}) ∩ (ℚ\sqrt{−p_{1}} + ℚ\sqrt{−p_{2}} ) = ℤ\sqrt{−p_{1}} + ℤ\sqrt{−p_{2}} .

Therefore, if ϵ ∈ ℤ[\sqrt{p_{1}p_{2}} ], then \sqrt{−ϵ} ∈ ℤ\sqrt{−p_{1}} + ℤ\sqrt{−p_{2}} . Otherwise \sqrt{−ϵ} lies in ℤ\sqrt{−p_{1}} + ℤ(\sqrt{−p_{1}} + \sqrt{−p_{2}} )/2 but not in ℤ\sqrt{−p_{1}} + ℤ\sqrt{−p_{2}} . Hence \sqrt{−ϵ} ≡ (\sqrt{−p_{1}} +\sqrt{−p_{2}} )/2 (mod ℤ\sqrt{−p_{1}} + ℤ\sqrt{−p_{2}} ) in this case. □

We return to the assumption that K = ℚ(\sqrt{−p} ,\sqrt{−3} ) with p ≡ 3 (mod 4) and p ≠ 3. Note that μ_{K} = ⟨ζ_{6}⟩ ⊂ R_{sp}^{×}, and R_{sp} ∩ (ℚ\sqrt{−p} + ℚ\sqrt{−3} ) = (ℤ\sqrt{−p} +ℤ\sqrt{−3} ). Let ϵ be the fundamental unit of F = ℚ(\sqrt{3p}). If ϵ ∈ ℤ[\sqrt{3p}], then \sqrt{−ϵ} ∈ R_{sp}, and hence R_{sp}^{×} = O_{K}^{×}. This holds in particular when p ≡ 3 (mod 8) and p ≠ 3 as remarked after (1.2). Assume that p ≡ 7 (mod 8) and ϵ ∉ ℤ[\sqrt{3p}]. Then (O_{F} /2O_{F})^{×} ≃ 𝔽_{4}^{×} and ϵ^{3} ∈ ℤ + 2O_{F} = ℤ[\sqrt{3p}]. On the other hand, [(O_{K}/2O_{K})^{×} : (R_{sp}/2O_{K})^{×}] = [(𝔽_{4}^{×})^{2} : 𝔽_{4}^{×}] = 3, so we have \sqrt{−ϵ} ∉ R_{sp} but (\sqrt{−ϵ})^{3} ∈ R_{sp}. In summary, we find that

[O_{K}^{×} : R_{sp}^{×}] = [O_{F}^{×} : ℤ[\sqrt{3p}]^{×}] = \begin{cases}1 & if ϵ ∈ ℤ[\sqrt{3p}]; \\ 3 & otherwise. \end{cases}

Therefore, we have h(R_{sp}) = ϖ_{3p}h(O_{K}), where ϖ_{3p} = 3/[O_{F}^{×} : ℤ[\sqrt{3p}]^{×}] as defined in (1.2).

Acknowledgements

J. Xue is partially supported by the 1000-plan program for young talents of PRC. He thanks Academia Sinica and NCTS for their hospitality and great working conditions. TC Yang and CF Yu are partially supported by the grants MoST 100-2628-M-001-006-MY4, 103-2811-M-001-142, 104-2115-M-001-001MY3 and 104-2811-M-001-066. The authors thank the referee for careful reading and helpful suggestions.

References

[1] Z. I. Borevič and D. K. Faddeev. Representations of orders with cyclic index. Trudy Mat. Inst. Steklov, 80:51–65, 1965.

[2] Wieb Bosma, John Cannon, and Catherine Playoust. The Magma algebra system. I. The user language. J. Symbolic Comput., 24(3-4):235–265, 1997. Computational algebra and number theory (London, 1993).

[3] D. A. Buell, H. C. Williams, and K. S. Williams. On the imaginary bicyclic biquadratic fields with class-number 2. Math. Comp., 31(140):1034–1042, 1977.

[4] Tommaso Giorgio Centeleghe and Jakob Stix. Categories of abelian varieties over finite fields, I: Abelian varieties over 𝔽_{p}. Algebra Number Theory, 9(1):225–265, 2015.

[5] Charles W. Curtis and Irving Reiner. Methods of representation theory. Vol. I. Wiley Classics Library. John Wiley & Sons, Inc., New York, 1990. With applications to finite groups and orders, Reprint of the 1981 original, A Wiley-Interscience Publication.

[6] M. Eichler. Über die Idealklassenzahl hyperkomplexer Systeme. Math. Z., 43(1):481–494, 1938.

[7] A. Fröhlich and M. J. Taylor. Algebraic number theory, volume 27 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1993.

[8] Kuniaki Horie and Mitsuko Horie. CM-fields and exponents of their ideal class groups. Acta Arith., 55(2):157–170, 1990.

[9] Loo Keng Hua. Introduction to number theory. Springer-Verlag, Berlin-New York, 1982. Translated from the Chinese by Peter Shiu.

[10] Tomoyoshi Ibukiyama, Toshiyuki Katsura, and Frans Oort. Supersingular curves of genus two and class numbers. Compositio Math., 57(2):127–152, 1986.

[11] Lawrence S. Levy and Roger Wiegand. Dedekind-like behavior of rings with 2-generated ideals. J. Pure Appl. Algebra, 37(1):41–58, 1985.

[12] Ke-Zheng Li and Frans Oort. Moduli of supersingular abelian varieties, volume 1680 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1998.

[13] Daniel A. Marcus. Number fields. Springer-Verlag, New York, 1977. Universitext.

[14] Jürgen Neukirch. Algebraic number theory, volume 322 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1999.

[15] Jean-Pierre Serre. Galois cohomology. Springer Monographs in Mathematics. Springer-Verlag, Berlin, english edition, 2002. Translated from the French by Patrick Ion and revised by the author.

[16] Sheng-Chi Shih, Tse-Chung Yang, and Chia-Fu Yu. Embeddings of fields into simple algebras over global fields. Asian J. Math., 18(2):365–386, 2014.

[17] John Tate. Endomorphisms of abelian varieties over finite fields. Invent. Math., 2:134–144, 1966.

[18] John Tate. Classes d’isogénie des variétés abéliennes sur un corps fini (d’après T. Honda). In Séminaire Bourbaki. Vol. 1968/69: Exposés 347– 363, volume 175 of Lecture Notes in Math., pages Exp. No. 352, 95–110. Springer, Berlin, 1971.

[19] Marie-France Vignéras. Arithmétique des algèbres de quaternions, volume 800 of Lecture Notes in Mathematics. Springer, Berlin, 1980.

[20] Lawrence C. Washington. Introduction to cyclotomic fields, volume 83 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1997.

[21] William C. Waterhouse. Abelian varieties over finite fields. Ann. Sci. École Norm. Sup. (4), 2:521–560, 1969.

[22] Jiangwei Xue, Tse-Chung Yang, and Chia-Fu Yu. Supersingular abelian surfaces and Eichler’s class number formula. arXiv:1404.2978v3. 29 pp.

[23] Jiangwei Xue, Tse-Chung Yang, and Chia-Fu Yu. Numerical invariants of totally imaginary quadratic ℤ[\sqrt{p}]-orders. Taiwanese J. Math., 20(4):723– 741, 2016.

[24] Chia-Fu Yu. On arithmetic of the superspecial locus. arXiv:1210.1120v2. 35 pp.

[25] Chia-Fu Yu. Simple mass formulas on Shimura varieties of PEL-type. Forum Math., 22(3):565–582, 2010.

[26] Chia-Fu Yu. Superspecial abelian varieties over finite prime fields. J. Pure Appl. Algebra, 216(6):1418–1427, 2012.

[27] Chia-Fu Yu. Endomorphism algebras of QM abelian surfaces. J. Pure Appl. Algebra, 217(5):907–914, 2013.

[28] Don Zagier. On the values at negative integers of the zeta-function of a real quadratic field. Enseignement Math. (2), 22(1-2):55–95, 1976.

Jiangwei Xue Collaborative Innovation Centre of Mathematics School of Mathematics and Statistics Wuhan University Luojiashan Wuhan, Hubei 430072 P.R. China. xue j@whu.edu.cn Tse-Chung Yang Institute of Mathematics Academia Sinica Astronomy-Mathematics Building 6F, No. 1, Sec. 4 Roosevelt Road Taipei 10617 TAIWAN tsechung@math.sinica.edu.tw

Chia-Fu Yu

Institute of Mathematics Academia Sinica and NCTS Astronomy-Mathematics Building No. 1, Sec. 4 Roosevelt Road Taipei 10617 TAIWAN chiafu@math.sinica.edu.tw 1644

Documenta Mathematica 21 (2016)