On superspecial abelian surfaces and type numbers of totally definite quaternion algebras
Abstract.
In this paper we determine the number of endomorphism rings of superspecial abelian surfaces over a field of odd degree over in the isogeny class corresponding to the Weil number . This extends earlier works of T.C. Yang and the present authors on the isomorphism classes of these abelian surfaces, and also generalizes the classical formula of Deuring for the number of endomorphism rings of supersingular elliptic curves. Our method is to explore the relationship between the type and class numbers of the quaternion orders concerned. We study the Picard group action of the center of an arbitrary order in a totally definite quaternion algebra on the ideal class set of said order, and derive an orbit number formula for this action. This allows us to prove an integrality assertion of Vignéras [Enseign. Math. (2), 1975] as follows. Let be a totally real field of even degree over , and be the (unique up to isomorphism) totally definite quaternion algebra unramified at all finite places of . Then the quotient of the class numbers is an integer.
Key words and phrases:
superspecial abelian surfaces, type numbers, class numbers, totally definite quaternion algebras.2010 Mathematics Subject Classification:
11R52, 11G101. Introduction
Throughout this paper denotes a prime number. Let denote the unique definite quaternion algebra up to isomorphism ramified exactly at and . The classical result of Deuring establishes a bijection between the set of isomorphism classes of supersingular elliptic curves over and the set of ideal classes of a maximal order in . The class number of is well known due to Eichler [6] (Deuring and Igusa gave different proofs of this result), and is given by
(1.1) 
where denotes the Legendre symbol. Under the correspondence , the type number of is equal to the number of nonisomorphic endomorphism rings of members in . An explicit type formula is also well known due to Deuring [5], which is given by
(1.2) 
for , and for . Here for any squarefree integer , we write for the class number of . Though these classical results were well established by 1950, different proofs with various ingredients such as mass formulas, Tamagawa numbers, theta series, cusp forms, algebraic modular forms, AtkinLehner involutions and traces of Hecke operators, have been generalized and revisited many times even until now. Different angles and approaches such as the EichlerShimizuJacquetLanglands correspondence and trace formulas also play important roles in the development. This paper is one instance of them, where we would like to generalize the explicit formulas (1.1) and (1.2) from to , which is the unique totally definite quaternion algebra (up to isomorphism) unramified at all finite places. Recall that a quaternion algebra over a totally real number field is said to be totally definite if is isomorphic to the Hamilton quaternion algebra for every embedding . Our interest for totally definite quaternion algebras stems from the study abelian varieties over finite fields. Note that and are the only two algebras that can occur as the endomorphism algebra of an abelian variety over a finite field [31], but do not satisfy the Eichler condition [27, Definition 34.3]. The cases where the endomorphism algebras satisfy the Eicher condition are easier to treat. Indeed, by a result of Jacobinski [16, Theorem 2.2], the class number of an order in such an algebra is equal to a ray class number of its center. Observe that the number in (1.1) (resp. in (1.2)) is also equal to the number of isomorphism classes (resp. nonisomorphic endomorphism rings) of supersingular elliptic curves in the isogeny class corresponding to the Weil number (or ), for any field containing . Thus, the present work may be also reviewed as an generalization of explicit formulas (1.1) and (1.2) in the arithmetic direction. A generalization of the geometric direction in the sense that the set is replaced by the superspecial locus of the Siegel moduli spaces has been inverstigated by Hashimoto, Ibukiyama, Katsura and Oort [10, 11, 12, 14, 17].
Now let be an odd power of , and let be the set of isomorphism classes of superspecial abelian surfaces over in the isogeny class corresponding to the Weil number . As a generalization of (1.1), T.C. Yang and the present authors [37, Theorem 1.2] (also see [39, Theorem 1.3]) show the following explicit formula for .
Theorem 1.1.
Let , and be its ring of integers.
(1) The cardinality of depends only on , and is denoted by .
(2) We have for , respectively.
(3) For and , one has
where is the class number of and is the Dedekind zeta function of .
(4) For and , one has
where and is the suborder of index in .
Let denote the set of isomorphism classes of endomorphism rings of abelian surfaces in . The cardinality of again depends only on the prime ([39, Theorem 1.3], see also Sect. 3), and is denoted by . In this paper we give an explicit formula for , which generalizes (1.2).
Theorem 1.2.
Let and .
(1) We have for , respectively.
(2) For and , we have
(1.3) 
(3) For and , we have
(1.4) 
It follows from Theorems 1.1 and 1.2 that except for the case where and . When , we actually prove this result first and use it to get formula (1.3). For the case where , we explain how this coindence arises in part (1) of Remark 4.3.
Similar to Deuring’s result for supersingular elliptic curves, the proof of Theorem 1.2 is reduced to the calculation of the type number of a maximal order in in , as well as those of certain proper orders and (see (4.1) and (4.2) for definition of these orders). We recall briefly the concept of proper orders. Henceforth all orders are assumed to be of full rank in their respective ambient algebras. Let be a number field, and be a order in (not necessarily maximal). A order in a finite dimensional semisimple algebra is said to be a proper order if . Unlike [4, Definition 23.1], we do not require to be a projective module. The class number of is denoted by . When is a totally real number field, a proper order in a CMextension (i.e. a totally imaginary quadratic extension) of is call a CM proper order. If coincides with , then we drop the adjective “proper” and simply write “order”.
It is well known that the type number of a totally definite Eichler order of squarefree level can be calculated by Eichler’s trace formula [7]; see also [18] for general orders. Some errors of Eichler’s formula were found and corrected later independently by M. Peters [23] and by A. Pizer [24]. Eichler’s trace formula contains a number of data which are generally not easy to compute. In [33, p. 212] M.F, Vignéras gave an explicit formula for the type number of any totally definite quaternion algebra , over any real quadratic field , that is unramified at all finite places. Her formula was based on the explicit formula for the class number in [32, Theorem 3.1] and the classtype number relationship ; see [33, p. 212]. Unfortunately, it was pointed out by Ponomarev in [26, Concluding remarks, p. 103] that the formula in [33, p. 212] is not a formula for in general. This conclusion is based on his explicit calculations for class numbers of positive definite quaternary quadratic forms [26, Sect. 5], and the correspondence between quaternary quadratic forms and types of quaternion algebras established in [25, Sect. 4]. The source of the difficulty is that the classtype number relationship may fail in general^{1}^{1}1In [26, p. 103] it reads “the class number of divided by the proper class number of is not, in general, the type number ”. However it should read “the class number” instead of “the proper class number” as the former one is what Vignéras’ formula is based on., even if is unramified at all finite places. To remedy this, we examine more closely the Picard group action described below.
Let be an arbitrary number field, a order in , and a proper order in a quaternion algebra . The Picard group acts naturally on the finite set of locally principal right ideal classes of by the map
(1.5) 
where (resp. ) denotes a locally principal fractional ideal in (resp. right ideal in ), and (resp. ) denotes its ideal class. Let be the set of orbits of this action, and be its cardinality:
(1.6) 
One of main results of the this paper is a formula for when is a totally definite quaternion algebra over a totally real field.
Theorem 1.3.
If , then the formula for can be simplified further (see Theorem 3.7).
Assume that is a totally real field of even degree over , and is the unique up to isomorphism quaternion algebra unramified at all finite places of . Let be a maximal order in . Then , and hence Theorem 1.3 leads to a type number formula for such in Corollary 3.11. In [32, Remarque, p. 82], Vigénras asserted that is always an integer. However, the assertion was mixed with the misconception that holds unconditionally on , and we could not locate a precise proof of this integrality elsewhere. As an application of our orbit number formula, we prove in Theorem 5.4 that for all . On the other hand, we give in Corollary 3.5 a necessary and sufficient condition on such that the relationship remains valid. In particular, for real quadratic fields satisfying this condition, Vigénras’s formula [33, p. 212] does give a formula for the type number . This approach of calculating the type number via the class number also paves the way to our proof of Theorem 1.2, where we treat certain orders (namely, and ) that does not contain . The type numbers of such orders are not covered by previous methods of EichlerPizer and Ponomarev.
This paper is organized as follows. In Section 2 we provide some preliminary studies on the action on . This is carried out more in depth for totally definite quaternion algebras in Section 3, and we derive the orbit number formula and its corollaries there. The calculations for Theorem 1.2 are worked out in Section 4, and we prove the integrality of in Section 5.
2. Preliminaries on the Picard group action
Let be a number field, its ring of integers, and a order in . Let be a quaternion algebra and a proper order in . This section provides a preliminary study of the action on in (1.5).
We follow the notation of [37, Sections 2.1–2.3]. Recall that admits a canonical involution such that and are respectively the reduced trace and reduced norm of . The reduced discriminant of is the product of all finite primes of that are ramified in . Let be the norm of over . More explicitly, is the submodule of spanned by the reduced norms of elements of . Clearly, is closed under multiplication, hence a suborder of containing . By [37, Lemma 3.1.1], is closed under the canonical involution if and only if . In particular, any order in is closed under the canonical involution. We have a the natural surjective map between the Picard groups:
(2.1) 
Given an ideal class , we study the stabilizer of the action on as in (1.5). Let be the inverse of , and the associated left order of . We have [34, Section I.4]. Thus
(2.2) 
where is the trivial ideal class of . Therefore, the study of often reduces to that of . We have
(2.3) 
Since is a nonzero twosided ideal, lies in the normalizer by [34, Exercise I.4.6]. To study more closely the relationship between and , we make use of the following lemma from commutative algebra.
Lemma 2.1.
Let be an extension of unital rings, with commutative and a finite module.

If is an ideal with , then .

Let be the totally quotient ring of . Suppose that the natural map is injective, and in . If is an submodule with for some , then .
Proof.
(i) Since is a finite module, the equality
implies that there exists such that
by [8, Corollary 4.7]. Necessarily since ,
and hence and .
(ii) Let . Then , which implies
that . We have , and hence
is an integral ideal of . Now it follows from part (i)
that , equivalently, .
∎
We return to the study of the action on .
Corollary 2.2.
Suppose that is a locally principal nonzero fractional ideal with for some .

If , then .

If , then with . In particular, belongs to the kernel of the canonical map .
Proof.
Part (i) follows directly from Lemma 2.1(ii) with , and , and part (ii) follows with , , and . ∎
We say a locally principal fractional ideal of capitulates in if the extended ideal is principal. The capitulation problems (for abelian extensions of number fields with and ) was studied by Hilbert (see Hilbert Theorem 94), and it continues to be a field of active research up to this day [15, 2, 30]. We will follow up on this line of investigation in Section 3, particularly in the derivation of the orbit number formula (see Theorem 3.7). However, our result does not explicitly depend on the works just cited.
Lemma 2.3.
For any ideal class , the stabilizer is contained in the kernel of the homomorphism
(2.4) 
Proof.
Clearly, commutes with any localization of . Thus for every proper order and locally principal fractional ideal . Since is locally principal, is a proper order with by [37, Section 2.3]. Suppose that for some . Taking on both sides, we get
(2.5) 
and hence . ∎
Corollary 2.4.
Let be the subgroup of consisting of the ideal classes that are perfect squares. Then the number is an integer.
Proof.
The map factors as . By Lemma 2.3, each orbit of has cardinality divisible by . The corollary follows. ∎
Corollary 2.5.
Suppose that is closed under the canonical involution and is odd. Then the action of on is free.
Proof.
As remarked before, is closed under the canonical involution if and only if . The two conditions imply that is trivial, so the corollary follows from Lemma 2.3. ∎
Remark 2.6.
In some cases, the Picard group action (1.5) can be utilized to calculated type numbers of certain orders. Concrete examples will be worked out in Section 4. To explain the ideas, we adopt the adelic point of view. Let be the profinite completion of , where the product runs over all primes . Given any finite dimensional vector space or finitely generated module , we set , and . Two orders and in the quaternion algebra are said to be in the same genus if , or equivalently, if there exists such that . They are said to be of the same type if for some . Let denote the set of conjugacy classes of orders in the genus of , and let denote the type number of . For example, it is well known that all maximal orders of lie in the same genus. If is a maximal order, then the set (resp. its cardinality ) depends only on and is denoted by (resp. ) instead.
Let be the normalizer of , which admits a filtration . It is easy to see that , and
(2.6)  
(2.7)  
(2.8) 
Denote the normal subgroup of by .
Lemma 2.7.
Keep the notation of (2.3). There is a canonical isomorphism
Proof.
There is a natural surjective map
(2.9) 
where denotes the isomorphism class of (equivalently, the conjugacy class of ). Clearly, factors through the projection .
Proposition 2.8.
If , then . If further is stable under the canonical involution and is odd, then
(2.10) 
Moreover, for any order in the genus of , we have
(2.11) 
Proof.
The assumption allows the canonical identification , and hence identifications of the orbits of the action with the fibers of . In particular, we have . By definition, for any order in the genus of . Hence .
3. Picard group action for totally definite quaternion algebras
Throughout this section, we assume that is a totally real field, and is a totally definite quaternion algebra. In particular, for any with , the algebra is a CMextension of . Let be a order in , and be a proper order in . The main goal of this section is to derive a orbit number formula for the action on . This also leads to a type number formula for when has even degree over and is unramified at all finite places of .
Notation 3.1.
Let be an extension of commutative Noetherian rings. The canonical homomorphism between the Picard groups is denoted by
If is a finite extension of number fields, we write for . Because of Corollary 2.2, we are interested in the following set of isomorphism classes of CM proper orders:
(3.1) 
where denotes the isomorphism class of .
Proposition 3.2.
The set is finite.
Proof.
Let be a CM proper order with and be its fractional field. Pick a nontrivial ideal class so that
(3.2) 
Necessarily , otherwise by part (ii) of Lemma 2.1.
There is a commutative diagram
If , then is a nontrivial ideal class in . By [3, Section 14], there are only finitely many CMextensions such that . On the other hand, suppose that , so that for some . We have , and hence for some . Note that since . There are only finitely many CMextensions such that . Indeed, let be the group of roots of unity in . There are only finitely many CMextensions such that . If , then if and only if is a CMextension of type II (i.e. ), and there are finitely many of these by [3, Lemma 13.3]. We conclude that the following set of CMextensions of is finite:
Now fix and a nontrivial ideal class . Pick such that . The CM proper orders such that are characterized in Lemma 3.3 below. For each with , we have , so forms an lattice in . There are only finitely many orders in containing . On the other hand, there is onetoone correspondence between and the set of submodules of the form in . Since is a CMextension, is a finite group. Therefore, there are only finitely many CM proper order in with . Since is finite as well, the proposition follows. ∎
Lemma 3.3.
Let be a CMextension with , and be a nontrivial ideal class in so that for some . Given a CM proper order , we have if and only if there exists such that^{2}^{2}2Up to multiplication by a unit in , the submodule depends only on the ideal class and not on the choice of . . In addition, we always have if such a unit exists.
Proof.
First, suppose that so that (3.2) holds. We have , and hence for some . It follows that
As remarked right after (3.2), since is assumed to be nontrivial.
Conversely, suppose that there exists such that . Let , the ideal of generated by . Then . It follows from Lemma 2.1 that , i.e. . ∎
When coincides with the maximal order in , we give a more precise characterization of orders in with in Lemma 5.1.
Let be an arbitrary CM proper order with fractional field , and be the unique nontrivial involution of . We define a symbol
(3.3) 
Note that for all orders . If is stable under the canonical involution of , then so is for every locally principal right ideal , in which case for every with .
Lemma 3.4.
If is a CM proper order with , then .
Proof.
This result is well known when and . In fact, the proof of [35, Theorem 10.3] applies, mutatis mutandis, to the current setting as well and shows that . ∎
Lemma 3.4 does not hold in general when . We exhibit in Example 4.4 an infinite family of pairs such that .
Denote by the set of optimal embeddings from into :
Fix a nontrivial ideal class . We define a few subsets of :
(3.4)  
(3.5)  
(3.6) 
Corollary 2.2 implies that for any ideal class ,
(3.7) 
Suppose that for some ideal class so that . It then follows from (2.5) that for some , where is the suborder of spanned over by the reduced norms of elements of . Note that is totally positive since is totally definite over . For any order in , Let be the quotient of the multiplicative group of locally principal fractional ideals by the subgroup of principal fractional ideals that are generated by totally positive elements. If , then is simply the narrow class group of . We denote the subgroup of 2torsions of by , and the canonical map by . For any order intermediate to , let us define
(3.8) 
If , then necessarily as well, and we simply write for , which coincides with the image of in . Clearly, is a subgroup of , where is the homomorphism defined in (2.4). By our construction,
(3.9) 
Using the norm map for CMextensions , one shows similarly that if . We claim that this is in fact an “if and only if” condition when . If is a nonprincipal fractional ideal such that for some totally positive , then is a CMextension of with (cf. [3, Theorem 14.2]). Thus if and only if . Combining this with Lemma 3.4, we obtain that
(3.10) 
Therefore, if and only if is trivial.
Corollary 3.5.
Let be a totally real number field of even degree over , and be a maximal order in the (unique up to isomorphism) totally definite quaternion algebra unramified at all finite places of . The following are equivalent:
(1) the group is trivial;
(2) the action of on is free;
(3) .
Moreover, if is odd, then the statements (1)–(3) hold.
Proof.
By [34, Section II.2], we have in this case, and hence . The equality holds if and only if the action of on is free, establishing the equivalence between (2) and (3).
For the equivalence between (1) and (2), it is enough to show that the latter holds if and only if . The sufficiency is clear by (3.7). As is unramified at all finite places, any CM extension of can be embedded into by the localglobal principle. If , then there exists a CMextension of such that . Fix an embedding from such a into . The image is contained in a maximal order . Set , which is a fractional right ideal with . We have , and hence by (3.7). Therefore, acts freely on if and only if . ∎
It is possible that is trivial despite being even. Indeed, one of such examples is given by (see [3, Section 12, p. 64]). We provide two applications of Corollary 3.5. First, if is a real quadratic field satisfying condition (1) of Corollary 3.5 and is a totally definite quaternion algebra unramified at all finite places of , then an explicit formula for of is given in [33, p. 212].
Second, recall that