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Extending the results of [24, Asian J. Math.], in [26, Doc. Math. 21, 2016] we calculated explicitly the number of isomorphism classes of superspecial abelian surfaces over an arbitrary finite field of odd degree over the prime field Fp. A key step was to reduce the calculation to the prime field case, and we calculated the number of isomorphism classes in each isogeny class through a concrete lattice description. In the present paper we treat the even degree case by a different method. We first translate the problem by Galois cohomology into a seemingly unrelated problem of computing conjugacy classes of elements of finite order in arithmetic subgroups, which is of independent interest. We then explain how to calculate the number of these classes for the arithmetic subgroups concerned, and complete the computation in the case of rank two. This complements our earlier results and completes the explicit calculation of superspecial abelian surfaces over finite fields.Date: October 4, 2018. 2010 Mathematics Subject Classification. 11R52, 11G10. 1 This is well known when the ground field k is algebraically closed, and is proved in [29] for arbitrary k 1 Lemma 6.1. Suppose that S × ⊆ N ( R), the normalizer of R in B × . Then the suborder R J := x Rx −1 ∩ B of S J is independent of the choice of x ∈ B × for J, and.Proof. Suppose that J = x ′ S for x ′ ∈ B × as well. Then there exists u ∈ S × such that x ′ = xu. Since S × ⊆ N ( R), we havewhich proves the independence of R J of the choice of x. If I is a locally principal right R-ideal such that IS = J, then R J = O l (I), the associated left order of I.Conjugating by x ∈ B × on the right hand side of (6.3), we obtainThe assumption S × ⊆ N ( R) also implies that R × S × , and hence R × J S × J and R × J S × J . The left action of S × J on the quotient group S × J / R × J factors through S × J /R × J ⊆ S × J / R × J , and its orbits are the right cosets of S × JRemark 6.2. The condition S × ⊆ N ( R) implies that R × S × . However, the converse does not hold in general. It is enough to provide a counterexample locally Corollary 6.3. Keep the notation and assumption of Lemma 6.1. If the natural homomorphism S ×Proof. It is enough to show that π is injective. The surjectivity of S × J → S × J / R × J implies that the monomorphism S × J /R × J ֒→ S × J / R × J is an isomorphism, and hence |π −1 ([J])| = [ S × J / R × J : S × J /R × J ] = 1.

Let A be a real quadratic order of discriminant p or 4p with a prime p. In this paper we classify all proper totally imaginary quadratic A-orders B with index w(B) = [B × : A × ] > 1. We also calculate numerical invariants of these orders including the class number, the index w(B) and the numbers of local optimal embeddings of these orders into quaternion orders. These numerical invariants are useful for computing the class numbers of totally definite quaternion algebras.

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Let F be a totally real field with ring of integers O F , and D be a totally definite quaternion algebra over F . A well-known formula established by Eichler and then extended by Körner computes the class number of any O F -order in D. In this paper we generalize the Eichler class number formula so that it works for arbitrary Z-orders in D. Our motivation is to count the isomorphism classes of supersingular abelian surfaces in a simple isogeny class over a finite prime field Fp. We give explicit formulas for the number of these isomorphism classes for all primes p.