Class number formulas for the norm one group of totally definite quaternion algebras

Abstract

We present two class number formulas associated to orders in totally definite quaternion algebras in the spirit of the Eichler class number formula. More precisely, let F be a totally real number field, D be a totally definite quaternion F -algebra, and O an O F -order in D. We derive explicit formulas for the following two class numbers associated to O: (1) the class number of the reduced norm one group with respect to O, namely, the cardinality of the double coset space D 1 \ D 1 / O 1 ; (2) the number of locally principal right O-ideal classes within the spinor class of the principal right O-ideals, that is, the cardinality ofBoth class numbers depend only on the spinor genus of O, hence the title of the present paper. The formulas work unconditionally for O F -orders with nonzero Eichler invariants at all finite places of F (e.g. Eichler orders of arbitrary level), and are extendable to other classes of orders as long as certain local condition for optimal embeddings is satisfied. The proofs are made possible by optimal spinor selectivity for quaternion orders.

Optimal spinor selectivity for quaternion Bass orders

“…Part (II) of Theorem 2.6 follows directly from [26,Theorem 2.11]. To prove the first part, we reduce it to local considerations as well.…”

“…If A satisfies the Eichler condition, then "optimal spinor selectivity" reduces to "optimal selectivity" by Remark 2.2. It was observed in [26] that understanding optimal spinor selectivity plays a crucial role in computing certain class numbers associated to orders in totally definite quaternion algebras. For this reason, we focus on optimal spinor selectivity in this paper.…”

Polarized superspecial simple abelian surfaces with real Weil numbers

“…Moreover, (4.26) T 1 [P, P + ] = Tp + (D) if [P, P + ] ∈ Pic + (O F ) 2 ; Tp − (D) otherwise.Proof. In light of (4.18) and Remarks 4.6 and 4.8, the bijectivity of Υ un : Λ un 1 [P, P + ] → T 1 [P, P + ] is just a restatement of[64, Lemma 4.7].To prove (4.26), let us fix a principally polarizable abelian surface X 0 /F p with π2 X0 = p and put O 0 := End(X 0 ). Then O 0 ∈ Tp + (D) by Corollary 3.3.7, and P(X) ≃ (O F , O F,+ ) by Corollary 4.3.…”