The spinor type number formula for totally definite quaternion orders

TL;DR

This paper refines the formula for counting quaternion orders within the same spinor genus, revealing divisibility properties of type numbers and class numbers related to the algebra's arithmetic invariants.

Contribution

It provides a new spinor genus refined type number formula for a broad class of quaternion orders, extending previous divisibility results to more general settings.

Findings

Type number $t(
)(0)$ divisible by a quotient of the Gauss genus group

Trace of Brandt matrix divisible by class number $h(F)$

Class number $h(
)$ divisible by $h(F)$ for certain orders

Abstract

Let $D$ be a totally definite quaternion algebra over a totally real number field $F$, and $\mathcal{O}$ be an $O_F$-order (of full rank) in $D$. The type number $t(\mathcal{O})$ is an important arithmetic invariant of $\mathcal{O}$ that counts the number of isomorphism classes of orders belonging to the same genus as $\mathcal{O}$ (i.e. locally isomorphic to $\mathcal{O}$ at every finite place $\mathfrak{p}$ of $F$). The type number formula has been studied by Eichler, Peters, Pizer, Vigneras, K\"orner and many others. As the genus of $\mathcal{O}$ further divides into spinor genera, one naturally seeks a finer type number formula for the number of isomorphism classes of orders belonging to the same spinor genus of $\mathcal{O}$. The main goal of this paper is to provide such a refinement for a large class of quaternion $O_F$-orders $\mathcal{O}$ that includes all Eichler orders. This enables us to prove that $t(\mathcal{O})$ is divisible by the order of a quotient group $\mathrm{WSG}(\mathcal{O})$ of the Gauss genus group $\mathrm{Cl}^+(O_F)/\mathrm{Cl}^+(O_F)^2$ naturally attached to $\mathcal{O}$. Similarly, we show that the trace of the $\mathfrak{n}$-Brandt matrix $\mathfrak{B}(\mathcal{O}, \mathfrak{n})$ is divisible by the class number $h(F)$ for any nonzero integral $O_F$-ideal $\mathfrak{n}$. In particular, the class number $h(\mathcal{O})=\mathrm{Tr}(\mathfrak{B}(\mathcal{O}, O_F))$ is always divisible by $h(F)$ for such quaternion orders. This generalizes the divisibility result of $h(\mathcal{O})$ proved in a different way by Chia-Fu Yu and the second named author [Indiana Univ. Math. J., Vol. 70, No. 2 (2021)] in the case when $\mathcal{O}$ is a maximal $O_F$-order in a totally definite quaternion algebra unramified at all the finite places.