The spinor type number formula for totally definite quaternion 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.