The spinor type number formula for totally definite quaternion orders
Yucui Lin, Jiangwei Xue
TL;DR
This work refines the classical type-number theory for totally definite quaternion $O_F$-orders by introducing a spinor-genus refinement and a spinor type-number formula. Central to the approach is the action of the central Picard group ${\mathrm{Picent}}({\mathcal{O}})$ on the ideal class set, combined with restricted optimal embeddings and the spinor selectivity framework, to compute the number of types within a spinor genus, $t_{\mathrm{sg}}({\mathcal{O}})$. Under a residually unramified hypothesis, the authors derive an explicit spinor type-number formula involving masses and local embedding data, and establish divisibility of the (spinor) type numbers by the wide spinor-genus group $|\mathrm{WSG}({\mathcal{O}})|$ (and by $|\mathrm{SG}({\mathcal{O}})|$ when $D$ ramifies at a finite place). They further connect these arithmetic invariants to the trace of Brandt matrices and, via the even Clifford correspondence, to ternary $O_F$-l lattices, showing that $t_{\mathrm{sg}}({\mathcal{O}}_L)$ equals the spinor class number of the lattice $L$, with analogous divisibility results for lattice class numbers. These results generalize earlier divisibility theorems for maximal or Eichler orders and illuminate deep links between quaternion orders, spinor genera, and ternary quadratic forms.
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, Körner 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.