Spinor class number formulas for totally definite quaternion orders

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 $mathcal{O}$ be an $O_F$-order in $D$. Assume that $mathcal{O}$ has nonzero Eichler invariants at all finite places of $F$ (e.g. $mathcal{O}$ is an Eichler order of arbitrary level). We derive explicit formulas for the following two class numbers associated to $mathcal{O}$: (1) the class number of the reduced norm one group with respect to $mathcal{O}$, namely, the cardinality of the double coset space $D^1backslashwidehat{D}^1/widehat{mathcal{O}}^1$; (2) the number of locally principal right $mathcal{O}$-ideal classes within the spinor class of the principal right $mathcal{O}$-ideals, that is, the cardinality of $D^timesbackslashbig(D^timeswidehat{D}^1widehat{mathcal{O}}^timesbig)/widehat{mathcal{O}}^times$. Both class numbers depend only on the spinor genus of $mathcal{O}$, hence the title of the present paper. The proofs are made possible by optimal spinor selectivity for quaternion orders.