On reduction of moduli schemes of abelian varieties with definite quaternion multiplications

In this paper we make an initial study on type D moduli spaces in positive characteristic $p\neq 2$, where we allow $p$ ramified in the definite quaternion algebra. We classify the isogeny classes of $p$-divisible groups with additional structures in question. We also study the reduction of the type $D$ moduli spaces of minimal rank.