The authors prove \(p\)-adic uniformization for Shimura curves attached to the group of unitary similitudes of certain binary skew Hermitian spaces \(V\) with respect to an arbitrary CM field \(K\) with maximal totally real subfield \(F\). For a place \(v \vert p\) of \(F\) that is not split in \(K\) and for which \(V_v\) is anisotropic, let \(\nu\) be an extension of \(v\) to the reflex field \(E\).

The authors define an integral model of the corresponding Shimura curve over \(\mathrm{Spec} O_{E,(\nu)}\)by means of a moduli problem for abelian schemes with suitable polarization and level structure prime to \(p\). The formulation of the moduli problem involves a Kottwitz condition, an Eisenstein condition, and an adjusted invariant. The uniformization of the formal completion of this model along its special fiber is given in terms of the formal Drinfeld upper half plane \(\widehat\Omega_{F_v}\) for \(F_v\).

The proof relies on the construction of the contracting functor which relates a relative Rapoport-Zink space for strict formal \(O_{F_v}\)-modules with a Rapoport-Zink space of \(p\)-divisible groups which arise from the moduli problem, where the \(O_{F_v}\)-action is usually not strict when \(F_v\ne \mathbb{Q}_p\). The authors' main tool is the theory of displays, in particular the Ahsendorf functor.