We 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
vβ£p of F that is not split in K and for which Vvβ is anisotropic, let
Ξ½ be an extension of v to the reflex field E. We define an integral
model of the corresponding Shimura curve over SpecOE,(Ξ½)β 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
first two conditions are conditions on the Lie algebra of the abelian
varieties; the last condition is a condition on the Riemann form of the
polarization. The uniformization of the formal completion of this model along
its special fiber is given in terms of the formal Drinfeld upper half plane for
Fvβ. The proof relies on the construction of the contracting functor which
relates a relative Rapoport-Zink space for strict formal OFvββ-modules with
a Rapoport-Zink space of p-divisible groups which arise from the moduli
problem, where the OFvββ-action is usually not strict when Fvβξ =Qpβ. Our main tool is the theory of displays, in particular the Ahsendorf
functor.Comment: 145 page