P -adic Uniformization of Unitary Shimura Varieties

Introduction Let Γ ⊂ PGUd−1,1(R) 0 be a torsion-free cocompact lattice. Then Γ acts on the unit ball B ⊂ C by holomorphic automorphisms. The quotient Γ\B is a complex manifold, which has a unique structure of a complex projective variety XΓ (see [Sha, Ch. IX, §3]). Shimura had proved that when Γ is an arithmetic congruence subgroup, XΓ has a canonical structure of a projective variety over some number field K (see [De1] or [Mi1]). For certain arithmetic problems it is desirable to know a description of the reduction of XΓ modulo w, where w is some prime of K. In some cases it happens that the projective variety XΓ has a p-adic uniformization. By this we mean that the Kw-analytic space (XΓ⊗K Kw) an is isomorphic to ∆\Ω for some p-adic analytic symmetric space Ω and some group ∆, acting on Ω discretely. Then a formal scheme structure on ∆\Ω gives us an OKw-integral model for XΓ ⊗K Kw. Cherednik was the first who obtained a result in this direction. Let F be a totally real number field, and let B/F be a quaternion algebra, which is definite at all infinite places, except one, and ramified at a finite prime v of F . Then Cherednik proved in [Ch2] that the Shimura curve corresponding to B has a p-adic uniformization by the p-adic upper half-plane Ω2Fv , constructed by Mumford (see [Mum1]), when the subgroup defining the curve is maximal at v. Cherednik’s proof is based on the method of elliptic elements, developed by Ihara in [Ih]. The next significant step was done by Drinfel’d in [Dr2]. First he constructed certain covers of Ω2Fv (see below). Then, when F = Q, he proved the existence of a p-adic uniformization by some of his covers for all Shimura curves, described in the previous paragraph, without the assumption of maximality at v. The basic idea of Drinfel’d’s proof was to invent some moduli problem, whose solution is the Shimura curve as well as a certain p-adically uniformized curve, showing, therefore, that they are isomorphic. Developing Drinfel’d’s method, Rapoport and Zink (see [RZ1, Ra]) obtained some higher-dimensional generalizations of the above results.