P-adic uniformization of unitary shimura varieties. II

In this paper we show that certain Shimura varieties, uniformized by the product of complex unit balls, can be p-adically uniformized by the product of Drinfeld upper half-spaces and their equivariant coverings. We also extend a p-adic uniformization to automorphic vector bundles. It is a continuation of our previous work [38] and contains all cases (up to a central modification) of a uniformization by known p-adic symmetric spaces. The idea of the proof is to show that an arithmetic quotient of the product of Drinfeld upper half-spaces cannot be anything else than a certain unitary Shimura variety. Moreover, we show that difficult theorems of Yau and Kottwitz appearing in [38] may be avoided.