1999/09/23 by Yakov Varshavsky
Mathematics · #math.NT #math.AG #msc:14G35 #msc:11G18
published as J. Differential Geom. 49 (1998) 75-113 · 30 pages
arxiv created 1999/09/23 · arxiv updated 2009/11/30
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 equivariant coverings) of Drinfeld upper half-spaces. We also extend a p-adic uniformization to automorphic vector bundles. It is a continuation of our previous work [V], 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 [V] may be avoided.