Hyperbolic Planes

Abstract

In this paper we consider a special class of arithmetic quotients of bounded symmetric domains which can roughly be described as higher- dimensional analogues of the Hilbert modular varities. The algebraic groups are defined as the unitary groups over two-dimensional right vector spaces over a division algebra with involution. If $d$ denotes the degree of the division algebra, then $d=1$ is essentially just case giving rise to Hilbert modular varieties. We determine the class number (number of cusps) of the arithmetic quotients, and find inter- esting modular subvarities whos existence derives from the algebraic structure of the division algebras. Also the moduli interpretation, given by Shimuras theory, is described.