Construction of arithmetic automorphic functions for special Clifford groups

Abstract

An important problem in the theory of arithmetic automorphic functions is to construct, for a reductive algebraic group overQwhich defines a bounded symmetric domain, a system of canonical models [2], [6], [18]. For the similitude group of a hermitian form over a quaternion algebra whose center is a totally real field, this is solved by Shimura [17], and for the similitude group of a hermitian form with respect to an involution of the second kind of a central division algebra over aCM-field, by Miyake [8], In this paper, we show that this also can be done for the special Clifford group of a quadratic formQover a totally real algebraic number field. (We have to impose certain conditions on the signature ofQin order thatGdefines a bounded symmetric domain, see 1.1.)