On Bost–Connes type systems and complex multiplication

Abstract

By using the theory of complex multiplication for general Siegel modular varieties we construct arithmetic subalgebras for BC-type systems attached to number fields containing a CM field. The abelian extensions obtained in this way are characterized by results of [Wei]. Our approach is based on a general construction of BC-type systems of Ha and Paugam [HP05] and extends the construction of the arithmetic subalgebra of Connes, Marcolli and Ramachandran [CMR05] for imaginary quadratic fields.