On Bost-Connes type systems and complex multiplication
In their fundamental paper [Sel. Math., New Ser. 1, No. 3, 411--457 (1995; Zbl 0842.46040)], \textit{J.-B. Bost} and \textit{A. Connes} constructed a quantum statistical mechanical system, known as the Bost-Connes or BC system, that recovers the explicit class field theory for \(\mathbb Q\). In this paper the author uses the theory of complex multiplication for Siegel modular varieties to construct arithmetic subalgebras for BC-type systems attached to number fields containing a CM field. The construction is based on the method used by \textit{E. Ha} and \textit{F. Paugam} [IMRP, Int. Math. Res. Pap. 2005, No. 5, 237--286 (2005; Zbl 1173.82305)] and extends the construction of arithmetic subalgebras for imaginary quadratic fields by \textit{A. Connes, M. Marcolli} and \textit{N. Ramachandran} [Sel. Math., New Ser. 11, No. 3-4, 325--347 (2005; Zbl 1106.58005)].
scientific article; zbMATH DE number 6045439
scientific article