(1) Collège de France, 3, rue d’Ulm, F-75005 Paris, France;(2) Max-Planck Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany;(3) Department of Mathematics, University of Maryland, College Park, MD, 20912, U.S.A
Abstract:
We construct a quantum statistical mechanical system which generalizes the Bost–Connes system to imaginary quadratic fields
K of arbitrary class number and fully incorporates the explicit class field theory for such fields. This system admits the
Dedekind zeta function as partition function and the idèle class group as group of symmetries. The extremal KMS states at
zero temperature intertwine this symmetry with the Galois action on the values of the states on the arithmetic subalgebra.
The geometric notion underlying the construction is that of commensurability of K-lattices.