KMS states and complex multiplication
  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 explict class field theory for such fields. This system admits the Dedekind zeta function as partition function and the Idele 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. We also give an interpretation of the original BC system and of the GL(2) system in terms of Shimura varieties, which motivates the construction for imaginary quadratic fields. The geometric notion underlying the construction is that of commensurability of K-lattices. 
Alain Connes (College de France), Matilde Marcolli (MPIM Bonn),   Niranjan Ramachandran (University of Maryland)