Selecta Mathematica

, 11:325

KMS states and complex multiplication

  • Alain Connes
  • Matilde Marcolli
  • Niranjan Ramachandran
Original Paper

DOI: 10.1007/s00029-005-0013-x

Cite this article as:
Connes, A., Marcolli, M. & Ramachandran, N. Sel. math., New ser. (2006) 11: 325. doi:10.1007/s00029-005-0013-x


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.

Mathematics Subject Classification (2000).

58B34 46L55 11R37 11G18 


KMS states K-lattices complex multiplication explicit class field theory quantum statistical mechanics 

Copyright information

© Birkhäuser Verlag, Basel 2005

Authors and Affiliations

  • Alain Connes
    • 1
  • Matilde Marcolli
    • 2
  • Niranjan Ramachandran
    • 3
  1. 1.Collège de FranceParisFrance
  2. 2.Max-Planck Institut für MathematikBonnGermany
  3. 3.Department of MathematicsUniversity of MarylandCollege ParkU.S.A

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