Bost–Connes system

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In mathematics, a Bost–Connes system is a quantum statistical dynamical system related to an algebraic number field, whose partition function is related to the Dedekind zeta function of the number field. Bost & Connes (1995) introduced Bost–Connes systems by constructing one for the rational numbers. Connes, Marcolli & Ramachandran (2005) extended the construction to imaginary quadratic fields.

Such systems have been studied for their connection with Hilbert's Twelfth Problem. In the case of a Bost–Connes system over Q, the absolute Galois group acts on the ground states of the system.


  • Bost, J.-B.; Connes, Alain (1995), "Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory" (PDF), Selecta Mathematica, New Series, 1 (3): 411–457, doi:10.1007/BF01589495, ISSN 1022-1824, MR 1366621, S2CID 116418599
  • Connes, Alain; Marcolli, Matilde; Ramachandran, Niranjan (2005), "KMS states and complex multiplication", Selecta Mathematica, New Series, 11 (3): 325–347, arXiv:math/0501424, Bibcode:2005math......1424C, doi:10.1007/s00029-005-0013-x, ISSN 1022-1824, MR 2215258, S2CID 10792121
  • Marcolli, Matilde (2005), Arithmetic noncommutative geometry, University Lecture Series, 36, With a foreword by Yuri Manin, Providence, RI: American Mathematical Society, ISBN 978-0-8218-3833-4, Zbl 1081.58005 CS1 maint: discouraged parameter (link)