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.
References
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, vol. 36, With a foreword by Yuri Manin, Providence, RI: American Mathematical Society , ISBN 978-0-8218-3833-4 , Zbl 1081.58005