Bost–Connes system

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) introduced Bost–Connes systems by constructing one for the rational numbers. (Connes Marcolli) 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", Selecta Mathematica, New Series 1 (3): 411–457, doi:10.1007/BF01589495, ISSN 1022-1824, https://cds.cern.ch/record/283504/files/SCAN-9506164.pdf 
• Connes, Alain; Marcolli, Matilde; Ramachandran, Niranjan (2005), "KMS states and complex multiplication", Selecta Mathematica, New Series 11 (3): 325–347, doi:10.1007/s00029-005-0013-x, ISSN 1022-1824, Bibcode: 2005math......1424C 
• 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