Selecta Mathematica

, Volume 1, Issue 3, pp 411–457 | Cite as

Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory

  • J. B. Bost
  • A. Connes


In this paper, we construct a naturalC*-dynamical system whose partition function is the Riemann ζ function. Our construction is general and associates to an inclusion of rings (under a suitable finiteness assumption) an inclusion of discrete groups (the associated ax+b groups) and the corresponding Hecke algebras of bi-invariant functions. The latter algebra is endowed with a canonical one parameter group of automorphisms measuring the lack of normality of the subgroup. The inclusion of rings ℤ⊂ℚ provides the desiredC*-dynamical system, which admits the ζ function as partition function and the Galois group Gal(ℚcycl/ℚ) of the cyclotomic extension ℚcycl of ℚ as symmetry group. Moreover, it exhibits a phase transition with spontaneous symmetry breaking at inverse temperature β=1 (cf. [Bos-C]). The original motivation for these results comes from the work of B. Julia [J] (cf. also [Spe]).


Dynamical System Partition Function Phase Transition Symmetry Breaking Symmetry Group 
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Copyright information

© Birkhäuser Verlag 1995

Authors and Affiliations

  • J. B. Bost
    • 1
  • A. Connes
    • 1
  1. 1.Institut des Hautes Études ScientifiquesBures-sur-YvetteFrance

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