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

@article{Bost1995HeckeAT,
  title={Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory},
  author={Jean-Beno{\^i}t Bost and Alain Connes},
  journal={Selecta Mathematica},
  year={1995},
  volume={1},
  pages={411-457}
}
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… 
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