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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
Article

Abstract

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]).

Keywords

Dynamical System Partition Function Phase Transition Symmetry Breaking Symmetry Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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