Selecta Mathematica

, Volume 1, Issue 3, pp 411–457

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]).


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  1. [A-W]
    H. Araki and E.J. Woods.A classification of factors. Publ. Res. Inst. Math. Sci. Kyoto Univ.,4 (1968), 51–130.Google Scholar
  2. [Bi]
    M. Binder.Induced factor representations of discrete groups and their type. Jour. Functional Analysis, to appear.Google Scholar
  3. [Bl]
    B.E. Blackadar.The regular representation of restricted direct product groups. Jour. Functional Analysis,25 (1977), 267–274.Google Scholar
  4. [Bos-C]
    J.-B. Bost and A. Connes. Produits eulériens et facteurs de type III. C.R. Acad. Sci. Paris,315 (I) (1992), 279–284.Google Scholar
  5. [Br-R]
    O. Bratteli and D.W. Robinson.Operator algebras and quantum statistical mechanics I, II. Springer-Verlag, New York-Heidelberg-Berlin, 1981.Google Scholar
  6. [Com]
    F. Combes. Poids associé à une algèbre hilbertienne à gauche. Compos. Math.23 (1971), 49–77.Google Scholar
  7. [C]
    A. Connes.Noncommutative geometry. Academic Press, 1994.Google Scholar
  8. [Co]
    A. Connes. Une classification des facteurs de type III. Ann. Sci. Ecole Norm. Sup., (4)6 (1973), 133–252.Google Scholar
  9. [C-T]
    A. Connes and M. Takesaki.The flow of weights on factors of type III. Tohoku Math. J.,29 (1977), 473–575.Google Scholar
  10. [Dir]
    P.A.M. Dirac.The quantum theory of the emission and absorption of radiation. Proc. Royal Soc. London,A114 (1927), 243–265.Google Scholar
  11. [G]
    A. Guichardet.Symmetric Hilbert spaces and related topics. Lecture Notes in Mathematics,261, Springer-Verlag, New York-Heidelberg-Berlin, 1972.Google Scholar
  12. [H]
    R. Haag.Local quantum physics. Springer-Verlag, New York-Heidelberg-Berlin, 1992.Google Scholar
  13. [J]
    B. Julia.Statistical theory of numbers. in Number Theory and Physics, Les Houches Winter School, J.-M. Luck, P. Moussa et M. Waldschmidt eds., Springer-Verlag, 1990.Google Scholar
  14. [P]
    G.K. Pedersen.C*-algebras and their automorphism groups. Academic Press, London-New York-San Francisco, 1979.Google Scholar
  15. [Ren]
    J. Renault.A groupoid approach to C*-algebras. Lecture Notes in Mathematics,793 (1980), Springer-Verlag, New York-Heidelberg-Berlin.Google Scholar
  16. [Ser1]
    J.-P. Serre.Cours d'arithmétique. P.U.F. Paris, 1970.Google Scholar
  17. [Ser2]
    J.-P. Serre.Arbres, amalgames, SL 2. Astérisque,46 (1977).Google Scholar
  18. [Sh]
    G. Shimura.Introduction to the arithmetic theory of automorphic functions. Princeton University Press, 1971.Google Scholar
  19. [Spe]
    D. Spector.Supersymmetry and the Möbius inversion function. Commun. Math. Phys.,127 (1990), 239–252.Google Scholar
  20. [T]
    J. Tate.Fourier analysis in number fields and Hecke's zeta function. in Algebraic Number Theory, J.W.S. Cassels et A. Fröhlich eds., Academic Press, 1967.Google Scholar
  21. [We1]
    A. Weil.Fonction zêta et distributions. Séminaire Bourbaki n0312, juin 1966.Google Scholar
  22. [We2]
    A. Weil.Basic number theory. Springer-Verlag, Berlin-Heidelberg-New York, 1974.Google Scholar

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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