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Inventiones mathematicae

, Volume 191, Issue 2, pp 383–425 | Cite as

On arithmetic models and functoriality of Bost-Connes systems. With an appendix by Sergey Neshveyev

  • Bora YalkinogluEmail author
Article

Abstract

This paper has two parts. In the first part we construct arithmetic models of Bost-Connes systems for arbitrary number fields, which has been an open problem since the seminal work of Bost and Connes (Sel. Math. 1(3):411–457, 1995). In particular our construction shows how the class field theory of an arbitrary number field can be realized through the dynamics of a certain operator algebra. This is achieved by working in the framework of Endomotives, introduced by Connes, Consani and Marcolli (Adv. Math. 214(2):761–831, 2007), and using a classification result of Borger and de Smit (arXiv:1105.4662) for certain Λ-rings in terms of the Deligne-Ribet monoid. Moreover the uniqueness of the arithmetic model is shown by Sergey Neshveyev in an appendix. In the second part of the paper we introduce a base-change functor for a class of algebraic endomotives and construct in this way an algebraic refinement of a functor from the category of number fields to the category of Bost-Connes systems, constructed recently by Laca, Neshveyev and Trifkovic (arXiv:1010.4766).

Mathematics Subject Classification

11R37 11R20 11M55 58B34 46L55 

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References

  1. 1.
    Borger, J.: The basic geometry of Witt vectors, I: the affine case. Algebra Number Theory 5(2), 231–285 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Borger, J., de Smit, B.: Lambda actions of rings of integers. arXiv:1105.4662
  3. 3.
    Bost, J.-B., Connes, A.: Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory. Sel. Math. New Ser. 1(3), 411–457 (1995) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Connes, A., Consani, C.: On the arithmetic of the BC-system. arXiv:1103.4672
  5. 5.
    Connes, A., Marcolli, M.: From physics to number theory via noncommutative geometry. In: Frontiers in Number Theory, Physics, and Geometry. I, pp. 269–347. Springer, Berlin (2006) CrossRefGoogle Scholar
  6. 6.
    Connes, A., Marcolli, M.: Noncommutative Geometry, Quantum Fields and Motives. American Mathematical Society Colloquium Publications, vol. 55. American Mathematical Society, Providence (2008) Google Scholar
  7. 7.
    Connes, A., Marcolli, M., Ramachandran, N.: KMS states and complex multiplication. Sel. Math. New Ser. 11(3–4), 325–347 (2005) MathSciNetzbMATHGoogle Scholar
  8. 8.
    Connes, A., Consani, C., Marcolli, M.: Noncommutative geometry and motives: the thermodynamics of endomotives. Adv. Math. 214(2), 761–831 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Connes, A., Consani, C., Marcolli, M.: Fun with \(\mathbb{F}_{1}\). J. Number Theory 129(6), 1532–1561 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Cornelissen, G., Marcolli, M.: Quantum statistical mechanics, L-series and Anabelian geometry. arXiv:1009.0736
  11. 11.
    Deligne, P., Ribet, K.A.: Values of abelian L-functions at negative integers over totally real fields. Invent. Math. 59(3), 227–286 (1980) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Ha, E., Paugam, F.: Bost-Connes-Marcolli systems for Shimura varieties. I. Definitions and formal analytic properties. Int. Math. Res. Pap. 5, 237–286 (2005) MathSciNetCrossRefGoogle Scholar
  13. 13.
    Laca, M.: Semigroups of -endomorphisms, Dirichlet series, and phase transitions. J. Funct. Anal. 152(2), 330–378 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Laca, M.: From endomorphisms to automorphisms and back: dilations and full corners. J. Lond. Math. Soc. (2) 61(3), 893–904 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Laca, M., Larsen, N.S., Neshveyev, S.: On Bost-{C}onnes type systems for number fields. J. Number Theory 129(2), 325–338 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Laca, M., Neshveyev, S., Trifkovic, M.: Bost-Connes systems, Hecke algebras, and induction. arXiv:1010.4766
  17. 17.
    Marcolli, M.: Cyclotomy and endomotives. P-Adic Numbers Ultrametric Anal. Appl. 1(3), 217–263 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Neshveyev, S.: Ergodicity of the action of the positive rationals on the group of finite adeles and the Bost-Connes phase transition theorem. Proc. Am. Math. Soc. 130(10), 2999–3003 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Neukirch, J.: Algebraic Number Theory. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 322. Springer, Berlin, (1999) zbMATHCrossRefGoogle Scholar
  20. 20.
    Yalkinoglu, B.: On Bost-{C}onnes type systems and complex multiplication. J. NCG (to appear). doi: 10.4171/JNCG/92. arXiv:1010.0879v1

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Max-Planck Institute for MathematicsBonnGermany

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