On arithmetic models and functoriality of Bost-Connes systems. With an appendix by Sergey Neshveyev
- 364 Downloads
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 Classification11R37 11R20 11M55 58B34 46L55
Unable to display preview. Download preview PDF.
- 2.Borger, J., de Smit, B.: Lambda actions of rings of integers. arXiv:1105.4662
- 4.Connes, A., Consani, C.: On the arithmetic of the BC-system. arXiv:1103.4672
- 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
- 10.Cornelissen, G., Marcolli, M.: Quantum statistical mechanics, L-series and Anabelian geometry. arXiv:1009.0736
- 16.Laca, M., Neshveyev, S., Trifkovic, M.: Bost-Connes systems, Hecke algebras, and induction. arXiv:1010.4766