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

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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).

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  1. Max-Planck Institute for Mathematics, Bonn, Germany

    Bora Yalkinoglu

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  1. Bora Yalkinoglu
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Correspondence to Bora Yalkinoglu.

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This work has been supported by the Marie Curie Research Training Network MRTN-CT-2006-031962 in Noncommutative Geometry, EU-NCG.

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Yalkinoglu, B. On arithmetic models and functoriality of Bost-Connes systems. With an appendix by Sergey Neshveyev. Invent. math. 191, 383–425 (2013). https://doi.org/10.1007/s00222-012-0396-1

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  • DOI: https://doi.org/10.1007/s00222-012-0396-1

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