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

• 成果类型: Article
• 署名作者: Yalkinoglu, Bora
• 署名单位: Max Planck Society
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-012-0396-1
• 发表日期: 2013
• 页码: 383-425
• 关键词: endomorphisms; geometry

摘要

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 Lambda-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).