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

TL;DR

This paper constructs arithmetic models of Bost-Connes systems for all number fields using Endomotives, linking class field theory with operator algebra dynamics, and introduces a base-change functor for algebraic endomotives.

Contribution

It provides the first construction of arithmetic models for Bost-Connes systems for arbitrary number fields and introduces a functorial framework connecting number fields and operator algebras.

Findings

Arithmetic models for all number fields constructed

Class field theory realized via operator algebra dynamics

A functorial relationship between number fields and Bost-Connes systems established

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 [3]. 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, Marcolli and Consani [5], and using a classification result of Borger and de Smit [1] 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 [13].