Bost-Connes systems for local fields of characteristic zero

TL;DR

This paper generalizes Bost-Connes systems to local fields of characteristic zero, exploring their properties and the absence of phase transitions, and examines the connection between global and local systems.

Contribution

It introduces a new framework for Bost-Connes systems in local fields of characteristic zero, extending previous global field models.

Findings

No phase transition occurs in local fields of characteristic zero

Established a relationship between global and local Bost-Connes systems

Extended the mathematical understanding of class field theory in this context

Abstract

The Bost-Connes system is a $C^*$-dynamical system which has a relation with the class field theory. The purpose of this paper is to generalize the notion of Bost-Connes systems to local fields of characteristic zero. The notable phenomenon in the case of local fields is the absence of the phase transition. We also investigate the relation between the global and local ones.