Fun with $\F_1$

TL;DR

This paper explores the algebraic and endomotive structures of the Bost--Connes quantum statistical system, linking them to the concept of the field with one element and its extensions, and analyzing their properties at primes.

Contribution

It provides an explicit integer model of the endomotive related to the Bost--Connes system, connecting it to the field with one element and Frobenius correspondences.

Findings

Endomotive arises from scalar extension of the field with one element.

Explicit integer model of the endomotive is constructed.

Analysis of reduction at primes of the endomotive and related algebra.

Abstract

We show that the algebra and the endomotive of the quantum statistical mechanical system of Bost--Connes naturally arises by extension of scalars from the "field with one element" to rational numbers. The inductive structure of the abelian part of the endomotive corresponds to the tower of finite extensions of that "field", while the endomorphisms reflect the Frobenius correspondences. This gives in particular an explicit model over the integers for this endomotive, which is related to the original Hecke algebra description. We study the reduction at a prime of the endomotive and of the corresponding noncommutative crossed product algebra.