# Bost–Connes type systems for function fields

### Benoît Jacob

Université Paris 7 Denis Diderot

## Abstract

We describe a construction which associates to any function field $k$ and any place $∞$ of $k$ a $C_{∗}$-dynamical system $(C_{k,∞},σ_{t})$ that is analogous to the Bost–Connes system associated to $Q$ and its archimedean place. Our construction relies on Hayes’ explicit class field theory in terms of sign-normalized rank one Drinfel’d modules. We show that $C_{k,∞}$ has a faithful continuous action of $Gal (K/k)$, where $K$ is a certain field constructed by Hayes such that $k_{ab,∞} ⊂K ⊂k_{ab}$. Here $k_{ab,∞}$ is the maximal abelian extension of $k$ that is totally split at $∞$. We classify the extremal KMS$_{β}$ states of $(C_{k,∞},σ_{t})$ at any temperature $0<1/β <∞$ and show that a phase transition with spontaneous symmetry breaking occurs at temperature $1/β =1$. At high temperature $1/β ≥1$, there is a unique KMS$_{β}$ state, of type III$_{q_{−β}}$, where $q$ is the cardinal of the constant subfield of $k$. At low temperature $1/β <1$, the space of extremal KMS$_{β}$ states is principal homogeneous under $Gal (K/k)$. Each such state is of type I$_{∞}$ and the partition function is the Dedekind zeta function $ζ_{k,∞}$. Moreover, we construct a $∗$-subalgebra $H$, we give a presentation of $H$ and of $C_{k,∞}$, and we show that the values of the low-temperature extremal KMS$_{β}$ states at certain elements of $H$ are related to special values of partial zeta functions.

## Cite this article

Benoît Jacob, Bost–Connes type systems for function fields. J. Noncommut. Geom. 1 (2007), no. 2, pp. 141–211

DOI 10.4171/JNCG/4