# 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 (Ck,∞,σt) 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 Ck,∞ has a faithful continuous action of Gal (K/k), where K is a certain field constructed by Hayes such that kab,∞ ⊂ K ⊂ kab. Here kab,∞ is the maximal abelian extension of k that is totally split at ∞. We classify the extremal KMSβ states of (Ck,∞,σt) 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 KMSβ state, of type IIIq−β, where q is the cardinal of the constant subfield of k. At low temperature 1/β < 1, the space of extremal KMSβ states is principal homogeneous under Gal (K/k). Each such state is of type I∞ and the partition function is the Dedekind zeta function ζk,∞. Moreover, we construct a *-subalgebra ℋ, we give a presentation of ℋ and of Ck,∞, and we show that the values of the low-temperature extremal KMSβ 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