# Bost–Connes systems associated with function fields

### Sergey Neshveyev

University of Oslo, Norway### Simen Rustad

University of Oslo, Norway

## Abstract

With a global function field $K$ with constant field $\mathbb{F}_q$, a finite set $S$ of primes in $K$ and an abelian extension $L$ of $K$, finite or infinite, we associate a C*-dynamical system. The systems, or at least their underlying groupoids, defined earlier by Jacob using the ideal action on Drinfeld modules and by Consani–Marcolli using commensurability of $K$-lattices are isomorphic to particular cases of our construction. We prove a phase transition theorem for our systems and show that the unique KMS$_\beta$-state for every $0<\beta\le1$ gives rise to an ITPFI-factor (ITPFI stands for “infinite tensor product of finite type I factors”) of type III$_{q^{-\beta n}}$, where $n$ is the degree of the algebraic closure of $\mathbb{F}_q$ in $L$. Therefore for $n=+\infty$ we get a factor of type III$_0$. Its flow of weights is a scaled suspension flow of the translation by the Frobenius element on Gal$(\bar{\mathbb{F}}_q/\mathbb{F}_q)$.

## Cite this article

Sergey Neshveyev, Simen Rustad, Bost–Connes systems associated with function fields. J. Noncommut. Geom. 8 (2014), no. 1, pp. 275–301

DOI 10.4171/JNCG/156