# Journal of Noncommutative Geometry

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**Volume 8, Issue 1, 2014, pp. 275–301**

**DOI: 10.4171/JNCG/156**

Published online: 2014-03-20

Bostâ€“Connes systems associated with function fields

Sergey Neshveyev^{[1]}and Simen Rustad

^{[2]}(1) University of Oslo, Norway

(2) University of Oslo, Norway

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$(\bar{\mathbb{F}}_q/\mathbb{F}_q)$.

*Keywords: *Bost–Connes systems, function fields, KMS-states, type III actions, Drinfeld modules

Neshveyev Sergey, Rustad Simen: Bostâ€“Connes systems associated with function fields. *J. Noncommut. Geom.* 8 (2014), 275-301. doi: 10.4171/JNCG/156