Bost–Connes systems associated with function fields

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 $0<\beta \le 1$ 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$({\bar{\mathbb{F}}}_q/{\mathbb{F}}_q)$.