Bost–Connes type systems for function fields

Abstract

We describe a construction which associates to any function field $k$ and any place $\infty$ of $k$ a ${\mathrm{C}}^{\ast }$-dynamical system $(C_{k,\infty },{\sigma }_t)$ 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 $C_{k,\infty }$ has a faithful continuous action of $\mathrm{Gal}(K/k)$, where $K$ is a certain field constructed by Hayes such that $k^{\mathrm{ab},\infty }\subset K\subset k^{\mathrm{ab}}$. Here $k^{\mathrm{ab},\infty }$ is the maximal abelian extension of $k$ that is totally split at $\infty$. We classify the extremal KMS${}_{\beta }$ states of $(C_{k,\infty },{\sigma }_t)$ at any temperature $0<1/\beta <\infty$ and show that a phase transition with spontaneous symmetry breaking occurs at temperature $1/\beta =1$. At high temperature $1/\beta \ge 1$, there is a unique KMS${}_{\beta }$ state, of type III${}_{q^{-\beta }}$, where $q$ is the cardinal of the constant subfield of $k$. At low temperature $1/\beta <1$, the space of extremal KMS${}_{\beta }$ states is principal homogeneous under $\mathrm{Gal}(K/k)$. Each such state is of type I${}_{\infty }$ and the partition function is the Dedekind zeta function ${\zeta }_{k,\infty }$. Moreover, we construct a $\ast$-subalgebra $\mathcal{H}$, we give a presentation of $\mathcal{H}$ and of $C_{k,\infty }$, and we show that the values of the low-temperature extremal KMS${}_{\beta }$ states at certain elements of $\mathcal{H}$ are related to special values of partial zeta functions.