Bost-Connes type systems for function fields

Details Bost-Connes type systems for function fields Bost-Connes type systems for function fields

2006-02-24

arxiv.org/abs/math/0602554v4

Mathematics

Operator Algebras

55 pages; index of notations; submitted to Journal of Noncommutative Geometry; v4: fixed font problem, more corrections, more

Scientific paper

We describe a construction which associates to any function field $k$ and any place $\infty$ of $k$ a $C^*$-dynamical system $(C_{k,\infty},\sigma_t)$ that is analogous to the Bost-Connes system associated to $\mathbb{Q}$ and its archimedian 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 $K/k$ is an extension such that $k^{\mathrm{ab},\infty}\subset K\subset k^{\mathrm{ab}}$, where $k^{\mathrm{ab},\infty}$ 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 $1/\beta\geqslant 1$, there is a unique KMS$_\beta$ state, of type $\mathrm{III}_{q^{-\beta}}$, 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 $\mathrm{I}_\infty$ and the partition function is the Dedekind zeta function $\zeta_{k,\infty}$. Moreover, we construct a ``rational'' $*$-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.

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