Bost-Connes type Systems for Function Fields

We describe a construction which associates to any function field k and any place ∞ of k a C-dynamical system (Ck,∞, σt) that is analogous to the Bost-Connes system associated to Q and its archimedian place. Our construction relies on Hayes’ explicit class field theory in terms of signnormalized rank one Drinfel’d modules. We show that Ck,∞ has a faithful continuous action of Gal(K/k), where K is a certain field constructed by Hayes, such that k ⊂ K ⊂ k, where k is the maximal abelian extension of k that is totally split at ∞. We classify the extremal KMSβ states of (Ck,∞, σt) at any temperature 0 < 1/β < ∞ and show that a phase transition with spontaneous symmetry breaking occurs at temperature 1/β = 1. At high temperature 1/β > 1, there is a unique KMSβ state, of type IIIq−β , where q is the cardinal of the constant subfield of k. At low temperature 1/β < 1, the space of extremal KMSβ states is principal homogeneous under Gal(K/k). Each such state is of type I∞ and the partition function is the Dedekind zeta function ζk,∞. Moreover, we construct a “rational” ∗-subalgebra H, we give a presentation of H and of Ck,∞, and we show that the values of the low-temperature extremal KMSβ states at certain elements of H are related to special values of partial zeta functions.