Bost-Connes systems associated with function fields

Abstract

With a global function field K with constant field 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 of type III_{q^{-\beta n}}, where n is the degree of the algebraic closure of 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 F_q/F_q).