Bost-Connes systems associated with function fields

TL;DR

This paper constructs and analyzes Bost-Connes systems linked to function fields, establishing phase transitions and classifying associated von Neumann algebras, extending prior work on ideal actions and K-lattices.

Contribution

It introduces a new class of Bost-Connes systems for function fields, proves phase transition theorems, and characterizes the resulting KMS states as type III factors.

Findings

Existence of phase transitions at specific inverse temperatures.

Unique KMS states induce type III factors with explicit classification.

Flow of weights described as a scaled suspension flow of Frobenius action.

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).