JournalsjncgVol. 8, No. 1pp. 275–301

Bost–Connes systems associated with function fields

  • Sergey Neshveyev

    University of Oslo, Norway
  • Simen Rustad

    University of Oslo, Norway
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With a global function field KK with constant field Fq\mathbb{F}_q, a finite set SS of primes in KK and an abelian extension LL of KK, 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 KK-lattices are isomorphic to particular cases of our construction. We prove a phase transition theorem for our systems and show that the unique KMSβ_\beta-state for every 0<β10<\beta\le1 gives rise to an ITPFI-factor (ITPFI stands for “infinite tensor product of finite type I factors”) of type IIIqβn_{q^{-\beta n}}, where nn is the degree of the algebraic closure of Fq\mathbb{F}_q in LL. Therefore for n=+n=+\infty we get a factor of type III0_0. Its flow of weights is a scaled suspension flow of the translation by the Frobenius element on Gal(Fˉq/Fq)(\bar{\mathbb{F}}_q/\mathbb{F}_q).

Cite this article

Sergey Neshveyev, Simen Rustad, Bost–Connes systems associated with function fields. J. Noncommut. Geom. 8 (2014), no. 1, pp. 275–301

DOI 10.4171/JNCG/156