Bost-Connes type dynamics and arithmetic of function fields

Abstract

The work of this thesis falls within two main themes: the analogy between number fields and function fi elds of an algebraic curve over a fi nite fi eld; and the recovery of arithmetic information from the Bost-Connes system associated to such global fields. We present a new construction for Bost-Connes systems for function fi elds based on the ring of S-integers and show that it provides a suitable framework to generalize the work of Takeishi to the functionfi eld case and thus recover class numbers. Further, we investigate an analogue of the Deligne-Ribet topological dynamical system that appears in Bost-Connes systems and show that isomorphisms that respect these dynamics are equivalent to arithmetic equivalences of the related global fields.