q-deformations of statistical mechanical systems and motives over finite fields

TL;DR

This paper explores q-deformations of Witt rings derived from motives over finite fields, constructing q-analogs of the Bost-Connes system and linking them to Habiro rings and categorifications.

Contribution

It introduces a novel approach to q-deformations of Witt rings based on geometric operations on zeta functions, connecting them to quantum statistical mechanics and algebraic structures.

Findings

Established q-analogs of the Bost-Connes system

Connected q-deformations to Habiro ring constructions

Linked deformations to categorifications of Bost-Connes systems

Abstract

We consider q-deformations of Witt rings, based on geometric operations on zeta functions of motives over finite fields, and we use these deformations to construct q-analogs of the Bost-Connes quantum statistical mechanical system. We show that the q-deformations obtained in this way can be related to Habiro ring constructions of analytic functions over the field with one element and to categorifications of Bost-Connes systems.