F-zeta geometry, Tate motives, and the Habiro ring

TL;DR

This paper introduces new concepts of F_zeta-geometry linked to roots of unity, connecting it with Tate motives and the Habiro ring, and explores their applications in algebraic and quantum contexts.

Contribution

It generalizes F_1-geometry to F_zeta-geometry, relating it to Tate motives and Habiro ring functions, and investigates examples from linear groups, matrix equations, and quantum modular forms.

Findings

F_zeta-geometry relates to formal roots of Tate motives.

Functions in the Habiro ring can be interpreted as counting functions.

Examples include structures from linear groups and quantum modular forms.

Abstract

In this paper we propose different notions of F_zeta-geometry, for zeta a root of unity, generalizing notions of F_1-geometry (geometry over the "field with one element") based on the behavior of the counting functions of points over finite fields, the Grothendieck class, and the notion of torification. We relate F_zeta-geometry to formal roots of Tate motives, and to functions in the Habiro ring, seen as counting functions of certain ind-varieties. We investigate the existence of F_zeta-structures in examples arising from general linear groups, matrix equations over finite fields, and some quantum modular forms.