𝔽ζ-geometry, Tate motives, and the Habiro ring

In this paper, we propose different notions of 𝔽ζ-geometry, for ζ a root of unity, generalizing notions of 𝔽1-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 𝔽ζ-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 𝔽ζ-structures in examples arising from general linear groups, matrix equations over finite fields, and some quantum modular forms.