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

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In this paper we propose different notions of Fzeta-geometry, for zeta 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 Fzeta-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 Fzeta-structures in examples arising from general linear groups, matrix equations over finite fields, and some quantum modular forms.