F_ζ-geometry, Tate motives, and the Habiro ring

In this paper, we propose different notions of F_zeta-geometry, for zeta a root of unity, generalizing notions of 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