  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.
