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In this paper we propose different notions of F_zeta-geometry, for zeta aroot of unity, generalizing notions of F_1-geometry (geometry over the "fieldwith one element") based on the behavior of the counting functions of pointsover finite fields, the Grothendieck class, and the notion of torification. Werelate F_zeta-geometry to formal roots of Tate motives, and to functions in theHabiro ring, seen as counting functions of certain ind-varieties. Weinvestigate the existence of F_zeta-structures in examples arising from generallinear groups, matrix equations over finite fields, and some quantum modularforms.

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