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.
© 2015 World Scientific Publishing Company. Received 28 October 2013. Accepted 25 May 2014. Published 1 July 2014. The first author is supported by a Summer Undergraduate Research Fellowship at Caltech. The second author is supported by NSF grants DMS-0901221, DMS-1007207, DMS-1201512, PHY-1205440.
Submitted - 1310.2261.pdf