𝔽_ζ-geometry, Tate motives, and the Habiro ring

Abstract

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

q-Deformations of statistical mechanical systems and motives over finite fields

“…Since #A ℓ (F q ) = q ℓ , this deformation generalizes the previous one, in the sense that the previous one occurs as first order term. We then show that, using these deformations one obtains interesting q-deformed Bost-Connes algebras that can be related to the constructions of [15], [17], and of [14], and also to the categorifications of Bost-Connes systems of [18]. We also discuss the role of q-analogs and a q-deformation of the Riemann zeta function in this context.…”

“…In terms of our geometric interpretation, these q-deformations arise from products with affine spaces. Equivalently, in motivic terms, they are given by products with powers of the Lefschetz motive L. Thus, we can frame an appropriate extension of the action of the Verschiebung in terms of the roots of Tate motives discussed in [14]. The idea of introducing roots of Tate motives was first suggested in [16].…”

“…A geometric construction of a square root Q(1/2) of the Tate motive Q(1) in terms of supersingular elliptic curves was given in [25]. A categorical construction of a square root of the Tate motive was given in §3.4 of [13] and generalized to arbitrary roots Q(r), for r ∈ Q + , in [14]. We recall here briefly this formal categorical definition.…”

“…The group G (n) is in turn the Galois group of a Tannakian category T (Q( 1 n )) which extends the category T by an n-th root Q(1/n) of the Tate motive Q (1). A more precise description of these categories and their properties is given in §4 of [14], where the construction of the roots Q(1/n) is explained in terms of the primary decomposition of n ∈ N. LetT be the Tannakian category that corresponds to the projective limitG of the groups G (n) , as in [14], that is, the Tannakian category obtained from Num Q (k) by adjoining roots of Tate motives of arbitrary order,…”

“…Manin proposed in [15] to use the Habiro ring of q-functions, [9], as a good notion of analytic geometry and analytic functions over F 1 , and analogs of the Bost-Connes system based on the Habiro ring were constructed in [17]. Relations between the Habiro ring and motives were discussed in [14].…”