Bost-Connes systems and F1-structures in Grothendieck rings, spectra, and Nori motives

Abstract

We construct geometric lifts of the Bost-Connes algebra to Grothendieck rings and to the associated assembler categories and spectra, as well as to certain categories of Nori motives. These categorifications are related to the integral Bost-Connes algebra via suitable Euler characteristic type maps and zeta functions, and in the motivic case via fiber functors. We also discuss aspects of F 1 -geometry, in the framework of torifications, that fit into this general setting.

Quantum Statistical Mechanics of the Absolute Galois Group

“…In [42] and [48] it was shown that the endomorphisms σ n and ρ n of the Bost-Connes algebra lift to various equivariant Grothendieck rings and further to the level of assemblers and homotopy theoretic spectra. The symmetries of the Bost-Connes system, given by Ẑ * in the original version, or by mGT according to Proposition 3.2 above, also can be lifted to these categorical and homotopy theoretic levels in a similar manner.…”

Motivic measures and $\mathbb{F}_1$-geometries

“…In analogy with this Hasse-Weil zeta function of varieties over F q , Lieber, Manin and Marcolli then define the F 1 -zeta function to be the ring morphism, by [15,Prop. 6.2]…”

“…One might speculate that the relevant counting measures µ : K 0 (Var Z ) ✲ Z are those which determine a ring-morphism ζ µ : K 0 (Var Z ) ✲ M(Z), with those factoring over W 0 (Z) corresponding to the nonarchimedean factors, and the remaining ones related to the Γ-factors. This is motivated by our description of the F 1 -zeta function of Lieber, Manin and Marcolli in [15]. Here, one considers integral schemes with a decomposition into tori G n m as F 1 -varieties and with morphisms respecting the decomposition and with all restrictions to tori being morphisms of group schemes.…”

“…with #X(F 1 m ) being the total number of m-th roots of unity in the tori making up X, see [15]. This ζ F1 is not a λ-ring morphism and does not factor through W 0 (Z).…”

“…Acknowledgements This paper owes much to recent work of Yuri I. Manin, Matilde Marcolli and co-authors, [23], [15] and [24]. Unconventional symbols are taken from the L A T E X-package halloweenmath [25], befitting the current topic.…”