2019
DOI: 10.48550/arxiv.1901.00020
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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.

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Cited by 4 publications
(14 citation statements)
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“…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.…”
Section: 7mentioning
confidence: 99%
“…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.…”
Section: 7mentioning
confidence: 99%
“…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]…”
Section: 1mentioning
confidence: 99%
“…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.…”
Section: Introductionmentioning
confidence: 99%
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