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

Marcolli, Matilde; Ren, Zhi

doi:10.1134/s2070046617030049

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Lemma There Are Endomorphisms σ1

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2022

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Cited by 2 publications

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“…They are geometric manifestations of the same relation between the maps σ n and ρn of the integral Bost-Connes system and the Frobenius and Verschiebung described in [CC14]. For other occurrences of the same relation see also [MaRe17], [MaTa17].…”

Section: Lemma There Are Endomorphisms σmentioning

confidence: 84%

Homotopy types and geometries below Spec(ℤ)

Manin¹,

Marcolli²

2020

Dynamics: Topology and Numbers

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After the first heuristic ideas about "the field of one element" F 1 and "geometry in characteristics 1" (J. Tits, C. Deninger, M. Kapranov, A. Smirnov et al.), there were developed several general approaches to the construction of "geometries below Spec Z". Homotopy theory and the "the brave new algebra" were taking more and more important places in these developments, systematically explored by B. Toën and M. Vaquié, among others.This article contains a brief survey and some new results on counting problems in this context, including various approaches to zeta-functions and generalised scissors congruences.

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Section: Lemma There Are Endomorphisms σmentioning

confidence: 84%

Homotopy types and geometries below Spec(ℤ)

Manin¹,

Marcolli²

2020

Dynamics: Topology and Numbers

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Bost–Connes Systems and F1-structures in Grothendieck Rings, Spectra, and Nori Motives

F.¹,

Manin²,

Marcolli³

2022

Facets of Algebraic Geometry

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

Cited by 2 publications

References 28 publications

Homotopy types and geometries below Spec(ℤ)

Homotopy types and geometries below Spec(ℤ)

Bost–Connes Systems and F1-structures in Grothendieck Rings, Spectra, and Nori Motives

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