Homotopy types and geometries below \(\mathrm{Spec}(\mathbb{Z})\). (English) Zbl 07217768

Manin, Yuri I.; Marcolli, Matilde

Moree, Pieter (ed.) et al., Dynamics: topology and numbers. Conference, Max Planck Institute for Mathematics, Bonn, Germany, July 2–6, 2018. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 744, 27-56 (2020).

Summary: After the first heuristic ideas about “the field of one element” \(\mathbb{F}_1\) and “geometry in characteristics 1” (J. Tits, C. Deninger, M. Kapranov, A. Smirnov et al.), were developed several general approaches to the construction of “geometries below \(\operatorname{Spec}\mathbb{Z}\)”. 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.
We introduce a notion of \(\mathbb{F}_1\) structures based on quasi-unipotent endomorphisms on homology. We also consider \(\mathbb{F}_1\) structures based on the integral Bost-Connes algebra and its endomorphisms. In both cases we consider lifts of these structures, via an equivariant Euler charactetristic, to the level of Grothendieck rings and further lifts, via the formalism of assembler categories, to homotopy theoretic spectra.
For the entire collection see [Zbl 1448.37001].

MSC:

14A23	Geometry over the field with one element
14-02	Research exposition (monographs, survey articles) pertaining to algebraic geometry
14G40	Arithmetic varieties and schemes; Arakelov theory; heights
14G15	Finite ground fields in algebraic geometry
14G10	Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
37P35	Arithmetic properties of periodic points
14C15	(Equivariant) Chow groups and rings; motives
14A22	Noncommutative algebraic geometry
11M41	Other Dirichlet series and zeta functions
55P43	Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.)
82B10	Quantum equilibrium statistical mechanics (general)

Keywords:

Weil numbers; quasi-unipotent endomorphisms; Bost-Connes systems; zeta functions; motivic integration; Grothendieck ring of varieties; geometries below \(\mathrm{Spec}\mathbb{Z}\)

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