• Corpus ID: 119713994

# Homotopy types and geometries below Spec Z

@article{Manin2018HomotopyTA,
title={Homotopy types and geometries below Spec Z},
author={Yu. I. Manin and Matilde Marcolli},
journal={arXiv: Algebraic Geometry},
year={2018}
}
• Published 28 June 2018
• Mathematics
• arXiv: Algebraic Geometry
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.~Toen and M.~Vaquie, among others. This article contains a brief survey and some new results on…
6 Citations
Homotopy Spectra and Diophantine Equations
• Mathematics
• 2020
Arguably, the first bridge between vast, ancient, but disjoint domains of mathematical knowledge, – topology and number theory, – was built only during the last fifty years. This bridge is the theory
Birational maps and Nori motives
• Mathematics
• 2020
The monograph [HuM-St17] contains a systematical exposition of Nori motives that were developed and studied as the “universal (co)homology theory” of algebraic varieties (or schemes), according to
Bost-Connes systems and F1-structures in Grothendieck rings, spectra, and Nori motives
• Mathematics
• 2018
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
Motivic measures and $\mathbb{F}_1$-geometries.
Right adjoints for the forgetful functors on $\lambda$-rings and bi-rings are applied to motivic measures and their zeta functions on the Grothendieck ring of $\mathbb{F}_1$-varieties in the sense of
Counting in times of fake fields
These are notes of a talk, given at the 'arithmetique en plat pays'-meeting in february 2020, on the potential uses of geometries over the fake field $\mathbb{F}_1$ to zeta functions and counting
Quantum Statistical Mechanics of the Absolute Galois Group
• Mathematics
Symmetry, Integrability and Geometry: Methods and Applications
• 2020
We present possible extensions of the quantum statistical mechanical formulation of class field theory to the non-abelian case, based on the action of the absolute Galois group on Grothendieck's

## References

SHOWING 1-10 OF 65 REFERENCES
Cyclotomy and analytic geometry over F_1
Geometry over non--existent "field with one element" $F_1$ conceived by Jacques Tits [Ti] half a century ago recently found an incarnation, in at least two related but different guises. In this paper
The basic geometry of Witt vectors
This is an account of the étale topology of generalized Witt vectors. Its purpose is to develop some foundational material needed in Λalgebraic geometry. The theory of the usual, “p-typical” Witt
∞-Categories for the Working Mathematician
• Mathematics
• 2018
homotopy theory C.1. Lifting properties, weak factorization systems, and Leibniz closure C.1.1. Lemma. Any class of maps characterized by a right lifting property is closed under composition,
The basic geometry of Witt vectors, I: The affine case
We give a concrete description of the category of etale algebras over the ring of Witt vectors of a given finite length with entries in an arbitrary ring. We do this not only for the classical
Local zeta factors and geometries under Spec Z
The first part of this note shows that the odd period polynomial of each Hecke cusp eigenform for full modular group produces via Rodriguez--Villegas transform ([Ro--V]) a polynomial satisfying the
Fun with $\F_1$
• Mathematics
• 2009
The basic geometry of Witt vectors. II: Spaces
This is an account of the algebraic geometry of Witt vectors and arithmetic jet spaces. The usual, “p-typical” Witt vectors of p-adic schemes of finite type are already reasonably well understood.
Derived $\ell$-adic zeta functions
• Mathematics
• 2017
We lift the classical Hasse--Weil zeta function of varieties over a finite field to a map of spectra with domain the Grothendieck spectrum of varieties constructed by Campbell and Zakharevich. We use