# Homotopy Spectra and Diophantine Equations

@inproceedings{Manin2020HomotopySA, title={Homotopy Spectra and Diophantine Equations}, author={Yu. I. Manin and Matilde Marcolli}, year={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 of spectra in the stable homotopy theory. In particular, it connects Z, the initial object in the theory of commutative rings, with the sphere spectrum S: see [Sc01] for one of versions of it. This connection poses the challenge: discover a new information in number theory using the developed…

## References

SHOWING 1-10 OF 49 REFERENCES

A First Approximation to Homotopy Theory.

- Mathematics, MedicineProceedings of the National Academy of Sciences of the United States of America
- 1953

This chapter discusses how, by passing to a direct limit of homotopy classes under suspension, it is possible to obtain a new category, called the S-category, that is simpler in structure than the Homotopy category of topological spaces and homOTopy classes of maps.

S-modules and symmetric spectra

- Mathematics
- 2001

We study a symmetric monoidal adjoint functor pair between the category of S-modules and the category of symmetric spectra. The functors induce equivalences between the respective homotopy categories…

Absolute algebra and Segal's Γ-rings: Au dessous de Spec(Z)‾

- Mathematics
- 2016

Abstract Text We show that the basic categorical concept of an S -algebra as derived from the theory of Segal's Γ-sets provides a unifying description of several constructions attempting to model an…

Homotopy types and geometries below Spec Z

- Mathematics
- 2018

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…

Homotopy types and geometries below
Spec(ℤ)

- Mathematics
- 2020

After the first heuristic ideas about 'the field of one element' F₁ and 'geometry in characteristics 1' (J. Tits, C. Deninger, M. Kapranov, A. Smirnov et al.), there were developed several general…

Brauer and Etale Homotopy Obstructions to Rational Points on Open Covers

- Mathematics
- 2020

In 2010, Poonen gave the first example of failure of the local-global principle that cannot be explained by Skorobogatov's etale Brauer-Manin obstruction. Motivated by this example, we show that the…

Witt vectors of non-commutative rings and topological cyclic homology

- Mathematics
- 1997

Classically, one has for every commutative ring A the associated ring of p-typical Wit t vectors W(A). In this paper we extend the classical construction to a functor which associates to any…

Existence of rational points as a homotopy limit problem

- Mathematics
- 2013

We show that the existence of rational points on smooth varieties over a field can be detected using homotopy fixed points of etale topological types under the Galois action. As our main example we…

The $K$-theory of assemblers

- Mathematics
- 2014

In this paper we introduce the notion of an assembler, which formally encodes "cutting and pasting" data. An assembler has an associated $K$-theory spectrum, in which $\pi_0$ is the free abelian…

The annihilator of the Lefschetz motive

- Mathematics
- 2017

In this paper we study a spectrum $K(\mathcal{V}_k)$ such that $\pi_0 K(\mathcal{V}_k)$ is the Grothendieck ring of varieties and such that the higher homotopy groups contain more geometric…