## 37 Citations

An arithmetic site of Connes-Consani type for imaginary quadratic fields with class number 1

- Mathematics
- 2017

We construct, for imaginary quadratic number fields with class number 1, an arithmetic site of Connes-Consani type. The main difficulty here is that the constructions of Connes and Consani and part…

MODEL THEORY AND GEOMETRY OF REPRESENTATIONS OF RINGS OF INTEGERS

- Mathematics
- 2015

The aim of this project is to attach a geometric structure to the ring of integers. It is generally assumed that the spectrum Spec(Z) dened by Grothendieck serves this purpose. However, it is still…

On the Etale Fundamental Group of Schemes over the Natural Numbers

- Mathematics
- 2019

The aims of the present thesis are to give a concrete description, in the modern language of arithmetic-algebraic geometry, of the Galois theory of Alexander Grothendieck (and the later generation of…

BC-system, absolute cyclotomy and the quantized calculus

- Mathematics
- 2021

We give a short survey on several developments on the BC-system, the adele class space of the rationals, and on the understanding of the ”zeta sector” of the latter space as the Scaling Site. The new…

Homological algebra in characteristic one

- Mathematics
- 2017

This article develops several main results for a general theory of homological algebra in categories such as the category of sheaves of idempotent modules over a topos. In the analogy with the…

Segal’s Gamma rings and universal arithmetic

- Mathematics
- 2020

Segal's Gamma-rings provide a natural framework for absolute algebraic geometry. We use Almkvist's global Witt construction to explore the relation with J. Borger F1-geometry and compute the Witt…

## References

SHOWING 1-10 OF 42 REFERENCES

Characteristic one, entropy and the absolute point

- Mathematics
- 2009

We show that the mathematical meaning of working in characteristic one is directly connected to the fields of idempotent analysis and tropical algebraic geometry and we relate this idea to the notion…

From monoids to hyperstructures: in search of an absolute arithmetic

- Mathematics
- 2010

We show that the trace formula interpretation of the explicit formulas expresses the counting functionN.q/ of the hypothetical curveC associated to the Riemann zeta function, as an intersection…

The cyclic and epicyclic sites

- Mathematics
- 2014

We determine the points of the epicyclic topos which plays a key role in the geometric encoding of cyclic homology and the lambda operations. We show that the category of points of the epicyclic…

Schemes over 𝔽1 and zeta functions

- MathematicsCompositio Mathematica
- 2010

Abstract We determine the real counting function N(q) (q∈[1,∞)) for the hypothetical ‘curve’ $C=\overline {\mathrm {Spec}\,\Z }$ over 𝔽1, whose corresponding zeta function is the complete Riemann…

Trace formula in noncommutative geometry and the zeros of the Riemann zeta function

- Mathematics
- 1998

Abstract. We give a spectral interpretation of the critical zeros of the Riemann zeta function as an absorption spectrum, while eventual noncritical zeros appear as resonances. We give a geometric…

Sheaves in geometry and logic: a first introduction to topos theory

- Mathematics
- 1992

This text presents topos theory as it has developed from the study of sheaves. Sheaves arose in geometry as coefficients for cohomology and as descriptions of the functions appropriate to various…

Tropical Algebraic Geometry

- Mathematics
- 2007

Preface.- 1. Introduction to tropical geometry - Images under the logarithm - Amoebas - Tropical curves.- 2. Patchworking of algebraic varieties - Toric geometry - Viro's patchworking method -…

The sieve topology on the arithmetic site

- Mathematics
- 2016

The induced topology on the points of the Connes–Consani "arithmetic site," which are the finite adele classes, is family coarse. In this note we define another topology on this set having properties…