# Geometry of the Arithmetic Site

@article{Connes2015GeometryOT,
title={Geometry of the Arithmetic Site},
author={Alain Connes and Caterina Consani},
year={2015},
volume={291},
pages={274-329}
}
• Published 19 February 2015
• Mathematics

## Figures from this paper

The Scaling Site
• Mathematics
• 2015
An arithmetic site of Connes-Consani type for imaginary quadratic fields with class number 1
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
Noncommutative Geometry, the Spectral Standpoint
• A. Connes
• Mathematics
New Spaces in Physics
• 2019
We report on the following highlights from among the many discoveries made in Noncommutative Geometry since year 2000: 1) The interplay of the geometry with the modular theory for noncommutative
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
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

## References

SHOWING 1-10 OF 42 REFERENCES
The Arithmetic Site
• Mathematics
• 2014
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
Noncommutative geometry, arithmetic, and related topics : proceedings of the Twenty-first Meeting of the Japan-U.S. Mathematics Institute
• Mathematics
• 2011
This valuable collection of essays by some of the world's leading scholars in mathematics presents innovative and field-defining work at the intersection of noncommutative geometry and number theory.
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
• Mathematics
Compositio 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
On a representation of the idele class group related to primes and zeros of L-functions
Let K be a global field. Using natural spaces of functions on the adele ring and the idele class group of K, we construct a virtual representation of the idele class group of K whose character is
Trace formula in noncommutative geometry and the zeros of the Riemann zeta function
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