The Arithmetic Site

@article{Connes2014TheAS,
  title={The Arithmetic Site},
  author={Alain Connes and Caterina Consani},
  journal={arXiv: Number Theory},
  year={2014}
}
View PDF on arXiv
49 Citations
Highly Influential Citations
5
Background Citations
18
Methods Citations
8

Figures from this paper

49 Citations

Geometry of the Arithmetic Site
The Scaling Site
The Scaling Site Le Site des Fréquences
We investigate the semi-ringed topos obtained by extension of scalars from the arithmetic site A of [3, 4], by replacing the smallest Boolean semifield B by the tropical semifield R + . The obtained…
The cyclic and epicyclic sites
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…
BC-system, absolute cyclotomy and the quantized calculus
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…
Segal’s Gamma rings and universal arithmetic
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…
Semiring Congruences and Tropical Geometry
One of the main motivations and inspirations for this thesis is the still open question of the definition of geometry in characteristic one. This is geometry over a structure, called an idempotent…
An arithmetic topos for integer matrices
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…
Universal Thickening of the Field of Real Numbers
We define the universal thickening of the field of real numbers. This construction is performed in three steps which parallel the universal perfection, the Witt construction and a completion process.…
...
...

References

SHOWING 1-10 OF 13 REFERENCES
From monoids to hyperstructures: in search of an absolute arithmetic
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…
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…
Algèbres de polynômes tropicaux
We continue, in this second article, the study of the algebraic tools which play a role in tropical algebra. We especially examine here the polynomial algebras over idempotent semi-fields. This work…
Schemes over 𝔽1 and zeta functions
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…
Sheaves in geometry and logic: a first introduction to topos theory
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…
Remarks on Semimodules
This is a study of universal problems for semimodules, in particular coequalizers, coproducts, and tensor products. Furthermore the structure theory of semiideals of the semiring of natural numbers…
Sheaves In Geometry And Logic
Sur une note de Mattuck-Tate.
1. Dans un travail recent [4], Mattuck et Täte deduisent Tinegalite fondamentale de A. Weil qui etablit Fhypothöse de Riemann pour les corps de fonctions [7] comme consequence facile du theoreme de…
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…
Noncommutative Geometry, Quantum Fields and Motives
Quantum fields, noncommutative spaces, and motives The Riemann zeta function and noncommutative geometry Quantum statistical mechanics and Galois symmetries Endomotives, thermodynamics, and the Weil…
...
...