The Arithmetic Site

Alain Connes; Caterina Consani

doi:10.1016/j.crma.2014.07.009

Number theory/Algebraic geometry

The Arithmetic Site

Presented by: the Editorial Board

Alain Connes ^1,^2,³; Caterina Consani ⁴

¹ Collège de France, 3, rue d'Ulm, 75005 Paris, France
² I.H.E.S., France
³ Ohio State University, USA
⁴ The Johns Hopkins University, Baltimore, MD 21218, USA

Comptes Rendus. Mathématique, Volume 352 (2014) no. 12, pp. 971-975.

Abstract
Résumé

We show that the non-commutative geometric approach to the Riemann zeta function has an algebraic geometric incarnation: the “Arithmetic Site”. This site involves the tropical semiring $\bar{N}$ viewed as a sheaf on the topos $\hat{N^{\times}}$ dual to the multiplicative semigroup of positive integers. We realize the Frobenius correspondences in the square of the “Arithmetic Site”.

Le « Site arithmétique » est l'incarnation en géométrie algébrique de l'espace non commutatif, de nature adélique, qui permet d'obtenir la fonction zêta de Riemann comme fonction de dénombrement de Hasse–Weil. Ce site est construit à partir du semi-anneau tropical $\bar{N}$ vu comme un faisceau sur le topos $\hat{N^{\times}}$ dual du semigroupe multiplicatif des entiers positifs. Nous réalisons les correspondances de Frobenius dans le carré du « Site arithmétique ».

Received: 2014-05-18
Accepted: 2014-07-07
Published online: 2014-10-08

DOI: 10.1016/j.crma.2014.07.009

Author's affiliations:

Alain Connes ^{1, 2, 3}; Caterina Consani ⁴

¹ Collège de France, 3, rue d'Ulm, 75005 Paris, France
² I.H.E.S., France
³ Ohio State University, USA
⁴ The Johns Hopkins University, Baltimore, MD 21218, USA

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     author = {Alain Connes and Caterina Consani},
     title = {The {Arithmetic} {Site}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {971--975},
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     year = {2014},
     doi = {10.1016/j.crma.2014.07.009},
     language = {en},
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Alain Connes; Caterina Consani. The Arithmetic Site. Comptes Rendus. Mathématique, Volume 352 (2014) no. 12, pp. 971-975. doi : 10.1016/j.crma.2014.07.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.07.009/

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Cited by Sources:

^☆ Both authors thank Ohio State University where this paper was written.

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