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Comptes Rendus Mathématique
Volume 352, n° 12
pages 971-975 (décembre 2014)
Doi : 10.1016/j.crma.2014.07.009
Received : 18 May 2014 ;  accepted : 7 July 2014
The Arithmetic Site
Le Site arithmétique
 

Alain Connes a, b, c , Caterina Consani d, 1
a Collège de France, 3, rue d'Ulm, 75005 Paris, France 
b I.H.E.S., France 
c Ohio State University, USA 
d The Johns Hopkins University, Baltimore, MD 21218, USA 

Abstract

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   viewed as a sheaf on the topos   dual to the multiplicative semigroup of positive integers. We realize the Frobenius correspondences in the square of the “Arithmetic Site”.

The full text of this article is available in PDF format.
Résumé

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   vu comme un faisceau sur le topos   dual du semigroupe multiplicatif des entiers positifs. Nous réalisons les correspondances de Frobenius dans le carré du « Site arithmétique ».

The full text of this article is available in PDF format.

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

1  Partially supported by the NSF grant DMS 1069218.


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