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”.
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 ».
Accepted:
Published online:
Alain Connes 1, 2, 3; Caterina Consani 4
@article{CRMATH_2014__352_12_971_0, author = {Alain Connes and Caterina Consani}, title = {The {Arithmetic} {Site}}, journal = {Comptes Rendus. Math\'ematique}, pages = {971--975}, publisher = {Elsevier}, volume = {352}, number = {12}, year = {2014}, doi = {10.1016/j.crma.2014.07.009}, language = {en}, }
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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☆ Both authors thank Ohio State University where this paper was written.
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