Geometry of the Arithmetic Site

By Alain Connes, C. Consani. 2015

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We introduce the Arithmetic Site: an algebraic geometric space deeply related to the non-commutative geometric approach to the Riemann Hypothesis. We prove that the non-commutative space quotient of the ad ele class space of the eld of rational numbers by the maximal compact subgroup of the id ele class group, which we had previously shown to yield the correct counting function to obtain the complete Riemann zeta function as Hasse-Weil zeta function, is the set of geometric points of the arithmetic site over the semield of tropical real numbers. The action of the multiplicative group of positive real numbers on the ad ele class space corresponds to the action of the Frobenius automorphisms on the above geometric points. The underlying topological space of the arithmetic site is the topos of functors from the multiplicative semigroup of non-zero natural numbers to the category of sets. The structure sheaf is made by semirings of characteristic one and is given globally by the semield of tropical integers. In spite of the countable combinatorial nature of the arithmetic site, this space admits a one parameter semigroup of Frobenius correspondences obtained as sub-varieties of the square of the site. This square is a semi-ringed topos whose structure sheaf involves Newton polygons. Finally, we show that the arithmetic site is intimately related to the structure of the (absolute) point in non-commutative geometry.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/87eeeedd8a79537585bfbd00b88e67a12c4b6af5