The Arithmetic Site

TL;DR

This paper introduces the 'Arithmetic Site', an algebraic geometric framework that models the non-commutative geometric approach to the Riemann zeta function and provides insights into the underlying space related to the Riemann Hypothesis.

Contribution

It establishes an algebraic geometric incarnation of the non-commutative space associated with the Riemann zeta function, connecting tropical geometry, topos theory, and number theory.

Findings

The arithmetic site is realized as a sheaf on a topos related to positive integers.

The points over the maximal compact subring correspond to a non-commutative space quotient of the adele class space.

Frobenius correspondences are realized and their compositions are computed.

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 which is the dual of the multiplicative semigroup of positive integers. We prove that the set of points of the arithmetic site over the maximal compact subring of the tropical semifield is the non-commutative space quotient of the adele class space of Q by the action of the maximal compact subgroup of the idele class group. We realize the Frobenius correspondences in the square of the "Arithmetic Site" and compute their composition. This note provides the algebraic geometric space underlying the non-commutative approach to RH.