  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 adele class space of the field
of rational numbers by the maximal compact subgroup of the idele 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 semifield of tropical real
numbers. The action of the multiplicative group of positive real numbers on the
adele 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 semifield 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.

Algebraic Geometry