  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.
