  We construct the scaling site S by implementing the extension of scalars on
the arithmetic site, from the smallest Boolean semifield to the tropical
semifield of positive real numbers. The obtained semiringed topos is the
Grothendieck topos semi-direct product of the Euclidean half-line and the
monoid of positive integers acting by multiplication, endowed with the
structure sheaf of piecewise affine, convex functions with integral slopes. We
show that the points of this topos coincide with the adele class space of the
rationals and that this latter space inherits the geometric structure of a
tropical curve. We restrict this construction to the periodic orbit of the
scaling flow associated to each prime and obtain a quasi-tropical structure
which turns this orbit into a variant C of the classical Jacobi description of
an elliptic curve. On C, we develop the theory of Cartier divisors, determine
the structure of the quotient of the abelian group of divisors by the subgroup
of principal divisors, develop the theory of theta functions, and prove the
Riemann-Roch formula which involves real valued dimensions, as in the type II
index theory. We show that one would have been led to the same definition of
the scaling site by analyzing the well known results on the localization of
zeros of analytic functions involving Newton polygons in the non-archimedean
case and the Jensen's formula in the complex case.
