Geometry of the scaling site

Abstract

In this paper we construct the scaling site ${\mathscr {S}}$ by implementing the extension of scalars on the arithmetic site ${\mathscr {A}}$, from the smallest Boolean semifield ${\mathbb B}$ to the tropical semifield ${\mathbb R}_+^\mathrm{max}$. The obtained semiringed topos is the Grothendieck topos ${[0,\infty )\rtimes {{\mathbb N}^{\times }}}$, semi-direct product of the Euclidean half-line and the monoid ${\mathbb N}^{\times }$ of positive integers acting by multiplication, endowed with the structure sheaf of piecewise affine, convex functions with integral slopes. We show that pointwise ${[0,\infty )\rtimes {{\mathbb N}^{\times }}}$ coincides with the adele class space of ${\mathbb Q}$ 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 p and obtain a quasi-tropical structure which turns this orbit into a variant $C_p={\mathbb R}_+^*/p^{\mathbb Z}$ of the classical Jacobi description ${\mathbb C}^*/q^{\mathbb Z}$ of an elliptic curve. On $C_p$, we develop the theory of Cartier divisors, determine the structure of the quotient $\mathrm{Div}(C_p)/{\mathcal P}$ 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 ${\mathscr {S}}$ 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 formula in the complex case.