The Scaling Site

TL;DR

This paper explores a tropical geometric structure called the scaling site, linking it to number theory, tropical curves, and analogues of elliptic curves with Riemann-Roch properties.

Contribution

It introduces the scaling topos as a tropical geometric extension of the arithmetic site, revealing new structures related to number theory and tropical geometry.

Findings

Points of the scaling site match the quotient of the adele class space of Q.

The structure sheaf defines a tropical curve over the arithmetic topos.

Riemann-Roch formula applies with real-valued dimensions and degrees.

Abstract

We investigate the semi-ringed topos obtained by extension of scalars from the arithmetic site of our previous work, by replacing the smallest Boolean semifield by the tropical semifield of real numbers with the max-plus operations. The obtained site is the semi-direct product of the Euclidean half-line by the action of the monoid of positive integers by multiplication. Its points are the same as the points of the arithmetic site over the tropical semifield of real numbers, and coincide with the quotient of the adele class space of Q by the action of the maximal compact subgroup of the idele class group. The structure sheaf of the scaling topos endows it with a natural structure of tropical curve over the arithmetic topos. The restriction of this structure to the periodic orbits of the scaling flow gives, for each prime p, an analogue of an elliptic curve whose Jacobian is a cyclic group of order p-1. The Riemann-Roch formula holds and involves real valued dimensions and real degrees for divisors.