The scaling site

Abstract

We investigate the semi-ringed topos obtained from the arithmetic site A of [3,4], by extension of scalars from the smallest Boolean semifield B to the tropical semifield R+max. The obtained site [0,∞)⋊N× is the semi-direct product of the Euclidean half-line and the monoid N× of positive integers acting by multiplication. Its points are the same as the points A(R+max) of A over R+max and form the quotient of the adele class space of Q by the action of the maximal compact subgroup Zˆ⁎ of the idèle class group. The structure sheaf of the scaling topos endows it with a natural structure of tropical curve over the topos N׈. The restriction of this structure to the periodic orbits of the scaling flow gives, for each prime p, an analogue Cp of an elliptic curve whose Jacobian is Z/(p−1)Z. The Riemann–Roch formula holds on Cp and involves real-valued dimensions and real degrees for divisors.