From monoids to hyperstructures: in search of an absolute arithmetic

We show that the trace formula interpretation of the explicit formulas expresses the counting function N(q) of the hypothetical curve C associated to the Riemann zeta function, as an intersection number involving the scaling action on the adèle class space. Then, we discuss the algebraic structure of the adèle class space both as a monoid and as a hyperring. We construct an extension Rconvex of the hyperfield S of signs, which is the hyperfield analogue of the semifield Rmax + of tropical geometry, admitting a one parameter group of automorphisms fixing S. Finally, we develop function theory over SpecK and we show how to recover the field of real numbers from a purely algebraic construction, as the function theory over SpecS.