  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 adele class space. Then, we discuss the algebraic structure of
the adele class space both as a monoid and as a hyperring. We construct an
extension R^{convex} of the hyperfield S of signs, which is the hyperfield
analogue of the semifield R_+^{max} of tropical geometry, admitting a one
parameter group of automorphisms fixing S. Finally, we develop function theory
over Spec(S) and we show how to recover the field of real numbers from a purely
algebraic construction, as the function theory over Spec(S).
