From monoids to hyperstructures: in search of an absolute arithmetic

TL;DR

This paper explores the algebraic and geometric structures underlying the Riemann zeta function, connecting trace formulas, hyperstructures, and tropical geometry to recover real numbers algebraically.

Contribution

It introduces a hyperring extension of the hyperfield of signs and develops a function theory over Spec(S), linking hyperstructures to real number reconstruction.

Findings

Trace formula interprets N(q) as an intersection number.

Constructs an extension R^{convex} of hyperfield S.

Recovers real numbers from algebraic structures over Spec(S).

Abstract

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).