The spherical Hall algebra of Spec(Z)

TL;DR

This paper introduces an arithmetic analog of the Hall algebra for Spec(Z), linking automorphic forms, quadratic relations, and the zeros of the Riemann zeta function within a novel algebraic framework.

Contribution

It constructs a new algebraic structure for automorphic forms on Spec(Z), identifying it with a shuffle algebra related to the Riemann zeta function and exploring its relations.

Findings

H is identified with a Feigin-Odesskii type shuffle algebra.

Quadratic relations encode the functional equation of Eisenstein-Maass series.

Cubic relations correspond to nontrivial zeros of the zeta function.

Abstract

We study an arithmetic analog of the Hall algebra of a curve, when the curve is replaced by the spectrum of the integers compactified at infinity. The role of vector bundles is played by lattices with quadratic forms. This algebra H consists of automorphic forms with respect to GL_n(Z), n>0, with multiplication given by the parabolic pseudo-Eisenstein series map. We concentrate on the subalgebra SH in H generated by functions on the Arakelov Picard group of Spec(Z). We identify H with a Feigin-Odesskii type shuffle algebra, with the function defining the shuffle algebra expressed through the Riemann zeta function. As an application we study relations in H. Quadratic relations express the functional equation for the Eisenstein-Maass series. We show that the space of additional cubic relations (lying an an appropriate completion of H and considered modulo rescaling), is identified with the space spanned by nontrivial zeroes of the zeta function.