Spherical Hall Algebra of $$\overline{\text{Spec }(\mathbb{Z})}$$

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 \(\mathit{GL}_{n}(\mathbb{Z}),n > 0\), 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 appropriate completion of H and considered modulo rescaling), is identified with the space spanned by nontrivial zeroes of the Riemann zeta function.KeywordsVector BundleMeromorphic FunctionAssociative AlgebraEisenstein SeriesAutomorphic FormThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.