Document Zbl 1357.11054 - zbMATH Open
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Spherical Hall algebra of \(\overline{\text{Spec}(\mathbb Z)}\). (English) Zbl 1357.11054
Castano-Bernard, Ricardo (ed.) et al., Homological mirror symmetry and tropical geometry. Based on the workshop on mirror symmetry and tropical geometry, Cetraro, Italy, July 2–8, 2011. Cham: Springer (ISBN 978-3-319-06513-7/pbk; 978-3-319-06514-4/ebook). Lecture Notes of the Unione Matematica Italiana 15, 153-196 (2014).
Summary: 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 \(\mathrm{GL}_n(\mathbb Z)\), \(n > 0\), 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 \(\mathrm{Spec}(Z)\). 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 zeros of the Riemann zeta-function.
For the entire collection see [Zbl 1300.14001].