Abstract
Let X be a smooth projective curve over a finite field. We describe H, the full Hall algebra of vector bundles on X, as a Feigin–Odesskii shuffle algebra. This shuffle algebra corresponds to the scheme S of all cusp eigenforms and to the rational function of two variables on S coming from the Rankin–Selberg L-functions. This means that the zeroes of these L-functions control all the relations in H. The scheme S is a disjoint union of countably many \({\mathbb G}_m\)-orbits. In the case when X has a theta-characteristic defined over the base field, we embed H into the space of regular functions on the symmetric powers of S.
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Kapranov, M., Schiffmann, O. & Vasserot, E. The Hall algebra of a curve. Sel. Math. New Ser. 23, 117–177 (2017). https://doi.org/10.1007/s00029-016-0239-9
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DOI: https://doi.org/10.1007/s00029-016-0239-9