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arXiv · 1201.6185

The Hall algebra of a curve

Mikhail Kapranov·Olivier Schiffmann·Eric Vasserot
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Abstract

Let X be a smooth projective curve over a finite field. We describe H, the full Hall algebra of vector bundles 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 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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Mikhail Kapranov, Olivier Schiffmann, Eric Vasserot. 2012-03-01. The Hall algebra of a curve. https://arxiv.org/abs/1201.6185

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