Resumen de The Hall algebra of a curve

Mikhail Kapranov, Olivier Schiffmann, Eric Vasserot

• 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 Gm-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.