We introduce a quantization of the graded algebra of functions on the
canonical cone of an algebraic curve C, based on the theory of formal
pseudodifferential operators. When C is a complex curve with Poincar\'e
uniformization, we propose another, equivalent construction, based on the work
of Cohen-Manin-Zagier on Rankin-Cohen brackets. We give a presentation of the
quantum algebra when C is a rational curve, and discuss the problem of
constructing algebraically "differential liftings"
