Operateurs differentiels globaux sur les courbes elliptiques

Let X a proper smooth curve over the field of complex numbers. Localization of the Heisenberg algebra gives the algebra of global sections of the ring of differential operators on the Jacobian J of X. It seems natural to ask for same kind of relations over a characteristic p > 0 field replacing the Heisenberg algebra by its entire form. We propose a conjecture about the constuction of an arrow between the two objects when X is an elliptic curve.