Ideals of rings of differential operators on algebraic curves (with an appendix by George Wilson)

Let X be a smooth affine irreducible curve over C and let D=D(X) be the ring of global differential operators on X. In this paper, we give a geometric classification of left ideals in D and study the natural action of the Picard group of D on the space of isomorphism classes of such ideals. Our results generalize the classification of left ideals of the first Weyl algebra A1(C) given in Berest and Wilson (2000, 2002) [15,16].