Let $\mathcal{D}=\mathcal{D}\_X$ be the sheaf of rings of differential operators on a smooth algebraic curve $X$.


Since $\dim X=1$, the D-modules of the form $\mathcal{D}/\mathcal{D}L$ are necessarily holonomic. Conversely, I've verified that a holonomic D-module over $\mathbb{A}^1$ or $\mathbb{G}\_m$ is always cyclic; i.e., of the form $\mathcal{D}/I$ for some (not necessarily principal) left-ideal $I$.


(As W. Sawin nicely remarked, this is obvious whenever the holonomic D-module $M$ is simple. But even then, it's not clear to me that the left-ideal $I$ in $M=\mathcal{D}/I$ is principal.)


**I wonder if it's true that every simple holonomic D-module over $X$, assumed to be a curve, is of the form $\mathcal{D}/\mathcal{D}L$.** If this is false in general, is it true for $X=\mathbb{A}^1$? For $X=\mathbb{G}\_m$? For elliptic curves?



When are simple holonomic D-modules of the form $\mathcal{D}/\mathcal{D}L$?
If $X$ is a smooth proper curve of positive genus, then there exist simple $D$-modules which have no global sections at all. This is because every degree $0$ line bundle admits a flat connection (since the obstruction to a connection is the Atiyah class which vanishes). The line bundle with connection gives a $D$-module, and as soon as the module is trivial, has no global sections.


Since it has no global sections, it admits no nontrivial map from $\mathcal D$ and hence can't be a quotient of $\mathcal D$.


If $X$ is an affine curve, a $\mathcal D$-module is an honest module and not a sheaf of modules. So it has plenty of global sections. For a holonomic $D$-module, the space of proper submodules is finite-dimensional and each one has infinite codimension, so one can choose an element that is outside all of them, showing that the module has the form $\mathcal D/\mathcal D I$.


However, $I$ need not be principal, unless $X$ has genus $0$ and the module is simple. This is because the quotient by the ideal generated by a differential operator of degree $d$ has generic rank $d$.


If we choose a $D$-module of generic rank $0$, supported at a single point, say, then the generic rank is $0$, so if the ideal is principal it must be generated by some function on $X$, necessarily vanishing only at that point.


It follows that every module of the form $\mathcal D/ DL$ supported at a single point is simple, which is why the simple assumption is necessary, and that this can only happen if the point generates a finite subgroup of the Picard group, which is why the genus $0$ assumption is necessary.


So your desired property can only be true for curves of genus $0$.


