**Disclaimer:** I know very little about both of the fields in question.


My question is pretty simple:



> 
> **What's the relation between differential Galois theory and D-modules
>  over algebraic curves?**
> 
> 
> 


Differential galois theory can't subsume D-modules obviously since the latter contains also information about behavior at singularities. So, in particular, **is differential Galois theory a "natural subset" of D-module theory over curves?** If so then in what precise sense? If not why? Still are there any methods/algorithms/ideas from differential Galois theory which are useful for studying D-modules over curves?


Finally, are there any sources discussing this?



D-modules over algebraic curves VS differential Galois theory
Consider what happens if you take a $D$-module on an algebraic curve (with field of fractions $K$) and remove all the information on the singularities. You can achieve this by tensoring over the structure sheaf with $K$, obtaining a module for the ring of differential operators on $K$. This ring is generated over $K$ by differentiation along a single meromorphic vector field (since it's generated by differentiation along all vector fields. So a module over it is just a $K$-vector space with a semilinear action of this differentiation. This will usually be finite-dimensional (I think always for holonomic $D$-modules).


To pass to the Galois theory, we pick a specific vector field and view it as a derivation $D$ on $K$, so we have a finite-dimensional vector space with an action of $D$.


From a finite-dimensional vector space with an action of $D$ one can make a differential field extension using Picard-Vessiot theory. Take a ring generated by independent transcendentals corresponding to basis of this vector space, with the $D$ action given by the $D$ action on the vector space, mod out by a maximal differential ideal, and take the field of fractions.


Any field extension generated by solutions of ODEs arises this way, because we can construct from an order $n$ ODE the vector space generated by a formal solution and its first $n-1$ derivatives and take the corresponding ring, which maps to the field, and the kernel is a differential ideal.




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I think this object, a vector space with an action of $D$ is one of the simplest objects one could study in the theory of ODEs, I guess other than an ODE itslf. To some extent, in differential Galois theory and D-module theory, we would take these objects and study them in different ways - in D-modules, one obviously passes from vector spaces to the richer $\mathcal O\_X$-modules, which we can study using commutative algebra, and also allow more than one differential operator to act at the same time, creating more interesting algebra, while in differential Galois theory, we pass to studying the differential field extensions and their automorphism groups, often including, as Avi notes, higher degree differential polynomials.


However there is a specific area where they remain close together. When we study $D$-modules whose underlying $\mathcal O\_X$-module is locally free, the space of analytic solutions is a representation of $\pi\_1(X)$. On the other hand the space of solutions in a differential field large enough to contain all the solutions is a representation of the differential Galois group. These representations can be identified, with the image of $\pi\_1$ inside $GL\_n$ a subgroup of the differential Galois group - this is simply because analytic continuation around a loop in $X$ always acts as an automorphism of the field of analytic solutions. In good cases (e.g. regular singularities, by the Riemann-Hilbert correspondence), the Zariski closure of $\pi\_1$ (the "monodromy group") is precisely equal to the differential Galois group, but not always - as in the case of $e^x$, which has no monodromy but a nontrivial differential Galois group.


So some aspects of the theory of $D$-modules, specifically their comparison to local systems / sheaves and the Riemann-Hilbert correspondence, are closely related to the representation theory of the relevant differential Galois group.


