Cohomology of the differential fundamental group of algebraic curves

Let X be a smooth projective curve over a field k of characteristic zero. The differential fundamental group of X is defined as the Tannakian dual to the category of vector bundles with (integrable) connections on X. This work investigates the relationship between the de Rham cohomology of a vector bundle with connection and the group cohomology of the corresponding representation of the differential fundamental group of X. Consequently, we obtain some vanishing and non-vanishing results for the group cohomology.