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 Xis 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. (c) 2025 Elsevier Masson SAS. All rights are reserved, including those for text and data mining, AI training, and similar technologies.