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We relate two different partial p-adic analogues of the classical Riemann-Hilbert correspondence on curves. The first one comes from Deninger-Werner and Faltings and is of algebraic nature. The second one comes from Andre and Berkovich and is defined on Berkovich analytic spaces.

We study a natural map from representations of a free (resp. free abelian) group of rank g in GL_r(C), to holomorphic vector bundles of degree zero over a compact Riemann surface X of genus g (resp. complex torus X of dimension g). This map defines what is called a Schottky functor. Our main result is that this functor induces an equivalence between the category of unipotent representations of Schottky groups and the category of unipotent vector bundles on X. We also show that, over a complex torus, any vector or principal bundle with a flat holomorphic connection is Schottky., Comment: In v3, the results are approached using standard cohomology and derived functor arguments, avoiding extensive use of Yoneda Extensions