Hilbert’s twenty-first problem, formulated to generalize Riemann’s work on hypergeometric equations, concerns the existence of linear differential equations of Fuchsian type on the complex plane with specified singular points and monodromic group.  Its modern solution, due to Deligne and known as the Riemann-Hilbert correspondence, establishes an equivalence between two different types of data on a complex algebraic manifold X: the representations of the fundamental group of X (topological data) and the linear systems of algebraic differential equations on X with regular singularieties (algebraic data).

I’ll review this classical theory, and discuss some recent progress towards similar problems for p-adic manifolds.