Reconstruction theorems for coadmissible D-cap-modules

We prove a Riemann-Hilbert correspondence for Ardakov-Wadsley's coadmissible D-cap-modules and, more generally, for Bode's $\mathcal{C}$-complexes. More precisely, we show that any given $\mathcal{C}$-complex can be reconstructed out of its solutions. As a corollary, we find that slight modifications of the solution and de Rham functors introduced by the author are fully faithful on $\mathcal{C}$-complexes and, in particular, on coadmissible D-cap-modules. One of the many steps of our proof is the explicit computation of the continuous Galois cohomology of a certain decompletion of $B_{dR}^{+}$ which we call the positive overconvergent de Rham period ring.