p-Adic differential equations and p-adic coefficients on curves

Let R be the ring of functions that are analytic at the edge of the disk D(0, 1) in some p-adic field and let A be the subring of those functions that are analytic in the entire disk. The aim of this paper is to prove, under some mild conditions over the p-adic monodromy, that an R-module free of finite rank with connection contains an A-lattice over which the connection has for unic singularity a meromorphic one in 0. This result is basic for local and global finitness properties in the theory of p-adic coefficients. Its proof uses all known results about p-adic differential equations giving the opportunity to present their theory.