Optimal bases for direct images of $p$-adic differential modules over discs

Abstract

Let (k, |•|) be a complete and algebraically closed valued field extension of (Qp, |•|p). Given a finite morphism ϕ : D 1 → D 2 of unit discs over k, a differential module (M, D) on D 1 and a point b ∈ D 2 (k), we construct explicitly an optimal basis of space of horizontal elements for the direct image ϕ * (M, D) at b in terms of the suitable chosen optimal bases of (M, D) at preimages of b by ϕ and ramification properties of the morphism.