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

Abstract

Let $(k,|\cdot|)$ be a complete and algebraically closed valued field extension of $(\mathbb{Q}_p,|\cdot|_p)$. Given a finite morphism $\varphi:\mathscr{D}_1\to \mathscr{D}_2$ of unit discs over $k$, a differential module $(M,D)$ on $\mathscr{D}_1$ and a point $b\in \mathscr{D}_2(k)$, we construct explicitly an optimal basis of space of horizontal elements for the direct image $\varphi_*(M,D)$ at $b$ in terms of the suitable chosen optimal bases of $(M,D)$ at preimages of $b$ by $\varphi$ and ramification properties of the morphism.