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 $\phi :{\mathcal{D}}_1\to {\mathcal{D}}_2$ of unit discs over $k$, a differential module $(M,D)$ on ${\mathcal{D}}_1$ and a point $b\in {\mathcal{D}}_2(k)$, we construct explicitly an optimal basis of space of horizontal elements for the direct image ${\phi }_{\ast }(M,D)$ at $b$ in terms of the suitable chosen optimal bases of $(M,D)$ at preimages of $b$ by $\phi$ and ramification properties of the morphism.

30 pages; comments are welcome