The convergence Newton polygon of a $p$-adic differential equation III : global decomposition and controlling graphs

Abstract

We deal with locally free $\mathcal{O}_X$-modules with connection over a Berkovich curve $X$. As a main result we prove local and global decomposition theorems of such objects by the radii of convergence of their solutions. We also derive a bound of the number of edges of the controlling graph, in terms of the geometry of the curve and the rank of the equation. As an application we provide a classification result of such equations over elliptic curves.Comment: 81 pages. This is a first draft, containing a maximum number of details. We plan to reduce its volume in a next versio