The convergence Newton polygon of a $p$-adic differential equation III : global decompositions

TL;DR

This paper develops local and global decomposition theorems for $p$-adic differential equations on Berkovich curves based on convergence radii, providing bounds on their structural complexity and classifying equations over elliptic curves.

Contribution

It introduces new decomposition theorems for $p$-adic differential modules on Berkovich curves using convergence radii, and offers bounds and classifications related to the curve's geometry.

Findings

Decomposition theorems for $p$-adic differential modules

Bound on the number of edges of the controlling graph

Classification of equations over elliptic curves

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.