The convergence Newton polygon of a $p$-adic differential equation IV : controlling graphs

TL;DR

This paper refines the understanding of the controlling graph for $p$-adic differential equations by providing bounds based on geometric and cohomological properties, leveraging super-harmonicity of convergence radii.

Contribution

It introduces bounds on the size of the controlling graph in terms of the curve's geometry and module rank, extending previous finiteness results.

Findings

Bound on controlling graph size related to curve geometry and rank

Relation between total height of Newton polygon and de Rham Euler characteristic

Super-harmonicity properties of convergence radii and partial heights

Abstract

In our previous works we proved a finiteness property of the radii of convergence functions associated with a vector bundle with connection on $p$-adic analytic curves. We showed that the radii are locally constant functions outside a locally finite graph in the curve, called controlling graph. In this paper we refine that finiteness results by giving a bound on the size of the controlling graph in terms of the geometry of the curve and the rank of the module. This is based on super-harmonicity properties of radii of convergence and partial heights of the Newton polygon. Under suitable assumptions, we relate the size of the controlling graph associated with the total height of the convergence Newton polygon to the Euler characteristic in the sense of de Rham cohomology.