Radius of convergence of p-adic connections and the Berkovich ramification locus

TL;DR

This paper investigates the radius of convergence for p-adic connections on curves, providing bounds and new proofs for ramification loci, and clarifies its relation to Kedlaya's intrinsic notion.

Contribution

It extends the theory of p-adic radii of convergence to direct images under finite morphisms and offers a new geometric proof of a p-adic Rolle theorem.

Findings

Lower bounds for the radius in etale coverings with good reduction

A new geometric proof of a variant of the p-adic Rolle theorem

Clarification of the relation between different notions of radius of convergence

Abstract

We apply the theory of the radius of convergence of a p-adic connection to the special case of the direct image of the constant connection via a finite morphism of compact p-adic curves, smooth in the sense of rigid geometry. In the case of an etale covering of curves with good reduction, we get a lower bound for that radius and obtain a new geometric proof of a variant of the p-adic Rolle theorem of Robert and Berkovich. We take this opportunity to clarify the relation between our notion of radius of convergencand the more intrinsic one used by Kedlaya