Appendix: proof of the Uniformity Conjecture

TL;DR

This paper proves a variant of the p-adic Rolle theorem using advanced theories of convergence radii and semistable reduction of p-adic curves, contributing to the understanding of p-adic differential equations.

Contribution

It introduces a new proof of the Uniformity Conjecture by comparing different notions of radius of convergence in p-adic geometry.

Findings

Established a variant of Alain Robert's p-adic Rolle theorem.

Compared and related different notions of radius of convergence.

Provided insights into the structure of p-adic differential modules.

Abstract

This paper originated as an appendix to the paper "Topology and Geometry of the Berkovich Ramification Locus for Rational Functions, II" by Xander Faber arXiv:1104.0943v2 [math.NT]. It may however be read independently. We prove a variant of Alain Robert's p-adic Rolle theorem, via the theory of the radius of convergence of p-adic connections and the theory of semistable reduction of p-adic curves. We carefully compare the present author's notion [Inv. Math. 182 (2010)] of radius of convergence, of a connection on a p-adic curve X, normalized by the choice of a semistable model of X, with Kedlaya's intrinsic generic radius of convergence of a differential module [Def. 9.4.7 in p-adic Differential Equations, Cambridge Studies in Adv. Math., vol. 125 (2010)].