The convergence Newton polygon of a p-adic differential equation I: Affinoid domains of the Berkovich affine line

Abstract

We prove that the radii of convergence of the solutions of a p-adic differential equation $\mathcal{F}$ over an affinoid domain X of the Berkovich affine line are continuous functions on X that factorize through the retraction of $X\to \mathrm{\Gamma }$ of X onto a finite graph $\mathrm{\Gamma }\subseteq X$. We also prove their super-harmonicity properties. This finiteness result means that the behavior of the radii as functions on X is controlled by a finite family of data.