Continuity of the radius of convergence of differential equations on $p$-adic analytic curves

TL;DR

This paper studies the behavior of the radius of convergence for $p$-adic differential equations on analytic curves, establishing its continuity and concavity, and characterizing Robba connections with constant solution sheaves.

Contribution

It introduces an intrinsic normalized radius of convergence on $p$-adic curves and proves its continuity and concavity, providing new insights into $p$-adic differential equations.

Findings

Proves continuity of the radius of convergence function.

Establishes logarithmic concavity of the radius function.

Characterizes Robba connections with constant solution sheaves.

Abstract

This paper deals with connections on $p$-adic analytic curves, in the sense of Berkovich. The curves must be compact but the connections are allowed to have a finite number of meromorphic singularities on them. For any choice of a semistable formal model of the curve, we define an intrinsic notion of normalized radius of convergence as a function on the curve, with values in $(0,1]$. For a sufficiently refined choice of the semistable model, we prove continuity and logarithmic concavity of that function. We characterize \emph{Robba connections}, that is connections whose sheaf of solutions is constant on any open disk contained in the curve.