Infinitesimal deformation of p-adic differential equations on Berkovich curves

TL;DR

This paper investigates how $p$-adic differential equations on Berkovich curves behave under automorphisms close to the identity, extending previous work and applying results to $p$-adic Gamma functions and $L$-functions.

Contribution

It generalizes the theory of $p$-adic differential equations by establishing semi-linear actions under near-identity automorphisms on Berkovich curves.

Findings

Established conditions for differential equations to acquire semi-linear automorphism actions.

Extended previous results from $p$-adic $q$-difference equations to a broader setting.

Applied theoretical results to analyze properties of $p$-adic Gamma and $L$-functions.

Abstract

We show that if a differential equations $\mathscr{F}$ over a quasi-smooth Berkovich curve $X$ has a certain compatibility condition with respect to an automorphism $\sigma$ of $X$, and if the automorphism is sufficiently close to the identity, then $\mathscr{F}$ acquires a semi-linear action of $\sigma$ (i.e. lifting that on $X$). This generalizes the previous works of Yves Andr\'e, Lucia Di Vizio, and the author about $p$-adic $q$-difference equations. We also obtain an application to Morita's $p$-adic Gamma function, and to related values of $p$-adic $L$-functions.