Spectrum of p-adic linear differential equations I: The shape of the spectrum

TL;DR

This paper analyzes the spectrum of p-adic linear differential equations on quasi-smooth curves, revealing its relation to spectral radii and providing a finer decomposition than existing methods.

Contribution

It extends previous work to generic points on curves, linking the spectrum to spectral radii and offering a more detailed spectral decomposition.

Findings

Spectrum relates to spectral radii of convergence.

Provides a finer spectral decomposition than Robba's.

Connects spectrum analysis to p-adic differential equations on curves.

Abstract

This paper extends our previous works arXiv:1802.07306 [math.NT], arXiv:1808.02382 [math.NT] on determining the spectrum, in the Berkovich sense, of ultrametric linear differential equations. Our previous works focused on equations with constant coefficients or over a field of formal power series. In this paper, we investigate the spectrum of $p$-adic differential equations at a generic point on a quasi-smooth curve. This analysis allows us to establish a significant connection between the spectrum and the spectral radii of convergence of a differential equation when considering the affine line. Furthermore, the spectrum offers a more detailed decomposition compared to Robba's decomposition based on spectral radii.