Resumen de Spectrum of p-adic linear differential equations I: the shape of the spectrum
This paper extends our previous works Azzouz (Math Z 296(3–4): 1613–1644, 2020;
Number Theory 231:139–157, 2022) 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 Robba (Ann Math 101(2):280–316, 1975).
