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

Tinhinane A. Azzouz

TL;DR

This paper develops a spectral theory for p-adic differential equations in the Berkovich setting, extending prior work from constant-coefficient and formal power series cases to differential modules at generic points on quasi-smooth curves. It introduces and analyzes the spectrum with respect to the derivation $(S-c)\frac{d}{dS}$, proving a deep link between the spectrum and radii of convergence, and showing how Frobenius and logarithm push-forwards yield a precise, intrinsic decomposition of the spectrum into spectral pieces $\{z_i\}+\mathbb{Z}_p$ and $\{a_j\}+\mathbb{Z}_p$. A spectral version of Young's theorem is established, allowing spectral decompositions even when radii are small, and the main result provides a concrete decomposition of any differential module into pure components controlled by the spectrum. The framework extends to differential equations on quasi-smooth curves, enabling local spectral analyses via finite étale covers and Frobenius, and connects spectral data with Robba-type radii, offering a refined diagnostic tool for p-adic differential systems with potential applications in p-adic cohomology and non-archimedean analytic geometry.

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.

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

TL;DR

This paper develops a spectral theory for p-adic differential equations in the Berkovich setting, extending prior work from constant-coefficient and formal power series cases to differential modules at generic points on quasi-smooth curves. It introduces and analyzes the spectrum with respect to the derivation , proving a deep link between the spectrum and radii of convergence, and showing how Frobenius and logarithm push-forwards yield a precise, intrinsic decomposition of the spectrum into spectral pieces and . A spectral version of Young's theorem is established, allowing spectral decompositions even when radii are small, and the main result provides a concrete decomposition of any differential module into pure components controlled by the spectrum. The framework extends to differential equations on quasi-smooth curves, enabling local spectral analyses via finite étale covers and Frobenius, and connects spectral data with Robba-type radii, offering a refined diagnostic tool for p-adic differential systems with potential applications in p-adic cohomology and non-archimedean analytic geometry.

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 -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.

Paper Structure

This paper contains 38 sections, 85 theorems, 209 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions and notations
  4. Differential modules
  5. Analytic spaces
  6. Universal points and fiber of a point under extension of scalars
  7. Sheaf of differential forms and étale morphisms
  8. Differential equations over an affinoid domain
  9. Spectrum of a differential module
  10. Push-forward and pull-back of a differential module and their spectra
  11. Spectrum of a differential module with regular singularities
  12. Spectrum of the derivation $S\frac{\mathrm{d}}{\mathrm{dS}}$
  13. The case of positive residual characteristic
  14. The case where $x\in (0,\infty)$
  15. The case where $x\not\in (0,\infty)$
  16. ...and 23 more sections

Key Result

Theorem 1.1

Assume that $\mathrm{char}(\tilde{k})=p>0$ and $x:=x_{0,r}\in {\mathbb A}^{1,\mathrm{an}}_{k}\setminus k$. Let $(M,\nabla)$ be a differential module $(\mathscr{H}(x), S\frac{\mathrm{d}}{\mathrm{dS}})$. Let $\operatorname{\mathbf{Frob}}_p:{\mathbb A}^{1,\mathrm{an}}_{k}\to {\mathbb A}^{1,\mathrm{an}}

Theorems & Definitions (173)

  • Theorem 1.1: Theorem \ref{['sec:bf-spectr-comp-3']}
  • Example 1.2
  • Theorem 1.3: Theorem \ref{['sec:spectr-diff-equat-1']}
  • Theorem 1.4: Theorem \ref{['sec:spectr-vers-youngs-9']}
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • Definition 2.8
  • Proposition 2.9
  • proof
  • ...and 163 more