Research
Thesis
Title: Spectrum of p-adic differential equations
Abstract
Differential equations constitute an important tool for the investigation of algebraic and analytic varieties, over the complex and the p-adic numbers. In the p-adic setting, they present phenomena that do not appear in the complex case. Indeed, the radius of convergence of the solutions of a linear differential equation may be finite, even without the presence of poles.
The knowledge of that radius permits us to obtain several interesting informations about the equation. More precisely, since the works of F. Baldassarri, we know how to associate a radius of convergence to all points of a p-adic curve in the sense of Berkovich endowed with a connection. Recent works of F. Baldassarri, K.S. Kedlaya, J. Poineau, and A. Pulita have proved that this radius behaves in a very controlled way. The radius of convergence can be refined using subsidiary radii, which are known to have similar properties. In order to push forward the study, we introduce a geometric object that refines also this radius, the spectrum in the sense of Berkovich of a differential equation.
In our thesis, we define the spectrum of a differential equation and provide its first properties. We also compute the spectra of some classes of differential modules: differential modules of a differential équation with constant coefficients, singular regular differential modules and at last differential modules over the field of Laurent power series.
Publications
Spectrum of a linear differential equation with constant coefficients, Mathematische Zeitschrift, 296, 1613–1644 (2020), DOI: 10.1007/s00209-020-02482-z, arXiv:1802.07306, hal-01715463.
Spectrum of a linear differential equation over a field of formal power series, Journal Of Number Theory , 231, 139-157 (2021), DOI: 10.1016/j.jnt.2020.11.021 arXiv:1808.02382, hal-01853797.
Spectrum of p-adic linear differential equations I: The shape of the spectrum, Selecta Mathematica 30, 13 (2024), DOI:10.1007/s00029-023-00904-4, PDF, arXiv:2111.03548.
Preprints
Spectrum of p-adic linear differential equations II: The variation of the spectrum, PDF, arXiv:2303.06014. (submitted)
In preparation
Wild finite étale coverings of p-adic annuli. ( with Andrea Pulita)
The spectrum of p-adic linear differential equations with respect to derivation (T − c)^n d/dT and the strong refined decomposition.
Techniques for computing the spectrum of an ultrametric operator.
Talks
August 2022: Sanya Mathematics Summer school, Tsinghua Sanya International Mathematics Forum, Sanya, China. Course title: Introduction to p-adic differential equations.
Jun 2022: BIMSA-YMSC Tsinghua Number Theory Seminar, Beijing, China. Title: Spectrum of p-adic differential equations.
July 2017 : Students’ Conference on Tropical and Non-Archimedean Geometry, Regensburg University, Regensburg, Germany. Title: Spectral theory and ultrametric differential equations.
Jun 2016 : École jeunes chercheurs en théorie des nombres , Université Blaise Pascal, Clermont-Ferrand, France. Title: Spectre d’un système différentiel aux coefficients constants.
Ph.D. Students seminar: 5 Talks.
Conferences and summer schools
2022
Sanya Mathematical Summer school, Tsinghua Sanya International Mathematics Forum, Sanya, China.
2019
Perfectoids, Institut Henri Lebesgue, Rennes, France.
2018
Berkovich Spaces 30 years, Institut Henri Poincaré, Paris, France.
2017
Students’ Conference on Tropical and Non-Archimedean Geometry , Université de Ratisbonne , Ratisbonne, Allemagne.
Berkovich Spaces, Tropical Geometry and Model Theory, Universided Los Andes, Bogtá, Colombie.
p-adic Analytic Geometry and Differential Equations CIRM, Marseille, France.
2016
École jeunes chercheurs en théorie des nombres , Université Blaise Pascal, Clermont-Ferrand, France.
2015
Students’ Conference on Tropical and Non-Archimedean Geometry,Université de Ratisbonne, Ratisbonne, Allemagne.
Géométrie et arithmétique sur les corps locaux et globaux, Université de Caen, Caen, France.
Conférence en hommage à Alexandre Grothendieck, Université de Montpellier, Montpellier, France.
2014
Special week: Berkovich spaces and applications, Université de Strasbourg, Strasbourg, France.