Spectrum of p-adic differential equations

Title: Spectrum of p-adic differential equations

Speaker: Tinhinane Amina Azzouz (BIMSA)

Time: 16:00-17:00 Beijing time, Jun 14, 2022

Zoom ID: 361 038 6975 Passcode: BIMSA

Room: 1110

Abstract: In the ultrametric setting, linear differential equations present phenomena that do not appear over the complex field. Indeed, the solutions of such equations may fail to converge everywhere, even without the presence of poles. This leads to a non-trivial notion of the radius of convergence, and its knowledge permits us to obtain several interesting information about the equation. Notably, it controls the finite dimensionality of the de Rham cohomology. In practice, the radius of convergence is really hard to compute and it represents one of the most complicated features in the theory of p-adic differential equations. The radius of convergence can be expressed as the spectral norm of a specific operator and a natural notion, that refines it, is the entire spectrum of that operator, in the sense of Berkovich. 
In our previous works, we introduce this invariant and compute the spectrum of differential equations over a power series field and in the p-adic case with constant coefficients.
In this talk we will discuss our last results about the shape of this spectrum for any linear differential equation, the strong link between the spectrum and all the radii of convergence, notably a decomposition theorem provided by the spectrum.

 

BIMSA-YMSC Tsinghua Number Theory Seminar