The accessory parameter problem in positive characteristic

TL;DR

This paper investigates the existence and properties of Fuchsian differential equations in positive characteristic, focusing on nilpotent p-curvature and local invariants, and determines minimal polynomial solutions for specific monodromy cases.

Contribution

It provides new results on the existence conditions of such differential equations and identifies minimal degrees of polynomial solutions in cases with logarithmic monodromy.

Findings

Existence criteria for Fuchsian equations with nilpotent p-curvature

Minimal degree of polynomial solutions in logarithmic monodromy cases

Characterization of local invariants in positive characteristic

Abstract

We study the existence of Fuchsian differential equations in positive characteristic with nilpotent p-curvature, and given local invariants. In the case of differential equations with logarithmic local mononodromy, we determine the minimal possible degree of a polynomial solution.