# Stratified Bundles on Curves and Differential Galois Groups in Positive Characteristic

## TL;DR

This paper constructs explicit stratifications on algebraic curves in positive characteristic and investigates their differential Galois groups, providing partial answers to the differential Abhyankar conjecture using p-adic and rigid analytic methods.

## Contribution

It introduces new constructions of stratifications on curves in positive characteristic and explores their differential Galois groups, advancing understanding of the differential Abhyankar conjecture.

## Key findings

- Constructed numerous stratifications on projective and affine curves.
- Determined possible differential Galois groups for these stratifications.
- Provided partial results related to the differential Abhyankar conjecture.

## Abstract

Stratifications and iterative differential equations are analogues in positive characteristic of complex linear differential equations. There are few explicit examples of stratifications. The main goal of this paper is to construct stratifications on projective or affine curves in positive characteristic and to determine the possibilities for their differential Galois groups. For the related "differential Abhyankar conjecture" we present partial answers, supplementing the literature. The tools for the construction of regular singular stratifications and the study of their differential Galois groups are $p$-adic methods and rigid analytic methods using Mumford curves and Mumford groups. These constructions produce many stratifications and differential Galois groups. In particular, some information on the tame fundamental groups of affine curves is obtained.