Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 071, 24 pages      arXiv:1812.02965      https://doi.org/10.3842/SIGMA.2019.071
Contribution to the Special Issue on Algebraic Methods in Dynamical Systems

Stratified Bundles on Curves and Differential Galois Groups in Positive Characteristic

Marius van der Put
Bernoulli Institute, University of Groningen, P.O. Box 407, 9700 AG Groningen, The Netherlands

Received December 10, 2018, in final form September 14, 2019; Published online September 21, 2019

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.

Key words: stratified bundle; differential equations; positive characteristic; fundamental group; Mumford curve; Mumford group; differential Galois group.

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