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Stratified Bundles on Curves and Differential Galois Groups in Positive Characteristic
Symmetry, integrability and geometry: methods and applications, Tome 15 (2019) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: stratified bundle, differential equations, positive characteristic, fundamental group, Mumford curve, Mumford group, differential Galois group. 
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     author = {Marius van der Put},
     title = {Stratified {Bundles} on {Curves} and {Differential} {Galois} {Groups} in {Positive} {Characteristic}},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2019_15_a70/}
}
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Marius van der Put. Stratified Bundles on Curves and Differential Galois Groups in Positive Characteristic. Symmetry, integrability and geometry: methods and applications, Tome 15 (2019). http://geodesic.mathdoc.fr/item/SIGMA_2019_15_a70/

[1] Battiston G., The base change of the monodromy group for geometric Tannakian pairs, arXiv: 1601.06519

[2] Borel A., Serre J.-P., “Théorèmes de finitude en cohomologie galoisienne”, Comment. Math. Helv., 39 (1964), 111–164 | DOI | MR | Zbl

[3] dos Santos J. P. P., “Fundamental group schemes for stratified sheaves”, J. Algebra, 317 (2007), 691–713 | DOI | MR | Zbl

[4] Ernst S., “Iterative differential embedding problems in positive characteristic”, J. Algebra, 402 (2014), 544–564, arXiv: 1107.1962 | DOI | MR | Zbl

[5] Esnault H., “On flat bundles in characteristic 0 and $p>0$”, European Congress of Mathematics, Eur. Math. Soc., Zürich, 2013, 301–313, arXiv: 1205.4884 | MR | Zbl

[6] Esnault H., Mehta V., “Simply connected projective manifolds in characteristic $p>0$ have no nontrivial stratified bundles”, Invent. Math., 181 (2010), 449–465, arXiv: 0907.3375 | DOI | MR | Zbl

[7] Gerritzen L., van der Put M., Schottky groups and Mumford curves, Lecture Notes in Math., 817, Springer, Berlin, 1980 | DOI | MR | Zbl

[8] Gieseker D., “Flat vector bundles and the fundamental group in non-zero characteristics”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 2 (1975), 1–31 | MR | Zbl

[9] Grothendieck A., “Éléments de géométrie algébrique. IV Étude locale des schémas et des morphismes de schémas IV”, Inst. Hautes Études Sci. Publ. Math., 32 (1967), 5–361 | DOI | MR

[10] Grothendieck A., Raynaud M., Revêtements étales et groupe fondamental, Lecture Notes in Math., 224, Springer-Verlag, Berlin–New York, 1971, arXiv: math.AG/0206203 | DOI | MR | Zbl

[11] Katz N. M., “On the calculation of some differential Galois groups”, Invent. Math., 87 (1987), 13–61 | DOI | MR | Zbl

[12] Kindler L., Regular singular stratified bundles in positive characteristic, Ph.D. Thesis, Universität Duisburg-Essen, 2012

[13] Kindler L., “Regular singular stratified bundles and tame ramification”, Trans. Amer. Math. Soc., 367 (2015), 6461–6485, arXiv: 1210.5077 | DOI | MR | Zbl

[14] Matzat B. H., Differential Galois theory in positive characteristic, Preprint 2001-35, IWR, 2001 | MR

[15] Matzat B. H., van der Put M., “Constructive differential Galois theory”, Galois Groups and Fundamental Groups, Math. Sci. Res. Inst. Publ., 41, Cambridge University Press, Cambridge, 2003, 425–467 | MR | Zbl

[16] Matzat B. H., van der Put M., “Iterative differential equations and the Abhyankar conjecture”, J. Reine Angew. Math., 557 (2003), 1–52 | DOI | MR | Zbl

[17] Röscheisen A., Iterative connections and Abhyankar's conjecture, Ph.D. Thesis, Heidelberg University, 2007 | DOI | Zbl

[18] Shiomi D., “On the Deuring–Shafarevich formula”, Tokyo J. Math., 34 (2011), 313–318 | DOI | MR | Zbl

[19] Springer T. A., Linear algebraic groups, Progress in Mathematics, 9, 2nd ed., Birkhäuser Boston, Inc., Boston, MA, 1998 | DOI | MR | Zbl

[20] Voskuil H. H., van der Put M., “Mumford curves and Mumford groups in positive characteristic”, J. Algebra, 517 (2019), 119–166, arXiv: 1707.03644 | DOI | MR | Zbl