Coverings of the affine line in characteristic \(p>0\) and Abhyankar's conjecture
type: Item
instance of: scholarly article
title: Coverings of the affine line in characteristic \(p>0\) and Abhyankar's conjecture
zbMATH Open document ID: 0798.14013
DOI: 10.1007/BF01231568
published in: Inventiones Mathematicae
publication date: 1994-11-03T00:00:00Z
full work available at URL: https://eudml.org/doc/144195
review text: Let \(p\) be a prime number, \(k\) an algebraically closed field with \(\text{char} k = p\) and \(D\) the affine line on \(k\); in 1957 the following conjecture was stated by \textit{S. Abhyankar}: Every quasi-\(p\)-group \(G\) (i.e. \(G\) is finite and generated by its Sylow \(p\)-subgroups) satisfies the property ``rev-\(p\)'' (i.e. \(G\) is the Galois group of a connected étale covering of \(D)\). In this paper Abhyankar's conjecture has finally been proved (until now only partial results were known), as a corollary of the following result: Let \(G\) be a quasi-\(p\)-group and \(S\) a Sylow \(p\)-group of \(G\). (1) For each invariant \(p\)-subgroup \(H\) of \(G\), if \(G/H\) satisfies rev-\(p\), also \(G\) does. (2) If every strict quasi-\(p\)- subgroup of \(G\) has rev-\(p\), then \(G(S)\) has rev-\(p\). (3) If \(G(S) \neq G\) and \(G\) has no non-trivial invariant \(p\)-subgroups, then \(G\) satisfies rev-\(p\). Here \(G(S)\) is the subgroup of \(G\) generated by all the strict subgroups of \(G\) which are quasi-\(p\)-groups and have a Sylow \(p\)-subgroup contained in \(S\). Point (1) above is a particular case of a result by \textit{J.-P. Serre} [C. R. Acad. Sci., Paris, Sér. I 311, No. 6, 341-346 (1990; Zbl 0726.14021)], while point (2) is an interesting application of an idea (first introduced by Harbater) of using rigid geometry to produce Galois coverings. Section 3 of the paper is dedicated to the links between rigid and formal geometry and it is interesting also for its own sake (a ``rigid geometry analog'' of Runge's theorem is proved). Point (3) follows in a sense, the more classical idea of specializing from characteristic 0 to char \(p\): a covering \(C\) with Galois group \(G\) of the projective line is built on a local field \(K\) with \(\text{char} K = 0\), and such that its residue characteristic is \(p\). A result by Hée allows then to build a covering of the affine line with Galois group \(G\) from the special fiber of a semistable reduction …
reviewed by: Alessandro Gimigliano
Mathematics Subject Classification ID: 14H30
Mathematics Subject Classification ID: 14G20
Mathematics Subject Classification ID: 14G15
zbMATH DE Number: 554024
zbMATH Keywords: quasi-\(p\)-groups
zbMATH Keywords: Sylow \(p\)-subgroup
zbMATH Keywords: Galois group of a connected étale covering
zbMATH Keywords: covering of the affine line
Wikidata QID: Q29030221
author: Michel Raynaud
MaRDI profile type: Publication
cites work: Coverings of Algebraic Curves
cites work: Galois theory on the line in nonzero characteristic
cites work: Alternating group coverings of the affine line for characteristic two
cites work: Mathieu-group coverings of the affine line
cites work: Étale cohomology for non-Archimedean analytic spaces
cites work: Q3325851
cites work: Formal and rigid geometry. I: Rigid spaces
cites work: Q5730198
cites work: The irreducibility of the space of curves of a given genus
cites work: Critères de platitude et de projectivité. Techniques de ''platification'' d'un module. (Criterial of flatness and projectivity. Technics of ''flatification of a module.)
cites work: Q3030914
cites work: Formal Patching and Adding Branch Points
cites work: Q4283540
cites work: Der Endlichkeitssatz für eigentliche Abbildungen in der nichtarchimedischen Funktionentheorie
cites work: Formal-algebraic and rigid-analytic geometry
cites work: Q3903107
cites work: Q4053758
cites work: Lectures on minimal models and birational transformations of two dimensional schemes
cites work: Q3277020
cites work: Cohomologie Galoisienne. Cours au College de France, 1962-1963. 4e ed
cites work: Q3348096
cites work: Rigid analytic spaces
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description: scientific article; zbMATH DE number 554024