Coverings of the affine line in characteristic \(p>0\) and Abhyankar's conjecture
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 of \(C\).
Galois group of a connected étale covering
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