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Coverings of the affine line in characteristic $$p>0$$ and Abhyankar’s conjecture. (Revêtements de la droite affine en caractéristique $$p>0$$ et conjecture d’Abhyankar.) (French) Zbl 0798.14013
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 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 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 $$C$$.

##### MSC:
 14H30 Coverings of curves, fundamental group 14G20 Local ground fields in algebraic geometry 14G15 Finite ground fields in algebraic geometry
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