# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Abhyankar, S.: Coverings of algebraic curves. Am. J. Math.79, 825-856 (1957) · Zbl 0087.03603 [2] Abhyankar, S.: Galois theory on the line in nonzero characteristics. Bull. Am. Math. Soc.27, 68-131 (1992) · Zbl 0760.12002 [3] Abhyankar, S.: Resolution of singularities of arithmetical surfaces. In: Schilling, O.F.G. (ed.) Proc. Conf. on Arithm. Alg. Geometry. Purdue 1963, pp. 111-152. New York: Harper & Row 1965 [4] Abhyankar, S., Ou, J., Sathaye, A.: Alternating group coverings of the affine line for characteristic two. (Preprint) · Zbl 0833.14017 [5] Abhyankar, S., Seiler, W., Popp, H.: Mathieu group coverings of the affine line. (Preprint) · Zbl 0788.14022 [6] Berkovich, V.: Etale cohomology of non-archimedean analytic spaces (à paraître) · Zbl 0804.32019 [7] Bosch, S., Güntzer, U., Remmert, R.: Non Archimedean Analysis. (Grundlehren Math. Wiss., vol. 261) Berlin Heidelberg New York: Springer 1984 · Zbl 0539.14017 [8] Bosch, S., Lütkebohmert, W.: Formal and rigid geometry. Math. Ann.295, 291-317 (1993) · Zbl 0808.14017 [9] Bosch, S., Lütkebohmert, W., Raynaud, M.: Néron Models. (Ergeb. Math., Grenzgeb., Bd. 21) Berlin Heidelberg New York: Springer 1990 [10] Bourbaki, N.: Algèbre commutative. Chapitre 3: graduations, filtrations et topologies. Paris: Hermann 1961 · Zbl 0119.03603 [11] Deligne, P., Mumford, D.: The irreducibility of the space of curves of given genus. Publ. Math., Inst. Hautes Étud. Sci.36, 75-110 (1969) · Zbl 0181.48803 [12] Grothendieck, A.: Eléments de géométrie algébrique (cité EGA). Publ. Math., Inst. Hautes Étud. Sci.4,11 (1960) [13] Grothendieck, A.: Revêtements étales et groupe fondamental. Séminaire de Géométrie Algébrique du Bois-Marie 1960/61 (cité SGA 1). (Lect. Notes Math., vol. 224) Berlin Heidelberg New York: Springer 1971 [14] Grothendieck, A.: Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux. Séminaire de géométrie algébrique du Bois-Marie 1962 (cité SGA 2). (Adv. Stud. Pure Math.) Amsterdam: North-Holland and Paris: Masson 1968 [15] Gruson, L., Raynaud, M.: Critères de platidude et de projectivité. Invent. Math.13, 1-89 (1971) · Zbl 0227.14010 [16] Harbater, D.: Galois covers of the arithmetic line. In: Chudnavsky, D.V. et al. (eds.) Number theory. (Lect. Notes Math., vol. 1240, pp. 185-195) Berlin Heidelberg New York: Springer 1987 · Zbl 0627.12015 [17] Harbater, D.: Formal patching and adding branch points. (Pré-publication) · Zbl 0790.14027 [18] Kambayashi, T.: Nori’s construction of Galois coverings in positive characteristics. In: Nagata, M. et al. (eds.) Algebraic and Topological Theories. Tokyo, 1985, pp. 640-647. Tokyo: Kinokuniya 1986 · Zbl 0800.14024 [19] Kiehl, R.: Der Endlichkeitssatz für eigentliche Abbildungen in der nichtarchimedischen Funktionentheorie. Invent. Math.2, 191-214 (1967) · Zbl 0202.20101 [20] Lütkebohmert, W.: Formal-algebraic and rigid-analytic geometry. Math. Ann.286, 341-371 (1990) · Zbl 0716.32022 [21] Mehlmann, F.: Flache Homomorphismus affinoïder Algebren. Schriftenr. Math. Inst. Univ. Münster, 2. Ser.19 (1981) · Zbl 0455.14014 [22] Raynaud, M.: Géométrie analytique rigide d’après Tate, Kiehl. ... Mém. Soc. Math. Fr.39-40, 319-327 (1974) · Zbl 0299.14003 [23] Raynaud, M.:p-groupes et réduction semi-stable des courbes. In: Cartier, P. et al. (eds.) The Grothendieck Festschrift, vol. 3, pp. 179-197. Boston: Birkhäuser 1990 [24] Shafarevich, I.R.: Lecture on minimal models and birational transformations of two dimensional schemes. Bombay: Tata Inst. Fund. Research 37 (1966) · Zbl 0164.51704 [25] Serre, J.-P.: Groupes algébriques et corps de classes. Paris: Hermann 1959 · Zbl 0097.35604 [26] Serre, J.-P.: Cohomologie galoisienne. (Lect. Notes Math., vol. 5) Berlin Heidelberg New York: Springer 1973 · Zbl 0259.12011 [27] Serre, J.-P.: Construction de revêtements étales de la droite affine en caractéristiquep. C.R. Acad. Sci., Paris, Sér. I311, 341-346 (1990) · Zbl 0726.14021 [28] Serre, J.-P.: Revêtements de courbes algébriques. Sérin. Bourbaki no 745 (1991-1992) (à paraître) [29] Steinberg, R.: Lectures on Chevalley groups. Notes prepared by John Faulkner and Robert Wilson. Yale University (1967) [30] Tate, J.: Rigid analytic spaces. Invent. Math.12, 257-289 (1971) · Zbl 0212.25601
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.