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Étale Galois covers of affine smooth curves. The geometric case of a conjecture of Shafarevich. On Abhyankar’s conjecture. (English) Zbl 0842.14017

The first problem considered in this paper is the Shafarevich conjecture: “The absolute Galois group over the maximal cyclotomic extension \(K^{\text{cycl}}\) of a global field \(K\) is profinite free”. The geometric version of this is a special case of the following conjecture: “Let \(K\mid \kappa\) be a function field of one variable over an algebraically closed field \(\kappa\). Then the absolute Galois group of \(K\) is profinite free”. This conjecture is known to be true if \(\text{char } \kappa=0\) and its proof (by A. Douady) uses the structure theorem of \(\pi_1 (U)\) for small affine opens \(U\) of the projective smooth model \(Z\) of \(K\mid \kappa\) (if \(r=\) number of points in \(Z- U\), \(\pi_1 (U)\) is the profinite group on \(2g (Z)+ r-1\) generators). This theorem is false for \(\text{char } \kappa=0\), and the precise structure of \(\pi_1 (U)\) is not known. – The conjecture is proved as a consequence of the following:
Theorem A: Any finite split embedding problem \((\gamma, \alpha)\) for \(G_K\) has proper solutions – where a finite split embedding problem \((\gamma, \alpha)\) for a profinite group \(G\) is a diagram \(B@> \alpha >> A@< \gamma <<G\) of profinite groups where \(\alpha\) and \(\gamma\) are surjective and \(B\) is finite, while a proper solutions is a surjective homomorphism \(\beta: G\to B\), with \(\alpha \beta= \gamma\). – Further a study of embedding problems is undertaken to give a second theorem which gives a finer description of the situation when \(\ker (\alpha)\) is a quotient of \(\pi_1 (\mathbb{A}^1_\kappa)\). The remarkable consequence of this second theorem is a proof of Abhyankar’s conjecture:
“A finite group \(B\) is a quotient of \(\pi_1(U)\) (notation as above) if and only if \(p(B)\) is a quotient of \(\pi_1 (U)\)” (where \(p(B)\) is the subgroup of \(B\) generated by all the Sylow subgroups of \(B\)).

MSC:

14H30 Coverings of curves, fundamental group
11R32 Galois theory
11R34 Galois cohomology
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References:

[1] Abhyankar, S.: Galois theory on the line in non-zero characteristics. Bul. AMS27 (1992) 68-131 · Zbl 0760.12002 · doi:10.1090/S0273-0979-1992-00270-7
[2] Bosch, S., Güntzer, U., Remmert, R: Non-archimedean analysis. Gr. Math. Wiss., Bd. 261. Berlin Heidelberg New York: Springer 1984 · Zbl 0539.14017
[3] Bell, J.-L., Slomson, A.-B.: Models and ultraproducts: an introduction. North Holland 1969 · Zbl 0179.31402
[4] Chatzidakis, Z.: Model theory of profinite groups having the Iwasawa property. Manuscript · Zbl 0906.03037
[5] Chevalley, C.: Introduction to the theory of algebraic functions of one variable. AMS, New York 1951 · Zbl 0045.32301
[6] Deligne, P.: Le groupe fondamental de la droite projective mois trois points, Galois groups over Q. Math. Sci. Res. Inst. Publ.16 79-297, Springer 1989 · Zbl 0742.14022
[7] Douady, A.: Détermination d’un groupe de Galois. C. R. Acad. Sci. Paris258 (1964) 5305-5308 · Zbl 0146.42105
[8] van den Dries, L., Ribenboim, P.: Application de la théorie des modèle aux groupes de Galois de corps de fonctions. C. R. Acad. Sci. Paris288 (1979) A789-A792 · Zbl 0426.12004
[9] Fresnel, J.: Geometrie Analytique Rigide. Notes, Bordeaux 1984
[10] Fried, M., Jarden, M.: Field Arithmetic. Springer Verlag Berlin Heidelberg 1986 · Zbl 0625.12001
[11] Gerritzen, L., van der Put, M.: Schottky groups and Mumford curves. LNM 817, Springer-Verlag Berlin Heidelberg New York 1980 · Zbl 0442.14009
[12] Grothendieck, A.: SGA 1. Revètements Étale et Groupe Fundamental. LNM 224, Springer-Verlag Berlin Heidelberg New York 1971 · Zbl 0234.14002
[13] Gruenberg, K. W.: Projective profinite groups. J. London Math. Soc.42 (1967) 155-165 · Zbl 0178.02703 · doi:10.1112/jlms/s1-42.1.155
[14] Harbater, D.: Abhyankar’s conjecture on Galois groups over curves, Invent. Math.117 (1994) 1-25 · Zbl 0805.14014 · doi:10.1007/BF01232232
[15] Harbater, D.: Fundamental groups and embedding problems in characteristicp,. Proc. of a Conf. on Inverse Galois Problem, ed M. Fried, AMS Contemp. Math. Series (to appear) · Zbl 0858.14013
[16] Iwasawa, K.: On solvable extensions of number fields. Ann. Math.58 (1953) 548-572 · Zbl 0051.26602 · doi:10.2307/1969754
[17] Kani, E.: Nonstandard Diophantine Geometry. Dissertation, Heidelberg 1978 · Zbl 0454.14010
[18] Kiehl, R.: Der Endlichkeitssatz für eigentliche Abbildungen in der nicht-archimedischen Funktionentheorie. Invent. Math.2 (1967) 191-214 · Zbl 0202.20101 · doi:10.1007/BF01425513
[19] Köpf, U.: Über eigentliche Familien algebraischer Varietäten über affinoiden Räumen. Schriftenreihe des Math. Inst. Univ. Münster, 2. Serie, Heft 7 (1974) · Zbl 0275.14006
[20] Krull, W., Neukirch, J.: Die Struktur der absoluten Galoisgruppe über dem KörperR(t). Math. Ann.193 (1971) 197-209 · Zbl 0236.12104 · doi:10.1007/BF02052391
[21] Lütkebohmert, W.: Riemann’s existence problem for ap-adic field. Invent. math.111 (1993) 309-330 · Zbl 0780.32005 · doi:10.1007/BF01231290
[22] Madan, M., Rosen, M.: The automorphism group of a function field. Proceedings of AMS,115 4 (1992) 923-929 · Zbl 0791.14011 · doi:10.1090/S0002-9939-1992-1088443-2
[23] Matzat, B.-H.: Der Kenntnisstand in der Konstruktiven Galoisschen Theorie, in: Progress in Mathematics, Vol. 95, Birkhäuser Verlag Basel (1991) · Zbl 0756.12004
[24] Mel’nicov, O.V.: Projective limits of free profinite groups. Dokl. Akad. Nauk BSSR24 (1980) 968-970, 1051
[25] Mumford, D.: An analytical construction of degenerating curves over complete local rings. Compos. Math.24 (1972) 129-174 · Zbl 0228.14011
[26] Nagata, M.: Local rings. Interscience Publishers New York London 1962 · Zbl 0123.03402
[27] Pop, F.: 1/2Riemann Existence Theorem with Galois Action, in: Algebra and Number Theory, (ed) G. Frey, J. Ritter: de Gruyter Proceedings in Mathematics, Berlin New York 1994
[28] Pop, F.: The geometric case of a conjecture of Shafarevich. Heidelberg-Mannheim, Preprint series Arithmetik, N0 8, Heidelberg 1993
[29] Popp, H.: Fundamentalgruppen algebraischer Mannigfaltigkeiten. LNM 176, Springer-Verlag Berlin-Heidelberg-New York 1970 · Zbl 0233.14001
[30] Raynaud, M.: Géométrie analytique rigide d’apres Tate, Kiehl, ..., Mémoire de Soc. Math. de France,39-40 (1974) 319-327
[31] Raynaud, M.: Revetements de la droite affine en caractéristiquep>0 et conjecture d’Abhyankar. Invent. Math.116 (1994) 425-462 · Zbl 0798.14013 · doi:10.1007/BF01231568
[32] Robinson, A., Roquette, P.: On the finiteness theorem of Siegel and Mahler concerning diophantine equations. J. Number Theory7 (1975) 121-176 · Zbl 0299.12107 · doi:10.1016/0022-314X(75)90013-X
[33] Roquette, P.: Some tendences in contemporary algebra, in: Perspectives in Mathematics. Anniversary of Oberwolfach 1984, Basel 1984, 393-422
[34] Serre, J.-P.: Cohomologie galoisienne. LNM 5, Springer Verlag 1973 · Zbl 0259.12011
[35] Serre, J.-P.: Revetements de courbes algébriques. Sém. Bourbaki, Vol. 1991/92, Exp.749, 167-182
[36] Serre, J.-P.: Topics in Galois Theory. Research notes in Math., Jones and Bartlett 1992 · Zbl 0746.12001
[37] Tate, J.: Rigid analytical spaces. Invent. Math.12 (1971) 257-289 · Zbl 0212.25601 · doi:10.1007/BF01403307
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