Embedding problems and open subgroups

Harbater, David; Stevenson, Katherine

doi:10.1090/S0002-9939-2010-10776-1

Embedding problems and open subgroups
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by David Harbater and Katherine Stevenson

Proc. Amer. Math. Soc. 139 (2011), 1141-1154

DOI: https://doi.org/10.1090/S0002-9939-2010-10776-1

Published electronically: November 17, 2010

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Abstract:

We study the properties of the fundamental group of an affine curve over an algebraically closed field of characteristic $p$ from the point of view of embedding problems. In characteristic zero the fundamental group is free, but in characteristic $p$ it is not even $\omega$-free. In this paper we show that it is “almost $\omega$-free” in the sense that each finite embedding problem has a proper solution when restricted to some open subgroup. We also prove that embedding problems can always be properly solved over the given curve if suitably many additional branch points are allowed in locations that can be specified arbitrarily; this strengthens a result of the first author.

References

Irene I. Bouw, Construction of covers in positive characteristic via degeneration, Proc. Amer. Math. Soc. 137 (2009), no. 10, 3169–3176. MR 2515387, DOI 10.1090/S0002-9939-09-10013-8
Adrien Douady, Détermination d’un groupe de Galois, C. R. Acad. Sci. Paris 258 (1964), 5305–5308. MR 162796
Michael D. Fried and Moshe Jarden, Field arithmetic, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 11, Springer-Verlag, Berlin, 2005. MR 2102046
A. Grothendieck, Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I, Inst. Hautes Études Sci. Publ. Math. 11 (1961), 167. MR 217085
Alexander Grothendieck and Jacob P. Murre, The tame fundamental group of a formal neighbourhood of a divisor with normal crossings on a scheme, Lecture Notes in Mathematics, Vol. 208, Springer-Verlag, Berlin-New York, 1971. MR 0316453
Revêtements étales et groupe fondamental, Lecture Notes in Mathematics, Vol. 224, Springer-Verlag, Berlin-New York, 1971 (French). Séminaire de Géométrie Algébrique du Bois Marie 1960–1961 (SGA 1); Dirigé par Alexandre Grothendieck. Augmenté de deux exposés de M. Raynaud. MR 0354651
Robert Guralnick and Katherine F. Stevenson, Prescribing ramification, Arithmetic fundamental groups and noncommutative algebra (Berkeley, CA, 1999) Proc. Sympos. Pure Math., vol. 70, Amer. Math. Soc., Providence, RI, 2002, pp. 387–406. MR 1935415, DOI 10.1090/pspum/070/1935415
David Harbater, Moduli of $p$-covers of curves, Comm. Algebra 8 (1980), no. 12, 1095–1122. MR 579791, DOI 10.1080/00927878008822511
David Harbater, Abhyankar’s conjecture on Galois groups over curves, Invent. Math. 117 (1994), no. 1, 1–25. MR 1269423, DOI 10.1007/BF01232232
David Harbater, Fundamental groups and embedding problems in characteristic $p$, Recent developments in the inverse Galois problem (Seattle, WA, 1993) Contemp. Math., vol. 186, Amer. Math. Soc., Providence, RI, 1995, pp. 353–369. MR 1352282, DOI 10.1090/conm/186/02191
David Harbater, Embedding problems and adding branch points, Aspects of Galois theory (Gainesville, FL, 1996) London Math. Soc. Lecture Note Ser., vol. 256, Cambridge Univ. Press, Cambridge, 1999, pp. 119–143. MR 1708604
David Harbater, Abhyankar’s conjecture and embedding problems, J. Reine Angew. Math. 559 (2003), 1–24. MR 1989642, DOI 10.1515/crll.2003.049
David Harbater, Patching and Galois theory, Galois groups and fundamental groups, Math. Sci. Res. Inst. Publ., vol. 41, Cambridge Univ. Press, Cambridge, 2003, pp. 313–424. MR 2012220
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
Kenkichi Iwasawa, On solvable extensions of algebraic number fields, Ann. of Math. (2) 58 (1953), 548–572. MR 0059314, DOI 10.2307/1969754
Moshe Jarden, On free profinite groups of uncountable rank, Recent developments in the inverse Galois problem (Seattle, WA, 1993) Contemp. Math., vol. 186, Amer. Math. Soc., Providence, RI, 1995, pp. 371–383. MR 1352283, DOI 10.1090/conm/186/02192
Manish Kumar, Fundamental group in positive characteristic, J. Algebra 319 (2008), no. 12, 5178–5207. MR 2423823, DOI 10.1016/j.jalgebra.2008.01.013
M. Kumar, The fundamental group of affine curves in positive characteristic, 2009 manuscript, available at arXiv:0903.4472.
Masayoshi Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0155856
Florian Pop, Étale Galois covers of affine smooth curves. The geometric case of a conjecture of Shafarevich. On Abhyankar’s conjecture, Invent. Math. 120 (1995), no. 3, 555–578. MR 1334484, DOI 10.1007/BF01241142
Brian Osserman, Linear series and the existence of branched covers, Compos. Math. 144 (2008), no. 1, 89–106. MR 2388557, DOI 10.1112/S0010437X0700303X
Rachel J. Pries, Wildly ramified covers with large genus, J. Number Theory 119 (2006), no. 2, 194–209. MR 2250044, DOI 10.1016/j.jnt.2005.10.013
M. Raynaud, Revêtements de la droite affine en caractéristique $p>0$ et conjecture d’Abhyankar, Invent. Math. 116 (1994), no. 1-3, 425–462 (French). MR 1253200, DOI 10.1007/BF01231568
Luis Ribes and Pavel Zalesskii, Profinite groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 40, Springer-Verlag, Berlin, 2000. MR 1775104, DOI 10.1007/978-3-662-04097-3
Jean-Pierre Serre, Cohomologie Galoisienne, Lecture Notes in Mathematics, Vol. 5, Springer-Verlag, Berlin-New York, 1973. Cours au Collège de France, Paris, 1962–1963; Avec des textes inédits de J. Tate et de Jean-Louis Verdier; Quatrième édition. MR 0404227
Jean-Pierre Serre, Construction de revêtements étales de la droite affine en caractéristique $p$, C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), no. 6, 341–346 (French, with English summary). MR 1071640
Akio Tamagawa, Finiteness of isomorphism classes of curves in positive characteristic with prescribed fundamental groups, J. Algebraic Geom. 13 (2004), no. 4, 675–724. MR 2073193, DOI 10.1090/S1056-3911-04-00376-5
Oscar Zariski, On the purity of the branch locus of algebraic functions, Proc. Nat. Acad. Sci. U.S.A. 44 (1958), 791–796. MR 95846, DOI 10.1073/pnas.44.8.791