The Oort conjecture on lifting covers of curves. (English) Zbl 1311.12003

This paper completes the proof of the Oort conjecture, which states that any cyclic branched cover of smooth projective curves in characteristic \(p\) lifts to characteristic zero, by reducing the general case to a special case that was proven in [the reviewer and S. Wewers, Ann. Math. (2) 180, No. 1, 233–284 (2014; Zbl 1307.14042)].
In particular, it is well-known that the Oort conjecture is equivalent to its local version, which states that if \(k\) is an algebraically closed field of characteristic \(p\), then for any cyclic \(G\)-extension \(k[[z]]/k[[t]]\), there is a DVR \(R\) with residue field \(k\) such that \(k[[z]]/k[[t]]\) lifts to a \(G\)-extension \(R[[Z]]/R[[T]]\). The result of Obus-Wewers above proves this conjecture under the assumption of no essential ramification, which says that the breaks \((u_1, \ldots, u_n)\) in the higher ramification filtration for the upper numbering of the extension satisfy \(u_i < u_{i-1} + p\) for all \(i\). This paper completes the proof of the (local) Oort conjecture by reducing it to the “no essential ramification” case.
The key result is a so-called “characteristic \(p\) Oort conjecture,” which loosely states that any cyclic extension \(k[[z]]/k[[t]]\) has an equicharacteristic deformation to a cyclic extension of \(k[[t]][[\varpi]]\) where the generic fiber has no essential ramification (although it is generally branched at multiple maximal ideals). This deformation is constructed explicitly using Artin-Schreier-Witt theory. The author then proves a global version of this conjecture, stating that a branched Galois cover of \(\mathbb{P}^1_k\) with cyclic inertia groups deforms to a branched cover of \(\mathbb{P}^1_{k((\varpi))}\) with cyclic inertia groups and no essential ramification. The result of Obus-Wewers shows that this (deformed) cover lifts to characteristic zero. A careful algebraic geometry argument of the author now shows that one can obtain a lift of the original cover from this.


12F10 Separable extensions, Galois theory
11G20 Curves over finite and local fields
13B05 Galois theory and commutative ring extensions
13F35 Witt vectors and related rings
14H25 Arithmetic ground fields for curves
14G32 Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory)


Zbl 1307.14042
Full Text: DOI arXiv Link


[1] J. Bertin and A. Mézard, ”Déformations formelles des revêtements sauvagement ramifiés de courbes algébriques,” Invent. Math., vol. 141, iss. 1, pp. 195-238, 2000. · Zbl 0993.14014 · doi:10.1007/s002220000071
[2] T. Chinburg, R. Guralnick, and D. Harbater, ”Oort groups and lifting problems,” Compos. Math., vol. 144, iss. 4, pp. 849-866, 2008. · Zbl 1158.12003 · doi:10.1112/S0010437X08003515
[3] T. Chinburg, R. Guralnick, and D. Harbater, ”The local lifting problem for actions of finite groups on curves,” Ann. Sci. Éc. Norm. Supér., vol. 44, iss. 4, pp. 537-605, 2011. · Zbl 1239.14024
[4] S. Corry, ”Galois covers of the open \(p\)-adic disc,” Manuscripta Math., vol. 131, iss. 1-2, pp. 43-61, 2010. · Zbl 1223.11140 · doi:10.1007/s00229-009-0301-4
[5] A. J. de Jong, ”Smoothness, semi-stability and alterations,” Inst. Hautes Études Sci. Publ. Math., vol. 83, pp. 51-93, 1996. · Zbl 0916.14005 · doi:10.1007/BF02698644
[6] M. A. Garuti, ”Prolongement de revêtements galoisiens en géométrie rigide,” Compositio Math., vol. 104, iss. 3, pp. 305-331, 1996. · Zbl 0885.14011
[7] Courbes semi-stables et groupe fondamental en géométrie algébriqueBasel: Birkhäuser, 2000. · Zbl 0978.14002
[8] M. A. Garuti, ”Linear systems attached to cyclic inertia,” in Arithmetic Fundamental Groups and Noncommutative Algebra, Providence, RI: Amer. Math. Soc., 2002, vol. 70, pp. 377-386. · Zbl 1072.14017 · doi:10.1090/pspum/070/1935414
[9] B. Green and M. Matignon, ”Liftings of Galois covers of smooth curves,” Compositio Math., vol. 113, iss. 3, pp. 237-272, 1998. · Zbl 0923.14006 · doi:10.1023/A:1000455506835
[10] D. Harbater, ”Moduli of \(p\)-covers of curves,” Comm. Algebra, vol. 8, iss. 12, pp. 1095-1122, 1980. · Zbl 0471.14011 · doi:10.1080/00927878008822511
[11] K. Kato, ”Vanishing cycles, ramification of valuations, and class field theory,” Duke Math. J., vol. 55, iss. 3, pp. 629-659, 1987. · Zbl 0665.14005 · doi:10.1215/S0012-7094-87-05532-3
[12] S. Lang, Algebra, third ed., New York: Springer-Verlag, 2002, vol. 211. · Zbl 0984.00001 · doi:10.1007/978-1-4613-0041-0
[13] A. Obus, The (local) lifting problem for curves, in Galois-Teichmüller theory and Arithmetic Geometry. · Zbl 1321.14028
[14] A. Obus and R. Pries, ”Wild tame-by-cyclic extensions,” J. Pure Appl. Algebra, vol. 214, iss. 5, pp. 565-573, 2010. · Zbl 1233.11125 · doi:10.1016/j.jpaa.2009.06.017
[15] A. Obus and S. Wewers, ”Cyclic extensions and the local lifting problem,” Ann. of Math., vol. 180, iss. 1, pp. 233-284, 2014. · Zbl 1307.14042 · doi:10.4007/annals.2014.180.1.5
[16] F. Oort, ”Lifting algebraic curves, abelian varieties, and their endomorphisms to characteristic zero,” in Algebraic Geometry, Bowdoin, 1985, Providence, RI: Amer. Math. Soc., 1987, vol. 46, pp. 165-195. · Zbl 0645.14017 · doi:10.1090/pspum/046.2
[17] F. Oort, Some questions in algebraic geometry. · Zbl 1182.01012
[18] T. Sekiguchi, F. Oort, and N. Suwa, ”On the deformation of Artin-Schreier to Kummer,” Ann. Sci. École Norm. Sup., vol. 22, iss. 3, pp. 345-375, 1989. · Zbl 0714.14024
[19] P. Roquette, ”Zur Theorie der Konstantenreduktion algebraischer Mannigfaltigkeiten. Invarianz des arithmetischen Geschlechts einer Mannigfaltigkeit und der virtuellen Dimension ihrer Divisoren,” J. Reine Angew. Math., vol. 200, pp. 1-44, 1958. · Zbl 0149.39202 · doi:10.1515/crll.1958.200.1
[20] P. Roquette, ”Abschätzung der Automorphismenanzahl von Funktionenkörpern bei Primzahlcharakteristik,” Math. Z., vol. 117, pp. 157-163, 1970. · Zbl 0194.35302 · doi:10.1007/BF01109838
[21] M. Saidi, Fake liftings of Galois covers between smooth curves.
[22] . J-P. Serre, Local Fields, New York: Springer-Verlag, 1979, vol. 67. · Zbl 0423.12016 · doi:10.1007/978-1-4757-5673-9
[23] T. Sekiguchi and N. Suwa, ”Théories de Kummer-Artin-Schreier-Witt,” C. R. Acad. Sci. Paris Sér. I Math., vol. 319, iss. 2, pp. 105-110, 1994. · Zbl 0845.14023 · doi:10.2748/tmj/1178207479
[24] T. Sekiguchi and N. Suwa, ”On the unified Kummer-Artin-Schreier-Witt theory,” Laboratoire de Mathématiques Pures de Bordeaux Preprint Series, vol. 111, 1999. · Zbl 0845.14023
[25] L. Thomas, ”Ramification groups in Artin-Schreier-Witt extensions,” J. Théor. Nombres Bordeaux, vol. 17, iss. 2, pp. 689-720, 2005. · Zbl 1207.11109 · doi:10.5802/jtnb.514
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.