Abstract
Let k be an algebraically closed field of characteristic p and let \({G \hookrightarrow {\rm Aut}_k(k\left[\kern-0.15em\left[ t \right]\kern-0.15em\right] )}\) be a faithful action on a local power series ring over k. Let R be a discrete valuation ring of characteristic 0 with residue field k. One asks, whether it is possible to find a faithful action \({G \hookrightarrow {\rm Aut}_R(R\left[\kern-0.15em\left[ t \right]\kern-0.15em\right] )}\) which reduces to the given action, i.e., a lift to characteristic 0. We show that liftable actions exists in the case that G = D 4 and p = 2. In fact we introduce a family, the supersimple D 4-actions, which can always be lifted to characteristic 0.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bertin, J., Mézard, A.: Déformations formelles des revêtements sauvagement ramifiés de courbes algébriques. Invent. Math. 141, 195–238 (2000)
Bouw, I., Wewers, S.: The local lifting problem for dihedral groups. Duke Math. J. 134, 421–452 (2006)
Chinburg, T.: Lifting local group actions on curves. In: Arithmetic and Differential Galois Theory. Oberwolfach Report 26/2007, Mathematisches Forschungsinstitut Oberwolfach (2007)
Gille, P.: Le groupe fondamental sauvage d’une courbe affine en caractéristique p > 0. In: Courbes Semistables et Groupe Fondamental en Géométrie Algébrique. Prog. Math. 187, 217–231 (2000)
Green, B.: Realizing deformations of curves using Lubin–Tate formal groups. Isr. J. Math. 139, 139–148 (2004)
Green, B., Matignon, M.: Liftings of Galois covers of smooth curves. Compos. Math. 113, 237–272 (1998)
Henrio, Y.: Arbres de Hurwitz et automorphismes d’orde p des disques et des couronnes p-adiques formels. arXiv:math.AG/0011098
Liu, Q.: Algebraic geometry and arithmetic curves. In: Oxford Graduate Texts in Mathematics, vol. 6. Oxford University Press, USA (2002)
Liu, Q., Lorenzini, D.: Models of curves and finite covers. Compos. Math. 118, 61–102 (1999)
Matignon, M.: p-groupes abéliens de type (p, . . . , p) et disques overts p-adiques. Manuscripta. Math. 99(1), 93–109 (1998)
Matignon, M.: Lifting Galois covers of smooth curve. In: Problems from the Workshop on Automorphisms of Curves, Leiden. Rend. Sem. Mat. Univ. Padova 113, 25–30 (2005)
Oort, F., Sekiguchi, T., Suwa, N.: On the deformation of Artin–Schreier to Kummer. Ann. Sci. École. Norm. Sup. 4e sér t. 22, 345–375 (1989)
Pagot, G.: \({\mathbb{F}_{p}}\) -espaces vectoriels de formes différentielles logarithmiques en caractéristique > 0 et automorphismes du disque ouvert p-adique. J Number Theory 97, 58–94 (2002)
Pagot, G: Relèvement en caractéristique zéro d’actions de groupes abéliens de type (p, . . . , p). Thèse Université Bordeaux 1 (12-02). http://www.math.u-bordeaux.fr/~matignon/preprints.html
Raynaud, M.: p-groupes et réduction semi-stables des courbes. In: The Grothendieck Festschrift III, pp. 179–197. Birkäuser Classics, Boston (1990)
Saïdi, M.: p-Rank and semi-stable reduction of curves II. Math. Ann. 312, 625–639 (1998)
Serre, J.P.: Local fields. In: Graduate Texts in Mathematics, vol. 67. Springer, New York (1979)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Brewis, L.H. Liftable D 4-covers. manuscripta math. 126, 293–313 (2008). https://doi.org/10.1007/s00229-008-0179-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00229-008-0179-6