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The local lifting problem for \(A_4\). (English) Zbl 1352.14019

Let \(k\) be an algebraically closed field of characteristic \(p\) and let \(G\) be a finite group. A local \(G\)-extension is a G-Galois extension \(k[[z]]/k[[s]]\). One of the most important topic in the study of Galois extensions in positive characteristic is the analysis of possible obstructions to lifting to characteristic \(0\) of the faithful continuous action of the finite group \(G\). This leading problem is the so-called local lifting problem.
Problem (The local lifting problem). Let \(k\) be an algebraically closed field of characteristic \(p\) and let \(G\) be a finite group. Does a given local \(G\)-Galois extension \(k[[z]] / k[[s]]\) lift in characteristic zero? In other word, does there exist a DVR \(R\) of characteristic zero with residue field \(k\) and a \(G\)-Galois extension \(R[[Z]] / R[[S]]\), which reduces to \(k[[z]] / k[[s]]\)?
When the answer is positive, the group \(G\) is called local Oort group (for \(p\)). The list of possible local Oort groups is quite limited according to well-known examples of obstructions; see for instance [J. Bertin, C. R. Acad. Sci., Paris, Sér. I, Math. 326, No. 1, 55–58 (1998; Zbl 0952.14018)], [T. Chinburg et al., Ann. Sci. Éc. Norm. Supér. (4) 44, No. 4, 537–605 (2011; Zbl 1239.14024)] and [L. H. Brewis and S. Wewers, Math. Ann. 345, No. 3, 711–730 (2009; Zbl 1222.14045)]. In particular from [Chinburg et al., loc. cit., Theorem 1.2] and [Brewis and Wewers, loc. cit.], a local Oort group in characteristic \(p\) is either cyclic, dihedral of order \(2p^n\), or the alternating group \(A_4\) (for \(p=2\)).
Cyclic group are known to be local Oort, this is the so called Oort conjecture; see [A. Obus and S. Wewers, Ann. Math. (2) 180, No. 1, 233–284 (2014; Zbl 1307.14042)] and [F. Pop, Ann. Math. (2) 180, No. 1, 285–322 (2014; Zbl 1311.12003)].
Dihedral groups of order \(2p\) are known to be local Oort for odd \(p\) in [I. I. Bouw and S. Wewers, Duke Math. J. 134, No. 3, 421–452 (2006; Zbl 1108.14025)] and for \(p=2\) in [G. Pagot, Relèvement en caractéristique zéro d’actions de groupes abéliens de type \((p,\dots,p)\). Bordeaux Cedex, FR: Université Bordeaux (PhD Thesis) (2002)]. Moreover the dihedral group \(D_9\) is local Oort by [A. Obus, “A generalization of the Oort conjecture”, Preprint, arXiv:1502.07623].
The main result of this paper is the proof of the following theorem.
Theorem 1. The group \(A_4\) is a local Oort group for \(p=2\).
A motivation of interest for the local lifting problem is the global lifting problem on deformations of Galois branched covers of algebraic curves.
Problem (The global lifting problem). Let \(\mathcal{X} / k\) be a smooth, connected, projective curve over an algebraically closed field of characteristic \(p\) and let \(G\) be a finite group acting on \(\mathcal{X}\). Does \((\mathcal{X},G)\) lift in characteristic zero?
According to the local-global principle, see [A. Grothendieck (ed.) and M. Raynaud, Séminaire de géométrie algébrique du Bois Marie 1960/61 (SGA 1), dirigé par Alexander Grothendieck. Augmenté de deux exposés de M. Raynaud. Revêtements étales et groupe fondamental. Exposés I à XIII. (Seminar on algebraic geometry at Bois Marie 1960/61 (SGA 1), directed by Alexander Grothendieck. Enlarged by two reports of M. Raynaud. Ètale coverings and fundamental group). Berlin-Heidelberg-New York: Springer-Verlag (1971; Zbl 0234.14002), Corollaire 2.12], \((\mathcal{X},G)\) lifts to characteristic zero over the DVR \(R\) if and only if the local lifting problem holds over \(R\) for each point (i.e. its complete local ring) of \(\mathcal{X}\) with nontrivial stabilizer in \(G\). Thus, the global lifting problem reduces to the local lifting problem. In this paper also the following corollary to Theorem 1 is obtained.
The groups \(A_4\) and \(A_5\) are Oort groups for every prime. That is, if \(G \in \{A_4,A_5\}\), then \((\mathcal{X},G)\) defined over an algebraically closed field of characteristic \(p\), lifts to characteristic zero.

MSC:

14H37 Automorphisms of curves
12F10 Separable extensions, Galois theory
13B05 Galois theory and commutative ring extensions
14B12 Local deformation theory, Artin approximation, etc.
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