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The local lifting problem for dihedral groups. (English) Zbl 1108.14025

Let \(k\) be an algebraically closed field of characteristic \(p\) and let \(G\) be a finite group. A local \(G\)-action is a faithful \(k\)-linear action \(\phi : G \hookrightarrow \text{Aut}_k(k[[y]])\). The authors prove the following
Theorem. Suppose that \(p\) is odd and that \(G\) is the dihedral group of order \(2p\). Then every local \(G\)-action \(\phi\) lifts to characteristic \(0\), i.e. there exists a complete discrete valuation ring \(R\) with residue field \(k\) and fraction field of characteristic \(0\) such that there is a lift of \(\phi\) to a \(R\)-linear action \(\phi_R : G \hookrightarrow \text{Aut}_R(R[[y]])\).
Their approach is based on a generalization of the method of B. Green and M. Matignon [J. Am. Math. Soc. 12, No. 1, 269–303 (1999; Zbl 0923.14007)] and Y. Henrio [arXiv:math.AG/0011098] (which treat the case \(G=\mathbb{Z}/p \mathbb{Z}\)). Let \(G=\mathbb{Z}/p \mathbb{Z} \rtimes \mathbb{Z}/m \mathbb{Z}\) be a semidirect product of a cyclic group \(C\) of order \(m\) by a cyclic group \(P\) of order \(p\) (with \(m,p\) coprime). Suppose that \(G\) acts on the open rigid disc over a \(p\)-adic field \(K\). To this action they associate a certain object called a Hurwitz tree. This object represents, in some sense, the reduction of the \(G\)-action on the disc to characteristic \(p\). Their first main result is that one can reverse this construction. As a consequence the local lifting problem for the group \(G\) can be reduced to the construction of certain Hurwitz trees. This can always been done for the dihedral group. This method involves the construction of certain differential forms on the projective line with very specific properties.
The authors state also the following interesting question : let \(\chi : C \to \mathbb{F}_p^*\) be a character and \(G=P \rtimes_{\chi} C\) (i.e. \(\tau \sigma \tau^{-1}=\sigma^{\chi(\tau)}\) for \(\sigma \in P\) and \(\tau \in C\)). Let \(\phi : G \hookrightarrow \text{Aut}_k(k[[y]])\) be a local \(G\)-action. Are the necessary conditions
1) \(\chi\) is either trivial or injective;
2) If \(\chi\) is injective then \(m| h+1\) where \(h=\text{ord}_y(\frac{\sigma(y)}{y}-1)\) (\(\sigma\) is a generator of \(P\)) is the conductor of \(\phi\) ; also sufficient conditions for \(\phi\) to lift?
Although they do not answer this question, the methods developed in the article reduce it to solving certain explicit equations over \(\mathbb{F}_p\).

MSC:

14H37 Automorphisms of curves
11G20 Curves over finite and local fields
14D15 Formal methods and deformations in algebraic geometry

Citations:

Zbl 0923.14007
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References:

[1] J. Bertin, Obstructions locales au relèvement de revêtements galoisiens de courbes lisses , C. R. Acad. Sci. Paris Ser. I Math. 326 (1998), 55–58. · Zbl 0952.14018
[2] J. Bertin and A. MéZard, Déformations formelles des revêtements sauvagement ramifiés de courbes algébriques , Invent. Math. 141 (2000), 195–238. · Zbl 0993.14014
[3] I. I. Bouw, Lifting covers of curves and differential data , conference lecture at “Meeting on arithmetic of fundamental groups,” Banff Centre, Banff, Alberta, Canada, September 2003.
[4] T. Chinburg, B. Guralnick, and D. Harbater, in preparation.
[5] B. Green and M. Matignon, Liftings of Galois covers of smooth curves , Compositio Math. 113 (1998), 237–272. · Zbl 0923.14006
[6] -, Order \(p\) automorphisms of the open disc of a \(p\)-adic field , J. Amer. Math. Soc. 12 (1999), 269–303. JSTOR: · Zbl 0923.14007
[7] Y. Henrio, Arbres de Hurwitz et automorphismes d’ordre \(p\) des disques et des couronnes \(p\)-adiques formels, \arxivmath.AG/0011098
[8] F. F. Knudsen, The projectivity of the moduli space of stable curves, II: The stacks \(M\sbg,n\), Math. Scand. 52 (1983), 161–199. · Zbl 0544.14020
[9] J. S. Milne, Étale Cohomology , Princeton Math. Ser. 33 , Princeton Univ. Press, Princeton, 1980. · Zbl 0433.14012
[10] F. Oort, “Lifting algebraic curves, abelian varieties, and their endomorphisms to characteristic zero” in Algebraic Geometry, Bowdoin, (Brunswick, Mass., 1985) , Proc. Sympos. Pure Math. 46 , Part 2, Amer. Math. Soc., Providence, 1987, 165–195. · Zbl 0645.14017
[11] -, Some questions in algebraic geometry , technical report, Utrecht University, Utrecht, Netherlands, 1995.
[12] G. Pagot, Relèvement en caractéristique zéro d’actions de groupes abéliens de type \((p, \ldots, p)\) , Ph.D. dissertation, Université Bordeaux I, Talence, France, 2002.
[13] J.-P. Serre, Corps locaux , 2nd ed., Hermann, Paris, 1968.
[14] S. Wewers, Formal deformation of curves with group scheme action , Ann. Inst. Fourier (Grenoble) 55 (2005), 1105–1165. · Zbl 1079.14006
[15] L. Zapponi, personal communication, September 2004.
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