The local lifting problem for dihedral groups (Q854583)

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 \(\phi_R : G \hookrightarrow \text{Aut}_R(R[[y]])\). \noindent Their approach is based on a generalization of the method of \textit{B. Green} and \textit{M. Matignon} [J. Am. Math. Soc. 12, No.~1, 269--303 (1999; Zbl 0923.14007)] and \textit{Y. Henrio} [\url{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 \(\mathbb{F}_p\).