Odd Dihedral Groups as Oort Groups

Definitions and mathematical statement

Let $p$ be an odd prime and let $n\geq 1$ be an integer. Define the dihedral group \[ D_{p^n}=\langle r,s\mid r^{p^n}=s^2=1,\ srs^{-1}=r^{-1}\rangle, \qquad |D_{p^n}|=2p^n. \] The restriction $n\geq 1$ is the conventional positive prime-power interpretation; the supplied statement does not explicitly specify the range of $n$.

For a precise global interpretation, call a finite group $G$ an Oort group in characteristic $p$ if the following lifting property holds. For every algebraically closed field $k$ of characteristic $p$, every smooth projective connected curve $X/k$, and every faithful action $\rho:G\hookrightarrow\operatorname{Aut}_k(X)$, there exist a complete discrete valuation ring $R$ of characteristic zero with residue field identified with $k$, a smooth proper relative curve $\mathcal X/R$, and an action of $G$ by $R$-automorphisms of $\mathcal X$, together with a $G$-equivariant isomorphism $\mathcal X\times_R k\cong X$ recovering $\rho$.

Question under this global interpretation. Is $D_{p^n}$ an Oort group in characteristic $p$ for every odd prime $p$ and every integer $n\geq 1$?

There is an unresolved terminology issue in the supplied record: its question says "Oort groups," whereas its literature assessment says "local Oort groups." For comparison, a finite group $G$ is a local Oort group in characteristic $p$ if, for every algebraically closed field $k$ of characteristic $p$ and every faithful continuous action $G\hookrightarrow\operatorname{Aut}_k(k[[t]])$, there exist a complete discrete valuation ring $R$ of characteristic zero with residue field $k$ and a continuous action of $G$ by $R$-algebra automorphisms of $R[[T]]$ whose reduction is equivariantly isomorphic to the given action on $k[[t]]$. The corresponding local question asks whether this property holds for $D_{p^n}$ for every odd prime $p$ and every $n\geq 1$. These two formulations are recorded separately; the supplied evidence does not identify which definition was intended or establish an equivalence between them.

Short English statement

The question asks whether every dihedral group of order $2p^n$, for an odd prime $p$ and a positive integer $n$, has the Oort lifting property from characteristic $p$ to characteristic zero. The source question uses global terminology, while the supplied literature assessment discusses local actions on a formal disc; the intended formulation requires review.

Sources

• Supplied local problem registry record, permanent UnsolvedMath ID $30002713$, source problem code catalogue entry, titled Odd Dihedral Groups as Oort Groups; collection: Oberwolfach Reports Open Problems. The supplied proposed year is 2014; no individual proposer is identified.
• Supplied report citation: Valuation Theory and Its Applications (2015), Oberwolfach Reports, pp. 2757--2823, DOI 10.4171/owr/2014/49. https://doi.org/10.4171/owr/2014/49. The background describes Oberwolfach Reports as workshop reports documenting mathematical discussions and open problems.
• Supplied literature source: Oberwolfach Report 49/2014. https://ems.press/content/serial-article-files/46540.
• Supplied literature source: Obus, A generalization of the Oort conjecture. https://arxiv.org/abs/1502.07623.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and states that the local Oort conjecture for all odd-prime-power dihedral groups has substantial special cases, including $D_p$ and $D_9$, but remains open in general. This is supplied dated curation derived from the OWR record, not an independently verified assessment. The source explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review of its cleanup. This supplied editorial label does not establish mathematical correctness or resolve the global/local terminology issue.