Classification of Local Oort Groups

Definitions and mathematical statement

The supplied question is to classify local Oort groups, described as groups for which every local $G$-extension lifts to characteristic zero. The source record does not specify the characteristic, the residue field, or the admissible class of groups. The following gives a parameterized formulation under explicitly stated conventions; these conventions require confirmation against the cited sources.

Let $p$ be a prime, let $k$ be an algebraically closed field of characteristic $p$, and let $G$ be a finite group. In this formulation, a local $G$-action is a faithful continuous action \[ \rho\colon G\hookrightarrow \operatorname{Aut}_{k}(k[[z]]), \] where continuity refers to the $z$-adic topology. Equivalently, it describes a local Galois extension $k[[z]]/k[[z]]^{G}$ with specified Galois group $G$.

Say that $\rho$ lifts to characteristic zero if there exist a complete discrete valuation ring $R$ of characteristic zero, with maximal ideal $\mathfrak m_R$ and a specified identification $R/\mathfrak m_R\cong k$, a faithful continuous action \[ \widetilde\rho\colon G\hookrightarrow \operatorname{Aut}_{R}(R[[Z]]), \] and a $G$-equivariant $k$-algebra isomorphism \[ R[[Z]]/\mathfrak m_R R[[Z]]\cong k[[z]]. \] Here continuity upstairs refers to the $(\mathfrak m_R,Z)$-adic topology, and the action on the left is induced by $\widetilde\rho$. The ring $R$ and the lifted action may depend on $\rho$.

For a specified class $\mathcal C$ of finite groups, define \[ \mathcal L(k;\mathcal C)= \left\{G\in\mathcal C:\text{every faithful continuous action of }G\text{ on }k[[z]]\text{ lifts as above}\right\}. \] Classification problem, subject to the unresolved conventions: determine $\mathcal L(k;\mathcal C)$ up to group isomorphism for the intended choices of $p$, $k$, and $\mathcal C$.

The record does not determine whether the lifting property is required over one fixed $k$, over every algebraically closed field of a fixed positive characteristic, or across all positive characteristics. It also does not specify whether admissible groups must occur as local Galois groups. If a group admits no such local action, the displayed universal lifting property holds vacuously unless an occurrence condition is imposed. These distinctions remain unresolved and prevent identifying a unique classification target from the supplied wording alone.

Short English statement

Classify the finite groups for which every action on a formal disc in positive characteristic can be lifted to an action over a complete discrete valuation ring of characteristic zero. The intended characteristic, residue-field quantifiers, and admissible groups need confirmation. The supplied record labels the classification open; this draft does not independently verify that status.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002712$, source problem code catalogue entry, title Classification of Local Oort Groups, in the Oberwolfach Reports Open Problems collection. The record gives proposed year $2014$ and names no individual proposer.
• Supplied report citation: Valuation Theory and Its Applications (2015), Oberwolfach Reports, DOI 10.4171/owr/2014/49, pp. 2757--2823. https://doi.org/10.4171/owr/2014/49. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference: Oberwolfach Report 49/2014. https://publications.mfo.de/bitstream/handle/mfo/3439/OWR_2014_49.pdf?isAllowed=y&sequence=1.
• Supplied literature reference: Chinburg--Guralnick--Harbater, Global Oort groups. https://www2.math.upenn.edu/~harbater/oortpaper3.pdf.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that cyclic groups are local Oort groups, that several additional cases are known, and that a complete classification remains unresolved. This is supplied dated curation derived from the OWR material, not an independently verified assessment; its status and cited claims require independent verification.