Admissible Groups over Global Function Fields

Definitions and mathematical statement

Let $k$ be a finite field and let $F/k$ be a finitely generated field extension of transcendence degree one, so that $F$ is the function field of an integral curve over $k$. Fix such a field $F$.

For this draft, use the customary admissibility convention that an $F$-division algebra $D$ is a finite-dimensional central division algebra: $D$ is an associative division algebra with center $F$ and $\dim_F D<\infty$. A subfield $L\subseteq D$ containing $F$ is maximal if no strictly larger commutative subfield of $D$ contains it. A finite group $G$ is admissible over $F$ if there exist such an algebra $D$ and a maximal subfield $L\subseteq D$ for which $L/F$ is Galois and \[ \operatorname{Gal}(L/F)\cong G. \]

Problem. For each fixed function field $F$ as above, characterize the finite groups $G$ admissible over $F$; that is, determine a necessary-and-sufficient condition for the existence of $D$ and $L$ satisfying these requirements.

Qualification. The supplied statement does not explicitly specify finite dimensionality or centrality of the division algebra. The convention used above requires confirmation against the cited report. The supplied record labels the problem open; this draft does not independently verify that status.

Short English statement

For a fixed function field of a curve over a finite field, characterize exactly which finite groups occur as Galois groups of Galois maximal subfields of finite-dimensional central division algebras over that field. The division-algebra convention and the supplied open-status assessment require review.

Sources

• Local UnsolvedMath registry, permanent ID $30001145$, title Admissible Groups over Global Function Fields, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The supplied record gives proposed year 2009 and names no individual proposer.
• The Arithmetic of Fields (2009), Oberwolfach Reports, pp. 285--354, DOI 10.4171/owr/2009/05: https://doi.org/10.4171/owr/2009/05. This is the supplied source report and the literature-source entry labeled "original OWR report."
• Supplied literature-source entry labeled "Harbater--Hartmann--Krashen, admissibility": https://arxiv.org/abs/1201.1938.
• Literature checked at: 2026-08-21. The supplied dated curation labels the problem open and states that the precise necessary-and-sufficient criterion remains open in the checked literature. This is a supplied dated assessment, not independently verified here; its time-sensitive status and cited claims require independent verification.