Density of Specializations of High-Genus Galois Covers

Definitions and mathematical statement

Let $T$ be transcendental over ${\mathbb{Q}}$, and let $E/{\mathbb{Q}}(T)$ be a finite Galois extension with Galois group $G$. The extension is ${\mathbb{Q}}$-regular if $E\cap\overline{{\mathbb{Q}}}={\mathbb{Q}}$. Let $C/{\mathbb{Q}}$ be the smooth projective geometrically integral curve with function field $E$, and write $g=g(C)$. Assume $g\geq 2$.

The recorded question asks whether, for every such extension, \[ \lim_{x\to\infty} \frac{\left|\operatorname{Sp}(E)\cap S(G,x)\right|} {\left|S(G,x)\right|}=0. \] Here the record calls $\operatorname{Sp}(E)$ the specialization set and describes the limit as density zero in $S(G)$, but does not define these sets or the ordering $S(G,x)$.

A standard interpretation of $S(G)$ is the set of ${\mathbb{Q}}$-isomorphism classes of finite Galois extensions of ${\mathbb{Q}}$ with group isomorphic to $G$. Under that interpretation, $\operatorname{Sp}(E)\cap S(G)$ consists of classes obtained from residue extensions at unramified rational specialization points with full Galois group $G$. These conventions require confirmation from the source; in particular, the supplied record does not specify the treatment of the point at infinity or of specializations with smaller Galois group.

Most importantly, no height, discriminant bound, or other rule defining $S(G,x)$ is supplied. The displayed limit therefore retains the source notation without selecting an ordering. A precise density assertion requires that rule and a counting convention, with $S(G,x)$ finite and its cardinality nonzero for all sufficiently large $x$. These are unresolved formulation requirements, not additional hypotheses asserted by the source.

Short English statement

For every ${\mathbb{Q}}$-regular Galois cover of function-field genus at least two, must its specializations have density zero among extensions of ${\mathbb{Q}}$ with the same Galois group? The supplied record leaves the counting order and some specialization conventions undefined, so the density statement requires source clarification.

Sources

• Local UnsolvedMath registry: data/unsolvedmath/problems.json, permanent ID $30004643$, title "Density of Specializations of High-Genus Galois Covers," source problem code catalogue entry. The record gives proposed year 2021 and supplies no individual proposer.
• Supplied report citation: Homotopic and Geometric Galois Theory (2022), Oberwolfach Reports, pp. 663--744, DOI 10.4171/owr/2021/12. https://doi.org/10.4171/owr/2021/12. The literature list also labels this source "Oberwolfach Report 12/2021." Both supplied year designations are retained without resolving their relationship.
• Supplied literature reference: "Density results for specialization sets." https://doi.org/10.1017/S1474748019000537.
• Supplied literature assessment, checked on exactly 2026-08-22: the status is labeled "partially solved." The assessment says that density zero is proved conditionally for sufficiently many branch points and in several unconditional families, while the genus-at-least-two question for a general cover remains unresolved. Its listed sources are https://doi.org/10.1017/S1474748019000537 and https://doi.org/10.4171/owr/2021/12. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment; its time-sensitive status and cited claims require independent verification.