Malle-Type Asymptotics for Refined Cover Counts

Definitions and mathematical statement

Under the usual interpretation of $S_n$, let $n$ be a positive integer and let $S_n$ be the group of permutations of $\{1,\ldots,n\}$. A subgroup $G\leq S_n$ is transitive if, for every $i,j\in\{1,\ldots,n\}$, some $g\in G$ satisfies $g(i)=j$. The source calls $G$ countable; with this interpretation of $S_n$, $G$ is necessarily finite. Let $c\subseteq G$ be conjugation-invariant, meaning that $gcg^{-1}=c$ for every $g\in G$.

The source denotes by $N^c_{G,q}(X)$ a refined count of geometrically connected $G$-covers whose ramification monodromy lies in $c$. For a scheme over a field $k$, geometrically connected means that its base change to an algebraic closure $\overline{k}$ is connected. However, the supplied record does not specify the base field or base curve, the meaning of $G$-cover, the equivalence relation or weights used in counting, or the quantity bounded by $X$. Consequently, this description does not yet define a counting function. In particular, interpreting $q$ as the cardinality of a finite field would require an additional hypothesis.

Proposed asymptotic, conditional on specifying the counting problem. For each admissible pair $(G,c)$, the source asks whether there are constants $a,b,C$, depending on $G$ and $c$, such that \[ N^c_{G,q}(X)\sim C X^a(\log X)^{b-1} \qquad (X\longrightarrow\infty). \] Here $f(X)\sim h(X)$ means that $h(X)$ is nonzero for all sufficiently large admissible $X$ and that $f(X)/h(X)\to1$. The source does not specify the domains or allowable values of $a,b,C$, the admissible values of $q$ and $X$, or whether $q$ is fixed during this limit. Its stated dependence of the constants on $G,c$ leaves their possible dependence on $q$, the permutation action, and other counting data unresolved. No uniformity in $q$ is specified.

The condition on ramification also requires clarification: the record does not specify which branch points are constrained, how monodromy elements are chosen, or hypotheses concerning characteristic, tame ramification, and wild ramification. It imposes no generating or nonemptiness condition on $c$, nor other conditions ensuring that the proposed asymptotic is applicable. These omissions prevent a fully specified mathematical conjecture; the displayed formula preserves the supplied target without resolving them.

Short English statement

The proposed refined function-field Malle-type question asks whether counts of geometrically connected covers, with ramification restricted by $c$, have a power-of-$X$ asymptotic with a logarithmic factor. The supplied record leaves the counting function and several hypotheses undefined, so this is a conditional statement template requiring mathematical review. The supplied partially_solved label is dated curation, not an independently verified conclusion.

Sources

• Local UnsolvedMath registry: permanent ID $30003182$, title "Malle-Type Asymptotics for Refined Cover Counts," exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the registry gives proposed year $2016$.
• Supplied report attribution: Topologie (2017), Oberwolfach Reports, DOI 10.4171/owr/2016/35, pp. 2009--2068. https://doi.org/10.4171/owr/2016/35. The background describes these reports as documenting mathematical workshops and their open-problem discussions. This authoritative report citation is supplied, but its support for the extracted formulation has not been independently checked here.
• Supplied literature reference: Ellenberg--Tran--Westerland. https://arxiv.org/abs/1701.04541.
• Supplied literature reference: Ellenberg--Venkatesh--Westerland. https://annals.math.princeton.edu/2016/183-3/p01.
• Supplied literature reference: Santens, function-field Malle. https://arxiv.org/abs/2512.12838.
• Literature assessment date: 2026-08-21. This is supplied dated curation, not independently verified. The supplied assessment reports upper-bound and special-family results for function-field analogues, together with recent large-field asymptotics, and describes the extracted unrestricted refined conjecture as not fully settled. The background explicitly characterizes this as dated literature triage derived from the supplied OWR curation and requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says that its mathematical content was preserved while notation, typography, or omitted definitions were repaired for self-contained publication. These are supplied editorial claims; they do not resolve the formulation gaps recorded here.