Non-Density of Bounded-Rank Galois Loci

Definitions and mathematical statement

The supplied record does not define the ambient object $X$, the parameters $\ell$ and $d$, the group $G_\ell$, the object $V_\ell$, or the loci $|X|_{V_\ell}$ and $|X|_{\leq d}$. Accordingly, the following is a conditional statement in the source notation, requiring the missing definitions and hypotheses before it constitutes a fully specified conjecture.

For the intended $X$, endowed with its Zariski topology, write \[ S_{\ell,d}:=|X|_{V_\ell}\cap |X|_{\leq d}. \] Here both loci are retained as source-designated subsets of points of $X$; their membership conditions are not supplied. A subset $S$ is non-Zariski-dense in $X$ if its Zariski closure $\overline{S}^{\,\mathrm{Zar}}$ is a proper subset of $X$. Write $G_\ell^{\mathrm{ab}}$ for the abelianization intended by the source; the record does not specify whether this uses an abstract commutator subgroup or its closure in a group topology.

Conjecture 1, as recorded. Under the intended ambient hypotheses and for the intended admissible parameters, assume that $G_\ell$ has finite abelianization. Then \[ \#G_\ell^{\mathrm{ab}}<\infty \quad\Longrightarrow\quad \overline{|X|_{V_\ell}\cap |X|_{\leq d}}^{\,\mathrm{Zar}}\subsetneq X. \] The ambient hypotheses, parameter ranges, and quantifiers over $X$, $V_\ell$, $\ell$, and $d$ are absent from the supplied excerpt and must be recovered before this schema can be made precise. In particular, neither the meaning of the bound indexed by $d$ nor the relationship between $G_\ell$ and $V_\ell$ is specified.

The supplied excerpt attributes the curve case to Cadoret--Tamagawa, with reference label [CT13], and describes the case $\dim X\geq 2$ as widely open. It also reports that the strategy of [CT13] gives a heuristic for $d=1$, referring to [Cad23, § 4]. The subsequent explanation is truncated after a reference to a diophantine consideration. These are supplied source qualifications, not independently verified mathematical or literature assessments. An extracted footnote says that an unspecified object is only well defined up to conjugacy; its referent cannot be recovered from this record.

Short English statement

The recorded conjecture says that, if $G_\ell$ has finite abelianization, the intersection of the two designated loci is not Zariski-dense in $X$. The record omits the definitions of these loci and the ambient hypotheses, so the statement remains a schema requiring review. Supplied curation labels the problem partially solved, attributing the curve case to Cadoret--Tamagawa and describing dimensions at least two as widely open; these claims have not been independently verified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005641$, exact source problem code catalogue entry, title "Non-Density of Bounded-Rank Galois Loci," in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2023, status partially_solved, and statement status corrected_verified. Its assertion that the statement was checked against the official report is supplied provenance, not an independent verification here.
• Supplied report citation: MFO--RIMS Tandem Workshop: Arithmetic Homotopy and Galois Theory (2024), Oberwolfach Reports, pp. 2377--2488, DOI 10.4171/owr/2023/42. Supplied source URL: https://doi.org/10.4171/owr/2023/42. The extracted contribution is titled "On the toric locus of $\ell$-adic local systems arising from geometry," Anna Cadoret, joint work with Jakob Stix. No separate proposer is supplied.
• Supplied literature source: "Cadoret--Stix report with the conjecture and known curve case," https://doi.org/10.4171/owr/2023/42. Literature checked at: 2026-08-22. This date and the accompanying partially solved assessment are supplied dated curation, not independently verified. The background explicitly requests independent verification of time-sensitive status and cited claims.
• Internal references preserved from the excerpt: [CT13], attributed in the supplied curation to Cadoret--Tamagawa for the curve case; and [Cad23, § 4], cited for a heuristic when $d=1$. Full bibliographic entries and separate links for these references are not supplied.