Non-Density of Toric Galois Loci

Definitions and mathematical statement

The supplied record refers to geometric $\ell$-adic local systems $V_\ell$ on a space $X$, but its excerpt begins with "With the above assumptions" and omits those assumptions. In particular, it does not specify the base field, the hypotheses on $X$, the geometric construction of $V_\ell$, the allowed primes, or the meaning of the point set denoted by $|X|$. The following is therefore a conditional transcription of the intended questions; the omitted setup must be recovered before this becomes a standalone mathematical statement.

Retain the source notation $|X|^{\mathrm{tor}}_{V_\ell}$ for its toric locus and set \[ T_\ell:=|X|^{\mathrm{tor}}_{V_\ell}\cap |X|. \] The supplied excerpt does not define the pointwise group $G_{\ell,x}$ or specify whether the toric condition concerns that whole group or its identity component. Accordingly, $T_\ell$ here names the source's intended locus without supplying an unverified definition.

Subject to all the omitted assumptions of the source's Conjecture 2, its non-density assertion is \[ \overline{T_\ell}^{\,\mathrm{Zar}}\ne X \qquad\text{for every prime $\ell$ allowed by that setup.} \] The associated independence question asks whether, for the systems arising from the same source setup, \[ T_\ell=T_{\ell'} \qquad\text{for every pair of allowed primes $\ell,\ell'$.} \] The common point domain and the relation between the systems must be specified in the missing setup. The intersection with $|X|$ is retained exactly as in the supplied clean statement; its role cannot be resolved from this excerpt.

The excerpt reports the following restricted result, which must be distinguished from the unrestricted conjecture. For a prime $\ell$, its level condition is \[ (\mathrm{Lev}_\ell):\quad \operatorname{im}\!\left(\pi_1(X_{\bar k})\longrightarrow \operatorname{GL}(V_\ell)\right) \text{ is a pro-$\ell$ group}. \] Here $k$, $\bar k$, the representation, and its fiber are part of the omitted setup. Assuming the hypotheses of Conjecture 2 and $(\mathrm{Lev}_{\ell_1})$, $(\mathrm{Lev}_{\ell_2})$ for two distinct primes $\ell_1\ne\ell_2$, the extracted Theorem 1 asserts the existence of a set $L$ of primes of positive Dirichlet density such that, for every $\ell\in L$, \[ \overline{T_\ell^{-}\cap X(k)}^{\,\mathrm{Zar}}\ne X. \] If the complex period map describing $V_\infty$ is finite-to-one, it further asserts that $T_\ell^{-}\cap X(k)$ is finite for these primes. The apparent reading of the damaged definition of the minus locus is \[ T_\ell^{-}=\{x\in T_\ell:\text{ there is an allowed prime }\ell_x\ne\ell \text{ with }x\in T_{\ell_x}\}. \] This reading requires source verification: superscripts and subscripts in the supplied extraction are corrupted. In particular, the reported theorem concerns a minus locus intersected with $X(k)$, whereas the conjecture is stated using $|X|$; these domains must not be identified without the missing assumptions.

The excerpt also reports that the solvable locus \[ |X|^{\mathrm{solv}}_{V_\ell} :=\{x\in|X|:G_{\ell,x}\text{ is solvable}\} \] is independent of $\ell$ and is contained in an exceptional locus denoted $|X|_{V_\ell}$, which the excerpt does not define. It attributes independence to class field theory, citing $[\mathrm{Se68}]$, and states that this holds more generally for any ${\mathbb{Q}}_\ell$-local system that is almost pointwise geometric in the sense of Fontaine--Mazur; that condition is not defined in the record.

The motivated toric locus, denoted $|X|^{\mathrm{tor}}_{V_{\mathrm{mot}}}$, is described as the set of points $x\in|X|$ for which the identity component of the motivated Galois group, in the sense of André, of the motivated motive $h^i(Y_x)(j)$ is a torus. The family $Y$, the indices $i,j$, and the required motivic setup are omitted. The excerpt asserts its inclusion in the toric locus for every prime $\ell$, and reports, under the restricted theorem's hypotheses, \[ \overline{|X|^{\mathrm{tor}}_{V_{\mathrm{mot}}}\cap X(k)}^{\,\mathrm{Zar}}\ne X. \] These are supplied background assertions, not independently verified results of this draft.

Finally, the excerpt says that Conjecture 2 follows from an unspecified Conjecture 1, and also from the Mumford--Tate and generalized André--Oort conjectures through a deep implication involving the average Colmez conjecture. It associates the intended toric locus with CM points of a polarizable rational variation of Hodge structure $V_\infty$, whose displayed construction and comparison are corrupted. It explicitly says that the general comparison with the CM locus, and even independence of $\ell$, are unknown in that discussion. Its final remark expresses hope for an extension to arbitrary $d\ge1$ without level conditions, still restricting $\ell$ to a positive-density set of primes, but the remark is truncated and $d$ is not defined.

Short English statement

Under the report's omitted geometric hypotheses, the conjecture asks whether the toric Galois locus is not Zariski dense in $X$. A related question asks whether that locus is independent of $\ell$. The supplied excerpt reports a restricted theorem involving level conditions at two distinct primes, a positive-density set of primes, and a smaller locus of $k$-rational points, with finiteness additionally requiring a finite-to-one period map. Missing hypotheses and damaged notation prevent a complete standalone formulation. This is a statement-only draft requiring human mathematical review.

Sources

• Local UnsolvedMath registry: supplied complete record for permanent ID $30005642$, title Non-Density of Toric Galois Loci, exact source problem code catalogue entry. The record supplies no individual proposer. Its proposed year is $2023$.
• 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 supplied title year $2024$ and the record's $2023$ year information are retained without reconciliation.
• Supplied literature source, labelled "Cadoret--Stix report": https://doi.org/10.4171/owr/2023/42. The background describes Oberwolfach Reports as records of mathematical workshops and their open-problem discussions.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem "partially solved" and says that a theorem under level conditions at two primes gives non-density and conditional finiteness conclusions while the unrestricted conjecture remains open. This is supplied dated curation derived from the OWR record, not an independently verified assessment; its time-sensitive status and cited claims require independent checking.
• The registry labels the statement corrected_verified and reports that it was checked against the official report. That verification claim is supplied metadata and was not independently repeated for this draft.
• The extracted discussion cites $[\mathrm{Se68}]$ for a class-field-theoretic assertion. No corresponding bibliographic entry is supplied, so this citation cannot be identified reliably from the record alone.