Conditional Homogeneous-Space Density Theorem

Definitions and mathematical statement

The supplied record does not recover a complete mathematical statement: the ground field, the precise class of spaces, the topology, and the content of the referenced Conjecture 9.1 are missing. The following describes the extracted conditional assertion and explicitly identifies the additional framework needed to interpret it.

In the usual number-field interpretation, let $k$ be a number field, let $\bar{k}$ be an algebraic closure, and let $G$ be a semisimple simply connected algebraic group over $k$. Let $X$ be a homogeneous $G$-space over $k$, meaning that $G_{\bar{k}}$ acts transitively on $X_{\bar{k}}$. The number-field hypothesis and this description of $X$ are contextual reconstructions, not explicitly supplied hypotheses of the extracted statement. For a geometric point $\bar{x}\in X(\bar{k})$, its geometric stabilizer is \[ H=\operatorname{Stab}_{G_{\bar{k}}}(\bar{x}). \] The supplied hypotheses require $H$ to be finite and solvable. Here solvability means that the derived series of its geometric-point group terminates in the trivial group.

Under this number-field interpretation, write ${\mathbb{A}}_k$ for the adele ring of $k$, and write \[ \operatorname{Br}(X)=H^2_{\mathrm{\acute{e}t}}(X,\mathbb G_m)_{\mathrm{tors}} \] for the cohomological Brauer group. The Brauer--Manin set is \[ X({\mathbb{A}}_k)^{\operatorname{Br}(X)} =\left\{(x_v)_v\in X({\mathbb{A}}_k): \sum_v\operatorname{inv}_v\left(b(x_v)\right)=0\text{ in }{\mathbb{Q}}/{\mathbb{Z}} \text{ for every }b\in\operatorname{Br}(X)\right\}, \] where $v$ ranges over the places of $k$ and $\operatorname{inv}_v$ is the local Brauer invariant.

The extracted conditional target asks whether, assuming the externally referenced Conjecture 9.1 and with $G$ and $H$ as above, the diagonal image of $X(k)$ satisfies \[ \overline{\operatorname{im}\left(X(k)\longrightarrow X({\mathbb{A}}_k)^{\operatorname{Br}(X)}\right)}^{\,\tau} =X({\mathbb{A}}_k)^{\operatorname{Br}(X)}. \] Here $\tau$ denotes the intended topology, which is not specified by the supplied record. The record also does not specify whether a compactification, a modified adelic space, or another approximation convention is intended. These choices cannot be silently supplied, particularly for a nonproper homogeneous space.

The supplied statement-verification note says that the report presents the conditional implication as a theorem following from an external Conjecture 9.1, rather than as a separate open problem. The same record retains an open label and marks the statement unrecoverable. These are supplied curation judgments, not independently verified mathematical conclusions. Neither the missing conjecture nor a precise replacement open problem is reconstructed here.

Short English statement

The extracted assertion concerns density of rational points in a Brauer--Manin set for a homogeneous space with semisimple simply connected acting group and finite solvable geometric stabilizer, assuming an unidentified Conjecture 9.1. Its field, precise space, and density topology remain unresolved. Supplied verification describes the conditional assertion as a theorem in the source, despite the retained open label; this draft requires human mathematical and source review.

Sources

• Local registry: data/unsolvedmath/problems.json, supplied record with permanent UnsolvedMath ID $30003768$, exact source problem code catalogue entry, title Conditional Homogeneous-Space Density Theorem, in the Oberwolfach Reports Open Problems collection. No individual proposer is named.
• Supplied report citation: Mini-Workshop: Arithmetic Geometry and Symmetries around Galois and Fundamental Groups (2019), Oberwolfach Reports, pp. 1021--1080, DOI 10.4171/owr/2018/17. Supplied source URL and literature-source link labeled "OWR 2018 report": https://doi.org/10.4171/owr/2018/17. The citation says 2019, whereas the proposed year, report label, and DOI contain 2018; this discrepancy is unresolved.
• Supplied literature source: Harpaz--Wittenberg, "homogeneous spaces": https://arxiv.org/abs/1802.09605.
• Supplied literature source: Harpaz--Wittenberg, "supersolvable descent": https://arxiv.org/abs/2206.13832.
• Supplied literature assessment, checked on 2026-08-21: Harpaz--Wittenberg are reported to prove the stated density for cyclic-filtration stabilizers and related supersolvable cases; the solvable assertion is reported to remain conditional on Conjecture 9.1, with no unconditional general theorem known. This is supplied dated curation derived from the OWR collection, not an independently verified literature assessment. The supplied verification note separately describes the conditional implication as a theorem in the report. None of these references or claims was independently consulted or verified for this draft.