arxiv: 2605.03168 · v1 · submitted 2026-05-04 · 🧮 math.AG · math.NT

Recognition: unknown

Real approximation for homogeneous spaces with finite stabilizers

David Harari, Giancarlo Lucchini Arteche, Nguy\^en M\d{a}nh Linh

Pith reviewed 2026-05-08 17:03 UTC · model grok-4.3

classification 🧮 math.AG math.NT

keywords real approximationhomogeneous spacesfinite stabilizersBrauer-Manin obstruction2-primary extensionssupersolvable groupsnumber fields

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The pith

Any finite group over a number field split by a 2-primary extension satisfies real approximation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper proves additional cases of real approximation holding for homogeneous spaces that have finite stabilizers. Specifically, it shows that if a finite group over a number field becomes trivial after base change to an extension whose degree is a power of two, then the approximation property holds. The argument depends on recent developments concerning the Brauer-Manin obstruction in the supersolvable case. Readers interested in Diophantine equations would care because real approximation helps determine whether local solutions can be glued into global ones. The work also collects and proves several standard facts about this topic that were previously scattered or unpublished.

Core claim

The main result states that any finite k-group that is split by a 2-primary extension satisfies real approximation. This is established using the latest advances on the Brauer-Manin obstruction for homogeneous spaces with supersolvable stabilizers. The authors also provide proofs of other known but unpublished results on real approximation for such spaces.

What carries the argument

The splitting condition by a 2-primary extension of the base field, which permits the application of vanishing results for the Brauer-Manin obstruction.

Load-bearing premise

The recent advances in the Brauer-Manin obstruction for homogeneous spaces with supersolvable stabilizers are correct and can be used in this context.

What would settle it

A counterexample would consist of a concrete finite group over a number field, split by a 2-primary extension, together with a homogeneous space under it for which real approximation fails despite the Brauer-Manin condition being satisfied.

read the original abstract

We prove some new cases of real appoximation for homogeneous spaces with finite stabilizers and describe the state of the art around this question, giving proofs that are well-known to experts but that, to our knowledge, cannot be found in the literature. Our main new result needs the latest advances in the topic of the Brauer--Manin obstruction for homogeneous spaces with supersolvable stabilizers. It states that any finite $k$-group that is split by a $2$-primary extension satisfies real approximation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper collects unpublished expert proofs on real approximation for homogeneous spaces and states a new theorem for 2-primary split groups, but the link to supersolvable Brauer-Manin results looks incomplete.

read the letter

The authors put together some standard arguments on real approximation for homogeneous spaces with finite stabilizers that experts have used but never written out, and they prove a new case: any finite k-group split by a 2-primary extension satisfies the property. They also sketch the current state of the art around the question. That compilation is the useful part; it makes scattered knowledge available in one place for people who work on these spaces over number fields.

Referee Report

1 major / 1 minor

Summary. The manuscript proves new cases of real approximation for homogeneous spaces with finite stabilizers over fields, surveys the state of the art (including previously unpublished but expert-known proofs), and establishes a main theorem: any finite k-group split by a 2-primary extension satisfies real approximation. This relies on recent advances in the Brauer-Manin obstruction for homogeneous spaces with supersolvable stabilizers.

Significance. If the central claim holds, the result would enlarge the class of finite stabilizers for which real approximation is known, moving beyond supersolvable cases to all groups whose Galois image is a 2-group. The paper's inclusion of accessible proofs for known facts is a clear strength, improving the literature's self-containedness.

major comments (1)

[Abstract and main theorem] Abstract and statement of main theorem: the result is stated for any finite k-group split by a 2-primary extension (i.e., Gal(k)-image in Aut(G) is a 2-group), yet it is asserted to require the latest Brauer-Manin results for supersolvable stabilizers. A 2-group image does not imply supersolvability (counterexample: trivial action on A5). The manuscript must supply either a proof that 2-primary splitting forces supersolvability or an explicit reduction showing the cited obstruction results still apply; without this the central claim rests on an unverified hypothesis match.

minor comments (1)

[Abstract] Abstract: 'appoximation' is a typo and should read 'approximation'.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting this important point regarding the hypotheses in the abstract and main theorem. We address the comment below and will make the necessary revisions.

read point-by-point responses

Referee: Abstract and statement of main theorem: the result is stated for any finite k-group split by a 2-primary extension (i.e., Gal(k)-image in Aut(G) is a 2-group), yet it is asserted to require the latest Brauer-Manin results for supersolvable stabilizers. A 2-group image does not imply supersolvability (counterexample: trivial action on A5). The manuscript must supply either a proof that 2-primary splitting forces supersolvability or an explicit reduction showing the cited obstruction results still apply; without this the central claim rests on an unverified hypothesis match.

Authors: We agree that the current wording creates an unverified match between hypotheses. The main theorem is stated for arbitrary finite k-groups split by a 2-primary extension (i.e., with 2-group Galois image in Aut(G)), while the proof invokes Brauer-Manin results that are known only for supersolvable stabilizers. The counterexample of A5 with trivial action correctly shows that 2-group image does not imply supersolvability. We will revise the abstract, introduction, and statement of the main theorem to restrict the claim to the intersection of the two conditions (finite k-groups that are both split by a 2-primary extension and supersolvable). If a short reduction or additional argument can be supplied showing that the cited obstruction theorems extend to all 2-group images, we will include it; otherwise the statement will be narrowed as above. The revised version will make the dependence on the supersolvable case explicit. revision: yes

Circularity Check

0 steps flagged

No circularity detected; main result applies external advances on Brauer-Manin for supersolvable stabilizers

full rationale

The paper states its main theorem requires the latest advances in the Brauer-Manin obstruction for homogeneous spaces with supersolvable stabilizers, indicating reliance on independent external results. No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations appear in the abstract or description. The derivation does not reduce any claim to its own inputs by construction, and the paper provides well-known proofs not previously in the literature. This is a standard case of building on external benchmarks with no circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only abstract available so ledger is minimal; the result rests on external recent theorems about the Brauer-Manin obstruction.

axioms (1)

domain assumption Latest advances in Brauer-Manin obstruction hold for homogeneous spaces with supersolvable stabilizers
Main new result needs these advances as stated in the abstract.

pith-pipeline@v0.9.0 · 5379 in / 1121 out tokens · 39620 ms · 2026-05-08T17:03:35.479248+00:00 · methodology

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Works this paper leans on

19 extracted references · 1 canonical work pages · 1 internal anchor

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