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Approximation theorems for classifying stacks over number fields

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Pith's one-line read This paper proves that the Brauer-Manin obstruction is the only obstruction to strong approximation for classifying stacks of connected linear algebraic groups over number fields.

desk verdict A genuinely new strong approximation theorem for BG that hinges on an unverified black-box application of Borovoi–Demarche; send to referees with a request to check hypotheses. read the letter →

arxiv 2507.13900 v2 pith:KKT3NNLF submitted 2025-07-18 math.NT

Ajneet Dhillon This is my paper · ORCID
classification math.NT MSC 11R3414D2314G0514F22
keywords classifyingstacksstrongapproximationBrauer-ManinobstructionalgebraicG-torsorsnumberfieldshomogeneousspacesBrauergroup
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper asks when finitely many local G-torsors for a connected linear algebraic group G over the completions of a number field can be approximated by a single G-torsor defined over the field. Its answer is that the Brauer-Manin obstruction is the only obstruction: if the local torsors are orthogonal to the Brauer group of the classifying stack BG, then a global G-torsor exists that is simultaneously close to all of them. To get there, the paper builds a topology on the adelic points of algebraic stacks, constructs a Brauer-Manin pairing for stacks, and identifies the Brauer group of BG with that of a homogeneous space of SL(V). The final theorem then follows by applying an equivariant strong approximation theorem for homogeneous spaces. A sympathetic reader would take away that strong approximation with Brauer-Manin obstruction holds for classifying stacks of connected linear algebraic groups.

What carries the argument

The argument is carried by a chain of reductions that ends at a known equivariant strong approximation result. First, since every connected linear algebraic group embeds into SL(V), the classifying stack BG is isomorphic to the quotient stack [G\SL(V)/SL(V)] (Lemma 2.1). The paper proves (Proposition 5.3) that the pullback map identifies the Brauer group of this quotient stack with the Brauer group of the homogeneous space G\SL(V), so the Brauer-Manin condition on BG becomes a condition on that space. At that point the proof invokes the equivariant strong approximation theorem for homogeneous spaces (cited as [5, Theorem 6.1]) to find a point of G\SL(V)(k) up to an SL(V)(k_S)-translate inside any given open neighbourhood of the adelic point; its image in BG is k-rational. The remaining machinery is the topologization of adelic points of stacks and the construction of the Brauer-Manin pairing for stacks, which make the approximations and the obstruction well-defined.

What would settle it

Find a connected linear algebraic group G and a finite set S of places (containing at least one finite place and all archimedean places) for which there are local G-torsors over the places of S that are orthogonal to the Brauer group of BG but cannot be approximated by any global G-torsor; this would refute Theorem 5.5. A more limited check is to verify the precise hypotheses of the equivariant strong approximation theorem cited in the proof on G\SL(V) and to identify one connected G for which a hypothesis fails.

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Extended reading notes

Core claim

The central discovery is Theorem 5.5: let G be a connected linear algebraic group over a number field k, and let S be a finite set of places containing at least one finite place and all archimedean places. Then strong approximation for the classifying stack BG off S with respect to the Brauer group Br(BG) holds: the diagonal image of BG(k) is dense in the adelic points of BG off S that are orthogonal to Br(BG) under the Brauer-Manin pairing. In concrete terms, a finite collection of local G-torsors over the completions at places of S can be approximated arbitrarily well by a global G-torsor precisely when the collection satisfies the Brauer-Manin condition. This extends the classical Brauer-Manin theory from varieties and schemes to stacks.

Load-bearing premise

The proof depends on the cited equivariant strong approximation theorem applying to the specific homogeneous space G\SL(V) under the action of SL(V) for every connected linear algebraic group G and every set S as in the theorem; if that theorem has additional hypotheses on S, stabilizers, or rational points that G\SL(V) does not satisfy, the density claim for BG does not follow.

Editorial extensions

If this is right

  • For G = PGL_n, the theorem gives a stack-level statement of the classical fact that the Brauer group of the number field controls which local PGL_n-torsors can be approximated by a global one.
  • The Brauer-Manin pairing for stacks is a new tool that can be applied to other stacks of the form [Y/G], not just classifying stacks, whenever the reduction to a special-group quotient works.
  • The result converts the existence question for global G-torsors into a finite computation: evaluate the Brauer-Manin pairing on the given local torsors; if it vanishes, approximation is possible.
  • For quotients of groupic varieties (including certain toric quotients), the paper's Theorem 5.4 yields strong approximation with Brauer-Manin obstruction as well, giving a family of examples beyond classifying stacks.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The theorem suggests that for any quotient stack [Y/G] whose associated SL(V)-bundle Y ×_G SL(V) satisfies strong approximation with Brauer-Manin, the same should hold for [Y/G]; the paper proves this in a special case, and checking mild hypotheses on Y could extend it further.
  • One natural test is to ask whether the condition that S contains a finite place is necessary; the equivariant input may require it, and exploring the archimedean-only case could reveal a genuinely different obstruction.
  • Because the Brauer-Manin pairing on stacks is constructed via spreading out, it might also be defined for stacks over global function fields, where an analogue of the theorem would be a further test of the method.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper develops a topology on the adelic points of a class of algebraic stacks over a number field, constructs a Brauer-Manin pairing for algebraic stacks, and proves a strong approximation theorem with Brauer-Manin obstruction for classifying stacks of connected linear algebraic groups. The main theorem (Theorem 5.5) asserts that if G is a connected linear algebraic group over a number field k and S is a finite set of places containing all archimedean places and at least one finite place, then the diagonal image of BG(k) is dense in BG(A^{S,k})_{Br(BG)}. The proof reduces the problem to an equivariant strong approximation theorem of Borovoi-Demarche applied to the homogeneous space G\SL(V) under the right action of SL(V). The paper also proves a more general quotient-stack theorem (Theorem 5.4) and establishes foundational results on spreading out stacks, on the Brauer group of quotient stacks, and on the Brauer-Manin pairing for stacks.

Significance. If correct, the main theorem answers a concrete question: a finite collection of local G-torsors can be approximated by a single global G-torsor whenever the local data is orthogonal to the Brauer group of BG, making the Brauer-Manin obstruction the only obstruction to strong approximation for classifying stacks. The paper also contributes a stack-theoretic framework for adelic topologies, correcting an error in the topologization of adelic points of stacks in [7], and extends the Brauer-Manin formalism to Artin stacks. The proof of the main theorem is short and cleanly reduces to the deep external results of Kneser, Sansuc, and Borovoi-Demarche. However, the two load-bearing steps—the verification of the exact hypotheses of [5,6.1] and the spectral sequence proof of Proposition 5.3—are not fully documented as written, so the current version requires revision before the central claim can be considered established.

major comments (2)
  1. [§5.2 and Theorem 5.5] The proof of Theorem 5.5 invokes [5, Theorem 6.1] as a black box, but the paper does not quote the theorem's precise hypotheses and does not verify that they hold for X=G\SL(V), H=SL(V), and the set S of Theorem 5.5. In particular, it is not documented whether [5,6.1] allows the stabilizer G to be an arbitrary connected linear algebraic group, whether S must satisfy any condition beyond containing all archimedean places and at least one finite place, whether X(k) must be nonempty, or whether the theorem's conclusion is stated for the right action of H on G\H rather than the left action on H/G. Since this external theorem is the decisive step that produces the rational point t and the element g∈SL(V)(k_S), the density claim for BG is unsupported unless the author supplies the exact statement of [5,6.1] and checks each hypothesis explicitly. This is a load-bearing verification gap rather than a demonstrated counterexample.
  2. [Proposition 5.3] The proof of Proposition 5.3 is a compressed spectral sequence argument that is not fully checkable. The text asserts that the spectral sequence for the presentation X→[X/SL(V)] and the spectral sequence for the projection X×SL(V)→X are related by an automorphism of X×SL(V), and that Proposition 5.1 then gives the desired isomorphism, but it does not spell out the identification of the E_1-terms, the shift in indices, the convergence, or why the isomorphisms in H^0, H^1, H^2 from Proposition 5.1 suffice to identify the H^2_tors groups. Because Proposition 5.3 is used in Theorem 5.5 to transfer the Brauer-Manin condition from BG to X, this proof should be expanded, for example by using the Leray spectral sequence for the SL(V)-torsor X→[X/SL(V)] together with the vanishings of R^1 and R^2 f_* G_m established in Corollary 5.2.
minor comments (5)
  1. [§5.2] The sentence 'If S is a finite set of places containing such that strong approximation holds for H off S' is missing words (presumably 'containing all archimedean places and such that'), and the phrase 'is dense is dense' contains a duplicate. Please correct both.
  2. [Theorem 5.5 proof] The cross-reference 'Corollary 5.3' should be 'Proposition 5.3', and the reference to 'Proposition 2.2' for the triviality of SL(V)-torsors should be to Corollary 2.3.
  3. [Throughout] There are several typographical errors: 'Knesser' should be 'Kneser', 'it's derived subgroup' should be 'its derived subgroup', 'diagrm' should be 'diagram', 'morhpism' should be 'morphism', and the notation for the off-S adeles appears inconsistently as 'A^{S,k}' and 'A S'; these should be unified.
  4. [§4.3] In the definition of the Brauer-Manin pairing for stacks, the sum is written over a finite set S, but the intended meaning is the sum over all v∈Ω_k, with S being a finite set containing the places where the given adelic point does not lift to O_v; the current wording is ambiguous and should be clarified.
  5. [§5.1] In the proof of Proposition 5.1, the phrase 'using the fact that SL(V) is equal to it's derived subgroup' should be corrected to 'its derived subgroup'; additionally, the citations to Sansuc's results could be made more precise by giving the specific statements used.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: Theorem 5.5 rests on the external Borovoi–Demarche equivariant strong approximation theorem; the sole self-citation [2] provides an elementary quotient-stack identification and is not load-bearing.

full rationale

The central derivation chain is independent of the paper's own prior work. Theorem 5.5 reduces strong approximation for BG to the equivariant statement for X = G\SL(V) under H = SL(V), invoking [5, Theorem 6.1] (Borovoi–Demarche) together with classical strong approximation for SL(V) (Kneser/Platonov). These are external results, not consequences of the paper's own framework. The only self-citation is [2] (Behrend–Dhillon), used in Sections 1 and 2 to motivate rewriting a G-quotient as an SL(V)-quotient via Lemma 2.1, [X/H] ≅ [X×^H G/G]. Lemma 2.1 is elementary and is proved directly in the text, so the citation is not load-bearing. No fitted parameter is relabelled as a prediction, no uniqueness result from the author's prior work is imported, and no definition is circular. One correctness flag, not a circularity flag: Section 5.2 paraphrases [5, Thm 6.1] without reproducing its precise hypotheses, so the applicability of that theorem to G\SL(V), with the required conditions on S and the stabilizer, should be checked against the original cited paper. This is a verification gap in an external citation rather than a self-referential reduction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No numerical parameters are fitted; the proof is a chain of references to established theorems. The main external inputs are Kneser's strong approximation for SL(V), Borovoi-Demarche's equivariant strong approximation for homogeneous spaces, Sansuc's Brauer group results, and standard spreading-out and descent technology. No new objects with independent evidence requirements are postulated.

assumptions (5)
  • domain assumption Kneser-Platonov strong approximation for SL(V) off a nonempty finite set of places
    Used in Theorem 5.5 to apply [5,6.1]; cited as [16] and [23].
  • domain assumption Borovoi-Demarche equivariant strong approximation for homogeneous spaces with connected stabilizer [5, Theorem 6.1]
    Transfers strong approximation from X=G\SL(V) to BG; exact hypotheses are not restated in the paper.
  • standard math Sansuc's cohomological results [27, 6.5, 6.6, 6.9] on Brauer groups of linear algebraic groups and homogeneous spaces
    Used in Proposition 5.1 to prove H^i(X,G_m) is unchanged by product with SL(V).
  • standard math Cohomological descent and spectral sequences for simplicial schemes over algebraic stacks [28, Tags 06XJ, 0D76, 09WJ]
    Used in Proposition 5.3 and in spreading out cohomology classes in Section 3.
  • standard math Spreading out results for schemes, algebraic spaces, and stacks [25, 3.2], [28, Tag 02WW], Theorems 4.1 and 4.2
    Underlies the topology on adelic points and the construction of the Brauer-Manin pairing.

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Pith. "Pith review of Approximation theorems for classifying stacks over number fields." pith.science (2026). https://pith.science/paper/KKT3NNLF

@misc{pith2026250713900,
  author       = {Pith},
  title        = {Pith review of: Approximation theorems for classifying stacks over number fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KKT3NNLF}},
  note         = {Machine review of arXiv:2507.13900}
}
abstract

Approximation theorems for algebraic stacks over a number field $k$ are studied in this article. For G a connected linear algebraic group over a number field we prove strong approximation with Brauer-Manin obstruction for the classifying stack $BG$. This result answers a very concrete question, given $G$-torsors $P_v$ over $k_v$, where $v$ ranges over a finite number of places, when can you approximate the $P_v$ by a $G$-torsor $P$ defined over $k$.

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