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Unramified Brauer groups of homogeneous spaces with finite stabilisers and the Grunwald Problem

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Pith's one-line read Over any characteristic-zero field, the unramified Brauer group of a homogeneous space with finite stabiliser is determined by restriction to bicyclic-procyclic subgroups of the étale fundamental group, and the paper turns this into an…

desk verdict Solid extension of Lucchini Arteche's formula to all char-0 fields with genuine algorithms; one missing proof in Theorem 3.3 but repairable. read the letter →

arxiv 2506.02600 v1 pith:RHY7IBTY submitted 2025-06-03 math.AG math.NT

Lucas Lagarde This is my paper
classification math.AGmath.NT MSC 14F2214M1711R3211R3414G05
keywords unramifiedBrauergrouphomogeneousspacesfinitestabiliserBrauer-ManinobstructionGrunwaldproblemBogomolovmultiplieralgorithmsupersolvablegroups
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

Over any field of characteristic zero, the paper claims, the unramified Brauer group of a homogeneous space of a semisimple simply connected group with finite geometric stabiliser is exactly the group of Brauer classes that vanish on every bicyclic-procyclic subgroup of the étale fundamental group: subgroups whose intersection with the stabiliser is generated by two commuting elements and whose image in the Galois group is generated by one element. This turns a transcendental obstruction class into a finite group-theoretic computation. For quotients $SL_{n,k}/G$ the paper gives a practical algorithm: list all central extensions of the finite $k$-group $G$ by $\mathbb{Q}/\mathbb{Z}(1)$, test the two conditions of Theorem 4.4, and read off the unramified classes. Over a number field, the same list computes the Brauer-Manin set on $SL_{n,k}/G$, and together with previous work on supersolvable descent it makes the Grunwald problem effective for supersolvable groups. A reader should care because the formula works uniformly, including over the real numbers, and it makes transcendental Brauer classes explicit enough to evaluate at local points.

What carries the argument

The load-bearing object is the subgroup $X^2_{\mathrm{bic},\mathrm{pcyc}}$ of $H^2(\pi_1^{\mathrm{\acute{e}t}}(X,x), \mathbb{Q}/\mathbb{Z}(1))$: classes that die on every closed subgroup $D$ whose intersection with the finite stabiliser is bicyclic (generated by two commuting elements) and whose projection to the absolute Galois group is procyclic. The proof uses the general criterion that an unramified class evaluates trivially on all Laurent-series points over perfect fields of cohomological dimension $1$, together with the identification of the normalised Brauer group $\operatorname{Br}^0(SL_{n,k}/G)$ with the group $\operatorname{Ext}^c_k(G(k), \mathbb{Q}/\mathbb{Z}(1))$ of $\Gamma_k$-equivariant central extensions. The two conditions in Theorem 4.4 turn this infinite criterion into a finite check on subgroups of the finite group $G(k)$.

What would settle it

For a small finite group with a bicyclic subgroup $B$ generated by commuting, Galois-stable elements $\tau$ and $\gamma$, take $L$ a field of cohomological dimension $1$ and search for a nonzero class in $H^1(L,\hat B)$ that vanishes on both $\langle\tau\rangle$ and $\langle\gamma\rangle$ but not in the associated semidirect-product cohomology group; finding one would break the proof of the second inclusion in Theorem 3.3.

Watch

Extended reading notes

Core claim

The central discovery is a complete description of $\operatorname{Br}_{\mathrm{nr}}(X)$ as $X^2_{\mathrm{bic},\mathrm{pcyc}}(\pi_1^{\mathrm{\acute{e}t}}(X,x), \mathbb{Q}/\mathbb{Z}(1))$, equal to the intersection $X^2_{\mathrm{bic},0}(\pi_1^{\mathrm{\acute{e}t}}(X,x), \mathbb{Q}/\mathbb{Z}(1)) \cap X^2_{\mathrm{cyc},\mathrm{pcyc}}(\pi_1^{\mathrm{\acute{e}t}}(X,x), \mathbb{Q}/\mathbb{Z}(1))$, valid for every field $k$ of characteristic $0$, where $X$ is homogeneous under a semisimple simply connected group with finite geometric stabiliser. In the pointed case $X \simeq SL_{n,k}/G$, a normalised class is unramified precisely when, as a central extension of $\Gamma_k$-groups, it splits on every bicyclic subgroup (the Bogomolov condition) and satisfies a Galois-compatibility condition involving twisted commutators. From this, the paper derives an algorithm that computes $\operatorname{Br}^0_{\mathrm{nr}}(SL_{n,k}/G)$ for any finite $k$-group $G$ over any characteristic-zero field and, over number fields, computes the Brauer-Manin set. The construction makes the computed classes explicit as central extensions, not just abstract abelian groups.

Load-bearing premise

The proof of the second inclusion in the main theorem relies on an unproved claim: a nonzero obstruction class attached to a group generated by two commuting, Galois-stable elements must remain visible in the cohomology of at least one of the two cyclic subgroups; if that claim fails, the proof collapses.

Editorial extensions

If this is right

  • For any field of characteristic zero, including real closed fields, the unramified Brauer group of such a homogeneous space is now described by a single cohomological formula; previously the formula was restricted to non-essentially real fields.
  • The unramified Brauer group of $SL_{n,k}/G$ is computable in the strong sense: the algorithm returns explicit central extensions representing the unramified classes, together with the constant classes from $\operatorname{Br}(k)$.
  • Over number fields, the Brauer-Manin set of $SL_{n,k}/G$ is computable for every finite $k$-group $G$, since the only places that matter form a finite explicit set and local pointed sets are finite and computable.
  • The Grunwald problem is effective for supersolvable groups: given a finite supersolvable group, a number field, and a finite set of places, the algorithm decides which prescribed local Galois extensions arise from a global Galois extension.

Reading between the lines

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

  • A natural extension beyond the paper's scope is to test whether the same bicyclic-procyclic restriction criterion describes unramified Brauer groups for homogeneous spaces of algebraic groups that are not semisimple simply connected; the proof's use of low-dimensional fields and fundamental-group sections would be the main obstacle.
  • The explicit description of transcendental classes for semidirect products $N \rtimes Q$ suggests that Brauer-Manin obstructions descend to smaller fields than the $N^2$-roots-of-unity construction requires; the paper already demonstrates one such descent, and extending it to other families seems feasible.
  • The algorithm's practical speed is governed by enumerating central extensions of a finite group, so improvements in computing the Bogomolov multiplier would directly accelerate the whole procedure.
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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

1 major / 5 minor

Summary. The paper establishes a formula for the unramified Brauer group Br_nr(X) of a homogeneous space X of a semi-simple simply connected group over a field k of characteristic 0 with finite geometric stabiliser: Br_nr(X) = X^2_{bic,pcyc}(pi_1^et(X,x), Q/Z(1)) = X^2_{bic,0} ∩ X^2_{cyc,pcyc}. It translates this into a concrete criterion for classes on SL_n,k/G represented by Gamma_k-equivariant central extensions (Theorem 4.4), proves an algorithm to compute Br^0_nr(SL_n,k/G) for any finite k-group G (Theorem 5.1), and an algorithm to compute the Brauer-Manin set over number fields (Theorem 6.1). Applications include the effectivity of the Grunwald problem for supersolvable groups via work of Harpaz-Wittenberg, and explicit transcendental Brauer-Manin obstructions, including an example over the real numbers.

Significance. If correct, this is a substantial advance: it removes the 'non-essentially real' restriction from Lucchini Arteche's formula, gives the first general algorithm for the full (including transcendental) unramified Brauer group in this setting, and makes the Brauer-Manin obstruction computable for these homogeneous spaces. Proposition 5.3, identifying Br^0_nr(SL_n,K/G) with the Bogomolov multiplier B_0(G) after adjoining N^2-th roots of unity, is a clean and useful structural statement. The paper also provides explicit examples rather than a purely existential algorithm, and the application to the Grunwald problem is clearly explained. The main theorem is not machine-checked and relies on a few asserted cohomological steps, but the identified gap in the proof of Theorem 3.3 is local and repairable.

major comments (1)
  1. [Theorem 3.3, proof] In the proof of the second inclusion, immediately after the construction of Y, W and Z, the text asserts that a non-zero class beta in H^1(L, hat B) cannot simultaneously restrict to zero in H^1(L, hat<tau>) and H^1(L, hat<gamma>). This is not proved, and for a general Gamma_L-module the analogous statement is false. In the present situation the assertion is valid: Gamma_L is procyclic, and the dual modules Hom(<tau>, Q/Z(1)) and Hom(<gamma>, Q/Z(1)) are trivial Gamma_L-modules because the cyclotomic action on Q/Z(1) cancels the inverse cyclotomic action on the cyclic subgroups, so H^1(Gamma_L, M_1 plus M_2) is Hom(Gamma_L, M_1 plus M_2) and the restriction map is injective. The proof should include this argument; as written, this is a load-bearing gap in the proof of the main criterion.
minor comments (5)
  1. [Section 3.2, proof of Theorem 3.3] The notation Gamma_L tilde sigma is used without definition; it presumably denotes the absolute Galois group of the field L^{<tilde sigma>} fixed by the topological generator tilde sigma. Please clarify.
  2. [Theorem 5.1 statement] The phrase 'given the datum of a field k' is imprecise; the proof shows that the input is the finite data Gal(K/k), G(L), and the action of Gal(K/k) on both G(L) and mu_{N^2}. The statement should be amended to match the proof.
  3. [Section 7.2.2, Proposition 7.12] The GAP computation asserting X^2_ab(G,Z/2) = 0 is reported without code or computational details. Since the paper is about algorithms, please include the script or a group-theoretic verification sufficient for reproducibility.
  4. [Proposition 7.1] The displayed 'exact sequence of lower terms' for the Hochschild-Serre spectral sequence is not the standard five-term sequence, whose middle term is ker[H^2(G,Q/Z) to H^2(N,Q/Z)^Q]. Since the identification B_0(G) subset H^1(Q, hat N) depends on this sequence, please provide a derivation or a precise reference.
  5. [Throughout] There are several typos and infelicities: 'rationnally' in the Introduction, 'explicitely' in the proof of Theorem 6.1, repeated 'a fortiori', and inconsistent spelling of 'Grunwald'/'Grunwald'. These should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main formula, algorithm, and Grunwald consequences are derived from external tools and self-contained arguments, with only a repairable proof gap that is not a circular reduction.

full rationale

The paper's central claim (Theorem 3.3) is proved by a direct ramification and vanishing argument: the inclusion X2_bic,pcyc ⊂ Br_nr is shown from Wittenberg's evaluation criterion and a diagram chase, while the reverse inclusion uses [LA19, Props 3.1 and 4.1] only as external lemmas about homogeneous spaces with prescribed stabilizers and about the abelian-stabilizer case; neither lemma states the target formula. Theorem 4.4 is a translation of Theorem 3.3 into central-extension language, not an assumption of it. Proposition 5.3 proves Br0_nr(SLn,K/G) ≃ B0(G) by checking the criterion of Corollary 4.7 for classes lifted from B0(G), rather than by defining B0(G) as that Brauer group. The algorithms in Theorems 5.1 and 6.1 enumerate finite lists and then verify the independently proved criteria; no fitted parameter is renamed as a prediction. The only internal weakness is an unproved injectivity assertion in the proof of Theorem 3.3: 'by functoriality of the Hochschild-Serre spectral sequence it cannot simultaneously die in both Br1(W) ... and Br1(Z)'. This is a proof gap rather than a circular step, and under the surrounding hypotheses (procyclic Γ_L, central σ' in the pullback, triviality of the relevant cyclic dual module) the needed injectivity follows from a standard long-exact-sequence argument, so it does not make the theorem self-referential. Citations to [LA19], [HW20], and [CT14] are prior external results, not results of the present paper, and are not used to assume the target equality.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claim rests on several deep theorems from the literature, none of which are proved in the paper. There are no fitted parameters and no invented physical or mathematical entities. One computational assertion (the GAP check) is not independently reproducible from the text.

assumptions (7)
  • domain assumption Proposition 3.1 (Wittenberg's criterion): a Brauer class of order prime to char(k) is unramified iff it evaluates to zero on X(L((t))) for every perfect field L of dimension at most 1 containing k.
    Invoked as the starting point of the proof of Theorem 3.3; cited from [CTS21, Theorem 10.5.12].
  • domain assumption Proposition 2.1: Br(X) ≅ H^2(π_1^{ét}(X,x), Q/Z(1)), functorially.
    Used throughout to translate Brauer classes into cohomology of the étale fundamental group; cited from [LA19, Proposition 3.2].
  • domain assumption Theorem 2.2 (Pál-Schlank): for a special k-group H, X(L)/H(L) ≅ Sec~(π_1^{ét}(X_L,x), Γ_L).
    Used to identify local points with H^1(L,G) in Lemma 6.3 and in the algorithm; cited from [PS22, Theorem 9.6].
  • domain assumption [LA19, Proposition 4.1]: if Γ_K is procyclic and the stabilizer is abelian, then Br_nr(Y) ≃ Br_0(Y).
    Used in the proof of the second inclusion in Theorem 3.3.
  • domain assumption [HW20, Théorème B]: for supersolvable finite k-groups, the Brauer-Manin obstruction controls weak approximation.
    Used in Corollary 1.7 to pass from computability of the Brauer-Manin set to effectivity of the Grunwald problem.
  • domain assumption [KT25, Lemma 3.2]: for H = N ⋊ B with N abelian and B bicyclic, the Bogomolov multiplier B_0(H) is trivial.
    Used in the proof of Proposition 7.1 to characterize B_0(G) for semidirect products of abelian groups.
  • ad hoc to paper GAP computation: X^2_ab(G,Z/2) = 0 for the 4-cover G of PSL_3(F_4) appearing in §7.2.2.
    A numerical group computation that is stated without code or output in the proof of Proposition 7.12; it is needed to prove Br^0_nr(SL_n,R/G)=0.

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Pith. "Pith review of Unramified Brauer groups of homogeneous spaces with finite stabilisers and the Grunwald Problem." pith.science (2026). https://pith.science/paper/RHY7IBTY

@misc{pith2026250602600,
  author       = {Pith},
  title        = {Pith review of: Unramified Brauer groups of homogeneous spaces with finite stabilisers and the Grunwald Problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RHY7IBTY}},
  note         = {Machine review of arXiv:2506.02600}
}
abstract

We provide an algorithm for calculating the unramified Brauer group of a homogeneous space $X$ of a semi-simple simply connected group $H$ with finite geometric stabiliser over any field of characteristic 0. When $k$ is a number field, we use the obtained description of the unramified Brauer group in order to study the Brauer-Manin obstruction to weak approximation on $X$. In particular, we provide an algorithm to compute the Brauer-Manin obstruction on $X$, which guarantees effectivity of the Grunwald problem for supersolvable groups thanks to previous work of Harpaz and Wittenberg.

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.