The Massey vanishing conjecture for number fields. (English) Zbl 1521.11070

Harpaz, Yonatan; Wittenberg, Olivier

Duke Math. J. 172, No. 1, 1-41 (2023).

Higher Massey products are invariants of a differential graded ring that do not factor through its cohomology. As such, they detect differences that are usually not seen by classical cohomological methods. The original example studied by Massey was that of the complement in \(\mathbb{R}^3\) of the Borromean ring configuration [W. S. Massey, in: Conf. algebr. Topol., Univ. Ill. Chicago Circle 1968, 174–205 (1969; Zbl 0212.55904); W. S. Massey, J. Knot Theory Ramifications 7, No. 3, 393–414 (1998; Zbl 0911.57009)]. Here, cohomology with coefficients in \(\mathbb{Z}\) is not able to detect the difference between this space and the complement of three completely unlinked rings, but higher Massey products in the corresponding differential complex do capture the essence of this nontrivial configuration.
These constructions were later studied in a more arithmetic context, considering étale cohomology of open subschemes of spectra of Dedekind rings, and finally taken to the “simpler” context of Galois cohomology with coefficients in \(\mathbb{F}_p\), where \(p\) is a prime number. Here, a higher Massey product takes classes \(\alpha_1,\ldots,\alpha_n\in H^1(k,\mathbb{F}_p)\), with \(n\geq 3\), and gives back a subset of \(H^2(k,\mathbb{F}_p)\), which could eventually be empty. It is in this context that Mináč and Tân conjectured that higher Massey products over an arbitrary field \(k\) contain the trivial class \(0\in H^2(k,\mathbb{F}_p)\) as soon as they are non-empty (in which case one says that the product vanishes). This is now known as the Massey vanishing conjecture.
In this article, the authors prove that the Massey vanishing conjecture holds for number fields (cf. Theorem 1.3). This was known to be true when \(n=3\) (and for an arbitrary field \(k\), cf. for instance [J. Mináč and N. D. Tân, J. Lond. Math. Soc., II. Ser. 94, No. 3, 909–932 (2016; Zbl 1378.12002)]) and for \(n=4\) and \(p=2\) [P. Guillot et al., Compos. Math. 154, No. 9, 1921–1959 (2018; Zbl 1455.12005)]; otherwise the question was completely open. This is thus a great breakthrough that hinges on:

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a translation of this conjecture in terms of embedding problems, due to W. G. Dwyer [J. Pure Appl. Algebra 6, 177–190 (1975; Zbl 0338.20057)];

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a translation of embedding problems in terms of homogeneous spaces of \(\mathrm{SL}_n\), due to A. Pál and T. M. Schlank [Int. J. Number Theory 18, No. 7, 1535–1565 (2022; Zbl 1500.12005)];

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recent developments by the authors of this paper in the study of the Brauer-Manin obstruction to the local-global principle for homogeneous spaces with finite supersolvable stabilizers [Y. Harpaz and O. Wittenberg, J. Am. Math. Soc. 33, No. 3, 775–805 (2020; Zbl 1469.14053)]; and

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known results on the Massey vanishing conjecture over local fields [J. Mináč and N. D. Tân, J. Eur. Math. Soc. (JEMS) 19, No. 1, 255–284 (2017; Zbl 1372.12004)] and refinements of these by the authors of this paper (cf. in particular Proposition 5.3).

More precisely, given classes \(\alpha_1,\ldots,\alpha_n\in H^1(k,\mathbb{F}_p)\), the first two tools provide a homogeneous space \(V\) of \(\mathrm{SL}_n\) with finite geometric stabilizers that is a splitting variety for the vanishing of the corresponding Massey product (cf. Proposition 2.5). This means that \(V(L)\neq\emptyset\) if and only if the Massey product vanishes when restricted to \(H^2(L,\mathbb{F}_p)\). Since the stabilizers of this homogeneous space turn out to be supersolvable, this enables the authors to apply their machinery on the Brauer-Manin obstruction and thus concentrate on whether this homogeneous space has local points (which it does, as a nice consequence of local duality) and whether these local points define adelic points that are orthogonal to the unramified Brauer group \(\mathrm{Br}_{\mathrm{nr}}(V)\). This second part requires delicate computations of the group \(\mathrm{Br}_{\mathrm{nr}}(V)\), which rely on the explicit description of the stabilizers of this homogeneous space and on previous results by F. A. Bogomolov (for the “geometric” part of the Brauer group, [Math. USSR, Izv. 30, No. 3, 455–485 (1988; Zbl 0679.14025); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 51, No. 3, 485–516 (1987)]) and by Harari, Demarche and Lucchini Arteche (for the “algebraic” part, cf. [D. Harari, Bull. Soc. Math. Fr. 135, No. 4, 549–564 (2007; Zbl 1207.11048); C. Demarche, Math. Ann. 346, No. 4, 949–968 (2010; Zbl 1297.14022); G. Lucchini Arteche, J. Algebra 411, 129–181 (2014; Zbl 1368.14033)]).
The last two sections of the article (6 and 7) are devoted to further computations of Brauer groups of these splitting varieties in particular cases, as well as generalizations of the Massey vanishing conjecture to cohomology with other coefficients (i.e., different from \(\mathbb{F}_p\)) for which the same methods apply.

Reviewer: Giancarlo Lucchini Arteche (Santiago)

MSC:

11R34	Galois cohomology
12G05	Galois cohomology
14M17	Homogeneous spaces and generalizations
14G05	Rational points
55S30	Massey products

Keywords:

Galois cohomology; higher Massey products; Massey vanishing conjecture; number fields

Citations:

Zbl 0212.55904; Zbl 0911.57009; Zbl 1378.12002; Zbl 1455.12005; Zbl 0338.20057; Zbl 1500.12005; Zbl 1469.14053; Zbl 1372.12004; Zbl 0679.14025; Zbl 1207.11048; Zbl 1297.14022; Zbl 1368.14033

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