Massey products in Galois cohomology and Pythagorean fields (Q6600718)

Let \(G\) be a pro-\(p\), for a given prime number \(p\), and let \(\mathbb{F}\sb p\) be the field with \(p\) elements, viewed as a trivial \(G\)-module. For \(n \ge 2\), the \(n\)-fold Massey product with respect to \(\mathbb{F}\sb p\) is a multi-valued map from the first \(\mathbb{F}\sb p\)-cohomology group \(H \sp 1 (G, \mathbb{F}\sb p)\) to the second \(\mathbb{F}\sb p\)-cohomology group \(H\sp 2(G, \mathbb{F}\sb p)\), which associates a sequence of length \(n\) of elements \(\alpha \sb 1, \dots , \alpha \sb n\) (not necessarily pairwise distinct) of \(H \sp 1(G, \mathbb{F}\sb p)\) to a (possibly empty) subset of \(H \sp 2(G, \mathbb{F}\sb p)\), denoted by \(\langle \alpha \sb 1, \dots , \alpha \sb n\rangle \). The \(n\)-fold Massey product is said to be defined at \(\alpha \sb 1, \dots , \alpha \sb n\) if the set \(\langle \alpha \sb 1, \dots , \alpha \sb n\rangle \) is nonempty, and it is said to vanish if \(0 \in \langle \alpha \sb 1, \dots , \alpha \sb n\rangle \). We say that \(G\) has the \(n\)-Massey vanishing property (abbr., \(n\)-MVP) with respect to \(\mathbb{F}\sb p\) if the \(n\)-fold Massey product vanishes whenever it is defined.\N\NFor any field \(\mathbb{K}\), let \(G \sb {\mathbb{K}}(p)\) be the maximal pro-\(p\) Galois group of \(\mathbb{K}\), that is, the maximal pro-\(p\) quotient of the absolute Galois group of \(\mathbb{K}\). The Mináč-Tân conjecture (stated in [\textit{J. Mináč} and \textit{Nguyen Duy Tân}, Adv. Math. 273, 242--270 (2015; Zbl 1334.12005)]) predicts that if \(\mathbb{K}\) contains a primitive \(p\)-th root of unity, then the pro-\(p\) group \(G\sb {\mathbb{K}}(p)\) possesses the \(n\)-MVP, for every \(n > 2\). The fact that \(G\sb {\mathbb{K}}(p)\) has the \(3\)-MVP has been established by \textit{E. Matzri} [``Triple Massey products in Galois cohomology'', Preprint, \url{arXiv:1411.4146}], see also the articles [\textit{I. Efrat} and \textit{E. Matzri}, J. Eur. Math. Soc. (JEMS) 19, No. 12, 3629--3640 (2017; Zbl 1425.12004); \textit{J. Mináč} and \textit{N. D. Tân}, J. Lond. Math. Soc., II. Ser. 94, No. 3, 909--932 (2016; Zbl 1378.12002)]. The active research concerning the Mináč-Tân conjecture, carried out by a number of authors since 2014, is incorporated in the search for cohomological obstructions which exclude series of pro-\(p\) groups from the class of maximal pro-\(p\) Galois groups of fields.\N\NThe paper under review considers a strong version of the MVP (introduced in [\textit{A. Pál} and \textit{E. Szabó}, ``The strong Massey vanishing conjecture for fields with virtual cohomological dimension at most 1'', Preprint, \url{arXiv:1811.06192}]). It is known that if the \(n\)-fold Massey product \(\langle \alpha \sb 1, \dots , \alpha \sb n\rangle \) is defined, for a pro-\(p\) group \(G\) and a sequence \(\alpha \sb 1, \dots , \alpha \sb n \in H \sp 1(G, \mathbb{F}\sb p)\), then one necessarily has \(\alpha \sb 1 \smallsmile \alpha \sb 2 = \dots = \alpha \sb {n-1} \smallsmile \alpha \sb n = 0\), where \(\smallsmile \) denotes the cup-product (see, e.g., Remark~2.2 in: [\textit{C. Quadrelli}, J. Number Theory 258, 40--65 (2024; Zbl 1537.12006)]). We say that \(G\) has the strong \(n\)-MVP if every sequence \(\alpha \sb 1, \dots \alpha \sb n \in H \sp 1(G, \mathbb{F}\sb p)\) satisfying the stated cup-product condition yields an \(n\)-fold Massey product which vanishes.\N\NThe first main result of the paper under review, stated as Theorem~1.2, asserts that if \(\mathbb{K}\) is a Pythagorean field (i.e. any sum of two squares of elements of \(\mathbb{K}\) is a square in \(\mathbb{K}\)) with finitely many square classes, and if \(L/\mathbb{K}\) is a finite extension, then the pro-\(2\) group \(G\sb L(2)\) has the strong \(n\)-MVP with respect to \(\mathbb{F}\sb 2\), for every \(n > 2\). The availability of the strong \(n\)-MVP of \(G \sb {\mathbb{K}}(2)\), for every \(n > 2\), has recently been obtained Pál and Quick in the case where \(\mathbb{K}\) is a field of virtual cohomological dimension at most \(1\)(see [\textit{A. Pal} and \textit{G. Quick}, ``Real projective groups are formal'', Preprint, \url{arXiv:2206.14645}]).\N\NThe proof of Theorem 1.2 relies on the description of the maximal pro-\(2\) Galois group of a Pythagorean fields with finitely many square classes (provided in: [\textit{J. Mináč}, C. R. Math. Acad. Sci., Soc. R. Can. 8, 103--108, Correction 261 (1986; Zbl 0595.12011); \textit{J. Mináč} et al., Adv. Math. 380, Article ID 107569, 50 p. (2021; Zbl 1483.12003)]).\N\NThe second main result of the reviewed paper, stated as Theorem~1.3, shows that if \(G\) be a pro-\(2\) group of (Efrat's) elementary type [\textit{I. Efrat}, Manuscr. Math. 95, No. 2, 237--249 (1998; Zbl 0902.12003)], then every open subgroup \(H\) of \(G\) has the strong \(n\)-MVP with respect to \(\mathbb{F}\sb 2\), for every \(n > 2\). Theorem 1.3 has already been proved by the author for every prime \(p\), but with restrictions for the case \(p = 2\), which exclude maximal pro-\(2\) groups of Pythagorean fields (see Theorem~1.2 of [Quadrelli, loc. cit.]). In fact, the second main result of the reviewed paper (where such restrictions are dropped) is complementary to the cited theorem.