Triple Massey products over global fields

Abstract

Let K be a global field which contains a primitive p-th root of unity, where p is a prime number. M. J. Hopkins and K. G. Wickelgren showed that for p = 2, any triple Massey product over K with respect to F p , contains 0 whenever it is defined. We show that this is true for all primes p.

The Massey Vanishing Conjecture for fourfold Massey products modulo 2

“…In contrast with the situation in Algebraic Topology, Hopkins-Wickelgren [HW15] showed that, if F is a number field, all triple Massey products in H * (F, Z/2Z) vanish as soon as they are defined. This result was extended to all fields F by Mináč-Tân [MT15b,MT16]. It motivated the following conjecture, known as the Massey Vanishing Conjecture, which first appeared in [MT17b] under an assumption on roots of unity, then in general in [MT16].…”

“…-when F is a number field, n = 3 and p = 2, by Hopkins-Wickelgren [HW15]; -when F is arbitrary, n = 3 and p = 2, by Mináč-Tân [MT15b,MT16]; -when F is arbitrary, n = 3 and p is odd, by Matzri [Mat14], followed by Efrat-Matzri [EM17] and Mináč-Tân [MT16]; -when F is a number field, n = 4 and p = 2, by Guillot-Mináč-Topaz-Wittenberg [GMT18]; -when F is a number field and n and p are arbitrary, by Harpaz-Wittenberg [HW19]. There are also results for specific classes of fields; for example, rigid odd fields [MT15a].…”

Generalized Bockstein maps and Massey products

“…The Massey vanishing conjecture was inspired by work of Hopkins–Wickelgren [HoWi]: using splitting varieties, they had proven that -fold Massey products over number fields that are defined contain zero when [HoWi]. Massey vanishing over number fields was extended to successively general n for arbitrary primes p : to in [MiTa2], to in [GMT] and to all n in work of Harpaz–Wittenberg [HrWt]. In each of these cases, the method is specific to number fields because it uses a local-to-global principal to prove the existence of rational points on a splitting variety.…”

Massey products and elliptic curves

“…Thus, $$\langle \chi _1,\ldots ,\chi _t\rangle$$ is not empty if and only if a homomorphism $$\vartheta$$ as in () exists, and $$\langle \chi _1,\ldots ,\chi _t\rangle$$ contains 0 if and only if there is such a $$\vartheta$$ that has a lift $$\rho (\vartheta )$$ to $$U_{t+1}(\mathbb {Z}/\ell )$$. For more details, see [5, Theorem 2.4] and also [20, Lemma 4.2].…”