Massey products in Galois cohomology and the elementary type conjecture

Abstract

Let p be a prime. We prove that a positive solution to Efrat's Elementary Type Conjecture implies a positive solution to a strengthened version of Minač–Tân's Massey Vanishing Conjecture in the case of finitely generated maximal pro-p Galois groups whose pro-p cyclotomic character has torsion-free image. Consequently, the maximal pro-p Galois group of a field K containing a root of 1 of order p (and also −1 if p=2) satisfies the strong n-Massey vanishing property for every n>2 (which is equivalent to the cup-defining n-Massey product property for every n>2, as defined by Minač–Tân) in several relevant cases.