On the Massey Vanishing Conjecture and Formal Hilbert 90

Abstract

Let $p$ be a prime number, let $G$ be a profinite group, let $\theta\colon G\to \mathbb{Z}_p^{\times}$ be a continuous character, and for all $n\geq 1$ write $\mathbb{Z}/p^n\mathbb{Z}(1)$ for the twist of $\mathbb{Z}/p^n\mathbb{Z}$ by the $G$-action. Suppose that $(G,\theta)$ satisfies a formal version of Hilbert's Theorem 90: for all open subgroups $H\subset G$ and every $n\geq 1$, the map $H^1(H,\mathbb{Z}/p^n\mathbb{Z}(1))\to H^1(H,\mathbb{Z}/p\mathbb{Z}(1))$ is surjective. We show that the Massey Vanishing Conjecture for triple Massey products and some degenerate fourfold Massey products holds for $G$. A key step in our proof is the construction of a Hilbert 90 module for $(G,\theta)$: a discrete $G$-module $M$ which plays the role of the Galois module $F_{\text{sep}}^\times$ for the absolute Galois group of a field $F$ of characteristic different from $p$.