Triple Massey Products with weights in Galois cohomology

TL;DR

This paper investigates the properties of triple Massey products in Galois cohomology, proving they contain zero under certain conditions and analyzing their structure using cohomological tools and algebraic theorems.

Contribution

It establishes new results on the vanishing of specific triple Massey products in Galois cohomology for arbitrary primes and explores their structure via symbols and algebraic methods.

Findings

Any defined 3MP of weight (n,1,m) with symbol entries contains zero.

For p=2, any defined 3MP of weight (1,k,1) with a symbol middle entry contains zero.

Uses algebraic tools to analyze the kernel of multiplication by a symbol in Galois cohomology.

Abstract

Fix an arbitrary prime $p$. Let $F$ be a field containing a primitive $p$-th root of unity, with absolute Galois group $G_F$, and let $H^n$ denote its mod $p$ cohomology group $H^n(G_F,\mathbb{Z}/p\mathbb{Z})$. The triple Massey product of weight $(n,k,m)\in \mathbb{N}^3$ is a partially defined, multi-valued function $\langle \cdot,\cdot,\cdot \rangle: H^n\times H^k\times H^m\rightarrow H^{n+k+m-1}.$ %(in the mod-$p$ Galois cohomology) In this work we prove that for an arbitrary prime $p$, any defined $3MP$ of weight $(n,1,m)$, where the first and third entries are assumed to be symbols, contains zero; and that for $p=2$ any defined $3MP$ of the weight $(1,k,1)$, where the middle entry is a symbol, contains zero. Finally, we use the description of the kernel of multiplication by a symbol to study general 3MP where the middle slot is a symbol. The main tools we will be using is Lemma 4.1 concerning the the annihilator of cup product with an $H^1$ element, and Theorem 5.2, generalizing a Theorem of Tignol on quaternion algebras with trivial corestriction along a separable quadratic extension.