Triple Massey Products with weights 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.