Extensions of unipotent groups, Massey products and Galois cohomology

We study the vanishing of four-fold Massey products in mod p Galois cohomology. First, we describe a sufficient condition, which is simply expressed by the vanishing of some cup-products, in direct analogy with the work of Guillot, Min'a\v{c} and Topaz for p=2. For local fields with enough roots of unity, we prove that this sufficient condition is also necessary, and we ask whether this is a general fact. We provide a simple splitting variety, that is, a variety which has a rational point if and only if our sufficient condition is satisfied. It has rational points over local fields, and so, if it satisfies a local-global principle, then the Massey Vanishing conjecture holds for number fields with enough roots of unity. At the heart of the paper is the construction of a finite group $\tilde U_5($\mathbb{F}$_p)$,whichhas$U_5($\mathbb{F}$_p)$asaquotient\mathrm{.}Here$U_n($\mathbb{F}$_p)$isthegroupofunipotent$n\times n$-matrices with entries in the field ${\mathbb{F}}_p$ with p elements; it is classical that $U_{n+1}({\mathbb{F}}_p)$ is intimately related to n-fold Massey products. Although ${\tilde{U}}_5({\mathbb{F}}_p)$ is much larger than $U_5({\mathbb{F}}_p)$, its definition is very natural, and for our purposes, it is easier to study.