LAUSR

Abstract 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, and is directly analogous to the work of Guillot, Minac 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 U ˜ 5 ( F p ) , which has U 5 ( F p ) as a quotient. Here U n ( F p ) is the group of unipotent n × n -matrices with entries in the field F p with p elements; it is classical that U n + 1 ( F p ) is intimately related to n-fold Massey products. Although U ˜ 5 ( F p ) is much larger than U 5 ( F p ) , its definition is very natural, and for our purposes, it is easier to study.