Extensions of unipotent groups, Massey products and Galois theory

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, Mináč 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(Fp), which has U5(Fp) as a quotient. Here Un(Fp) is the group of unipotent n×n-matrices with entries in the field Fp with p elements; it is classical that Un+1(Fp) is intimately related to n-fold Massey products. Although U˜5(Fp) is much larger than U5(Fp), its definition is very natural, and for our purposes, it is easier to study.