Counting Galois ${\mathbb{U}}_4({\mathbb{F}}_p)$-extensions using Massey products

Abstract

We use Massey products and their relations to unipotent representations to parametrize and find an explicit formula for the number of Galois extensions of a given local field with the prescribed Galois group ${\mathbb{U}}_4({\mathbb{F}}_p)$ consisting of unipotent four by four matrices over ${\mathbb{F}}_p$. Further applications of this method involve the counting of certain Galois extensions with restricted ramifications, and counting the numbers of Galois ${\mathbb{U}}_4({\mathbb{F}}_p)$-extensions of some other fields. For each Demushkin pro-$p$-group, we find a very simple version of the condition when the $n$-fold Massey product of one-dimensional cohomological elements of $G$ with coefficients in ${\mathbb{F}}_p$, is defined. As an easy consequence, we determine those ${\mathbb{U}}_n({\mathbb{F}}_p)$ which occur as an epimorphic image of any given Demushkin group.

33 pages, minor corrections and some changes in exposition. This article will appear in the Journal of Number Theory

References in corpus (3)

• Triple Massey products in Galois cohomology
• Construction and classification of some Galois modules
• Triple Massey products and absolute Galois groups