3-fold Massey products in Galois cohomology -- The non-prime case

TL;DR

This paper extends the understanding of 3-fold Massey products in Galois cohomology to non-prime cases, showing they are non-essential under certain conditions, using unitriangular representations.

Contribution

It generalizes previous results by proving non-essentiality of 3-fold Massey products for composite m, not just prime, via new methods involving unitriangular representations.

Findings

3-fold Massey products are non-essential for composite m

The proof applies to fields containing roots of unity of order m

Unitriangular representations are key to the argument

Abstract

For $m\geq2$, let $F$ be a field of characteristic prime to $m$ and containing the roots of unity of order $m$, and let $G_F$ be its absolute Galois group. We show that the 3-fold Massey products $\langle\chi_1,\chi_2,\chi_3\rangle$, with $\chi_1,\chi_2,\chi_3\in H^1(G_F,\mathbb{Z}/m)$ and $\chi_1,\chi_3$ $\mathbb{Z}/m$-linearly independent, are non-essential. This was earlier proved for $m$ prime. Our proof is based on the study of unitriangular representations of $G_F$.