Chasing maximal pro-p Galois groups via 1-cyclotomicity

TL;DR

This paper investigates conditions under which certain pro-$p$ groups can be realized as maximal pro-$p$ Galois groups of fields, showing specific obstructions and proposing conjectures related to 1-cyclotomicity.

Contribution

It demonstrates that certain amalgamated free pro-$p$ products of Demushkin groups cannot be maximal pro-$p$ Galois groups and explores cohomological obstructions and properties related to 1-cyclotomicity.

Findings

Amalgamated free pro-$p$ products of Demushkin groups are not 1-cyclotomic Galois groups.

Quadraticity of cohomology and Massey product vanishing fail for these groups.

Minač-Tân pro-$p$ group cannot be 1-cyclotomic.

Abstract

Let $p$ be a prime. We prove that certain amalgamated free pro-$p$ products of Demushkin groups with pro-$p$-cyclic amalgam cannot give rise to a 1-cyclotomic oriented pro-$p$ group, and thus do not occur as maximal pro-$p$ Galois groups of fields containing a root of 1 of order $p$. We show that other cohomological obstructions which are used to detect pro-$p$ groups that are not maximal pro-$p$ Galois groups - the quadraticity of $\mathbb{Z}/p\mathbb{Z}$-cohomology and the vanishing of Massey products - fail with the above pro-$p$ groups. Finally, we prove that the Mina\v{c}-T\^an pro-$p$ group cannot give rise to a 1-cyclotomic oriented pro-$p$ group, and we conjecture that every 1-cyclotomic oriented pro-$p$ group satisfy the strong $n$-Massey vanishing property for $n>2$.