[Paper Review] The Kernel Unipotent Conjecture and the vanishing of Massey products for odd rigid fields

Ján Mináč, Nguyêñ Duy Tân+1 more|arXiv (Cornell University)|Dec 10, 2013

Algebraic Geometry and Number Theory24 references9 citations

TL;DR

This paper proves the Kernel Unipotent Conjecture and the Vanishing $n$-Massey Product Conjecture for odd rigid fields and Demushkin groups of rank 2, establishing these properties for a significant new class of fields beyond those with free pro-$p$ Galois groups. It constructs pro-$p$ groups that fail the kernel $n$-unipotent property, demonstrating the sharpness of these conjectures and providing new filtration results and automatic realization theorems for Galois groups.

ABSTRACT

A major difficult problem in Galois theory is the characterization of profinite groups which are realizable as absolute Galois groups of fields. Recently the Kernel $n$-Unipotent Conjecture and the Vanishing $n$-Massey Conjecture for $n\geq 3$ were formulated. These conjectures evolved in the last forty years as a byproduct of the application of topological methods to Galois cohomology. We show that both of these conjectures are true for odd rigid fields. This is the first case of a significant family of fields where both of the conjectures are verified besides fields whose Galois groups of $p$-maximal extensions are free pro-$p$-groups. We also prove the Kernel Unipotent Conjecture for Demushkin groups of rank 2, and establish a number of further related results.

Motivation & Objective

• To verify the Kernel $n$-Unipotent Conjecture and the Vanishing $n$-Massey Product Conjecture for odd rigid fields.
• To extend the known families of fields satisfying these conjectures beyond those with free pro-$p$ Galois groups of $p$-maximal extensions.
• To construct examples of pro-$p$ groups that do not satisfy the kernel $n$-unipotent property, showing the conjectures are not universally valid for all pro-$p$ groups.
• To compare Zassenhaus filtrations with the descending $p$-central series and establish new automatic Galois realization theorems.

Proposed method

• Use of the $p$-Zassenhaus filtration and the descending $p$-central series to analyze the structure of pro-$p$ groups.
• Construction of a free pro-$p$ group $S$ on $N$ generators and quotienting by a closed normal subgroup $R$ such that $RS_{(3)} = R_0S_{(3)}$, where $R_0$ is generated by commutator relations $[x_1,x_2][x_i,x_j]^{-1}$.
• Definition of $G_{\mathcal{L}} = \bigcap_{H \in \mathcal{L}} \bigcap_{\rho} \ker(\rho)$, where $\rho$ runs over all homomorphisms $G \to H$, to test the kernel $n$-unipotent property.
• Application of linear independence of commutators modulo $S_{(3)}$ to show $[\bar{x}_1, \bar{x}_2] \not\in G_{(3)}$, establishing non-vanishing in the filtration.
• Use of group homomorphisms to finite groups $H$ with $|H| < N$ to show $[\bar{x}_1, \bar{x}_2] \in G_{\mathcal{L}}$, demonstrating that $G_{\mathcal{L}}$ escapes the $n$-th term of the $p$-central series.
• Leveraging known results from [Vo], [EM2], and [Ef1] to show that constructed groups are not realizable as maximal pro-$p$ Galois groups of fields containing a $p$-th root of unity.

Experimental results

Research questions

• RQ1 Do the Kernel $n$-Unipotent Conjecture and the Vanishing $n$-Massey Product Conjecture hold for odd rigid fields for all $n \geq 3$ and primes $p$?
• RQ2 Can the kernel $n$-unipotent property be violated by pro-$p$ groups that are not realizable as maximal pro-$p$ Galois groups of fields with a $p$-th root of unity?
• RQ3 To what extent do Zassenhaus filtrations and the descending $p$-central series coincide in pro-$p$ groups arising from Galois theory?
• RQ4 Are the structural properties of absolute Galois groups, such as those in [EM1, MS2, EM2], valid for all torsion-free pro-$p$ groups or only for specific classes?
• RQ5 Can new automatic Galois realization theorems be established based on the kernel $n$-unipotent and vanishing Massey product properties?

Key findings

• The Kernel $n$-Unipotent Conjecture and the Vanishing $n$-Massey Product Conjecture are both verified for odd rigid fields, marking the first such verification for a significant family of fields beyond free pro-$p$ groups.
• For each $n \geq 3$, examples of pro-$p$ groups are constructed that do not satisfy the kernel $n$-unipotent property, and these groups are not realizable as maximal pro-$p$ Galois groups of fields containing a $p$-th root of unity.
• The group $G = S/R$ with $R = R_0 S_{(3)}$ is torsion-free and satisfies $[\bar{x}_1, \bar{x}_2] \not\in G_{(3)}$, showing that the $p$-central series does not capture all normal subgroups in such groups.
• When $N > |H|$ for all $H \in \mathcal{L}$, the element $[\bar{x}_1, \bar{x}_2]$ lies in $G_{\mathcal{L}}$ but not in $G_{(n)}$ for $n \geq 3$, proving $G_{\mathcal{L}} \not\subseteq G_{(n)}$ and violating the kernel $n$-unipotent condition.
• The results show that properties of absolute Galois groups derived in [EM1, MS2, EM2] do not extend to general torsion-free pro-$p$ groups, as demonstrated by counterexamples in the appendix.
• The paper establishes that the kernel $n$-unipotent property fails for $G$ when $N > p^{(n-1)n/2}$, providing a quantitative threshold for failure.