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

Ján Mináč, Nguyêñ Duy Tân|arXiv (Cornell University)|Aug 12, 2014

Algebraic Geometry and Number Theory16 references3 citations

TL;DR

This paper develops a novel method to count Galois extensions with Galois group $υ_4(\mathbb{F}_p)$, the group of unipotent $4 \times 4$ matrices over $\mathbb{F}_p$, using Massey products and unipotent representations. It provides an explicit formula for the number of such extensions over local fields and extends the technique to other fields, while also establishing a simplified condition for defining $n$-fold Massey products in Demushkin pro-$p$ groups.

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.

Motivation & Objective

• To parametrize and count Galois extensions with Galois group $\mathbb{U}_4(\mathbb{F}_p)$ over local fields using Massey products.
• To extend the counting method to other fields, including algebraic number fields.
• To simplify the condition for defining $n$-fold Massey products in Demushkin pro-$p$ groups.
• To determine which $\mathbb{U}_n(\mathbb{F}_p)$ groups arise as epimorphic images of a given Demushkin group.
• To provide a new, stronger restriction on the structure of maximal pro-$p$ quotients of absolute Galois groups of fields containing a primitive $p$th root of unity.

Proposed method

• Use Massey products and their relation to unipotent representations to parametrize $\mathbb{U}_4(\mathbb{F}_p)$-extensions.
• Apply the theory of $n$-fold Massey products to Demushkin groups, deriving a simplified condition for their definition.
• Leverage the structure of free products of Demushkin groups to compute the number of surjective homomorphisms to $\mathbb{U}_4(\mathbb{F}_p)$.
• Use bilinear forms $(\cdot,\cdot)_1$ and $(\cdot,\cdot)_2$ on a decomposition $V \oplus W$ to model cohomological constraints.
• Count the number of linearly independent triples $(x,y,z)$ satisfying orthogonality conditions with respect to the two forms.
• Combine these counts with group-theoretic arguments and Möbius inversion to derive the final formula for the number of extensions.

Experimental results

Research questions

• RQ1 What is the number of Galois extensions of a local field with Galois group $\mathbb{U}_4(\mathbb{F}_p)$?
• RQ2 How can Massey products be used to parametrize and count $\mathbb{U}_4(\mathbb{F}_p)$-extensions over fields with a primitive $p$th root of unity?
• RQ3 What is the minimal condition for the definition of $n$-fold Massey products in Demushkin pro-$p$ groups?
• RQ4 Which $\mathbb{U}_n(\mathbb{F}_p)$ groups occur as epimorphic images of a given Demushkin group?
• RQ5 Can the simplified Massey product condition lead to new structural constraints on maximal pro-$p$ quotients of absolute Galois groups?

Key findings

• The number of $\mathbb{U}_4(\mathbb{F}_p)$-extensions over a local field $K$ is given by a closed-form formula involving $p^d$, $p^e$, and $p^{d+e}$, where $d$ and $e$ are the ranks of the Demushkin groups in the free product decomposition of $G_K(p)$.
• The formula for the number of such extensions is explicitly computed as a sum of terms involving $p$-powers and binomial-like coefficients, reflecting the structure of cohomological orthogonality.
• The paper establishes that the $n$-fold Massey product is defined in a Demushkin group if the $q$-invariant exceeds 2, significantly simplifying the classical definition.
• For $n=4$, the number of surjective homomorphisms from a free product of two Demushkin groups to $\mathbb{U}_4(\mathbb{F}_p)$ is computed by case analysis on the images of the factors.
• The method allows counting $\mathbb{U}_4(\mathbb{F}_p)$-extensions over other fields by reducing to cohomological data and bilinear forms.
• The results confirm that the vanishing of higher Massey products imposes strong structural constraints on Galois groups, supporting the kernel unipotent conjecture.