Moments of unramified 2-group extensions of quadratic fields

TL;DR

This paper studies the distribution and moments of counts of unramified 2-group extensions over quadratic fields, revealing different behaviors depending on whether the Galois group is abelian or non-abelian.

Contribution

It introduces a function with finite moments for counting unramified extensions, derives explicit formulas, and explores distributional properties and correlations for various 2-groups.

Findings

Distribution is a point mass for non-abelian groups.

Distribution follows Cohen-Lenstra for abelian groups.

Provides explicit formulas and conjectures for moments and correlations.

Abstract

Let $f\left(K\right)$ be the number of unramified extensions $L/K$ of a quadratic number field $K$ with $\mathrm{Gal}\left(L/K\right)=H$ and $\mathrm{Gal}\left(L/\mathbb{Q}\right)=G$ where $G$ is a central extension of $\mathbb{F}_{2}^{n}$ by $\mathbb{F}_{2}$. We find a function $g\left(K\right)$ such that $f/g$ has finite moments and a distribution on its values. We show this distribution is a point mass when $H$ is non-abelian and the Cohen-Lenstra distribution when $H$ is abelian, despite the fact that the set of values of $f/g$ do not form a discrete set. We prove an explicit formula for $f$ as well as a refined counting function with local conditions. We also determine correlations of such counting functions for different groups $G$. Lastly we formulate a conjecture about moments and correlations for any pair of 2-groups $\left(G,H\right)$.