The imaginary case of the nonabelian Cohen--Lenstra heuristics

TL;DR

This paper investigates the distribution of certain Galois groups in imaginary extensions, computes moments in the function field case, and proposes a conjecture based on random group models to understand their probability distribution.

Contribution

It introduces an imaginary analog of nonabelian Cohen--Lenstra heuristics, computes moments via Hurwitz stacks, and develops random group models to predict distributions.

Findings

Computed moments of Galois group distributions in function fields.

Proposed a random group model to simulate Galois group behavior.

Conjectured the distribution of Galois groups in imaginary extensions.

Abstract

For a finite group $\Gamma$, we study the distribution of the Galois group $G_{\emptyset}^{\#}(K)$ of the maximal unramified extension of $K$ that is split completely at $\infty$ and has degree prime to $|\Gamma|$ and $\textit{Char}(K)$, as $K$ varies over imaginary $\Gamma$-extensions of $\mathbb{Q}$ or $\mathbb{F}_q(t)$. In the function field case, we compute the moments of the distribution of $G_{\emptyset}^{\#}(K)$ by counting points on Hurwitz stacks. In order to understand the probability of the distribution, we prove that $G_{\emptyset}^{\#}(K)$ admits presentations of a specific form, then use this presentation to build random groups to simulate the behavior of $G_{\emptyset}^{\#}(K)$, and make the conjecture to predict the distribution using the probability measures of these random groups. Our results provide the imaginary analog of the work of Wood, Zureick-Brown, and the first author on the nonabelian Cohen--Lenstra heuristics.