Nonabelian Cohen–Lenstra moments

Abstract

In this article, we give a conjecture for the average number of unramified $G$-extensions of a quadratic field for any finite group $G$. The Cohen–Lenstra heuristics are the specialization of our conjecture to the case in which $G$ is abelian of odd order. We prove a theorem toward the function field analogue of our conjecture and give additional motivations for the conjecture, including the construction of a lifting invariant for the unramified $G$-extensions that takes the same number of values as the predicted average and an argument using the Malle–Bhargava principle. We note that, for even $\left|G\right|$, corrections for the roots of unity in $\mathbb{Q}$ are required, which cannot be seen when $G$ is abelian.