Nonabelian Cohen-Lenstra Moments

Abstract: In this paper 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 that $G$ is abelian of odd order. We prove a theorem towards the function field analog 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 $|G|$, corrections for the roots of unity in $\mathbb{Q}$ are required, which can not be seen when $G$ is abelian.