# Nonabelian Cohen-Lenstra Moments

## TL;DR

This paper proposes a conjecture for the average number of unramified G-extensions of quadratic fields, generalizing Cohen-Lenstra heuristics to nonabelian groups, supported by partial results and motivations from function fields and invariants.

## Contribution

It introduces a new conjecture extending Cohen-Lenstra heuristics to nonabelian groups and provides partial proofs and motivations for this conjecture.

## Key findings

- Proved a theorem for the function field analog of the conjecture.
- Constructed a lifting invariant matching the predicted averages.
- Identified the need for corrections involving roots of unity for even |G|.

## 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.