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arXiv · 1702.04644

Nonabelian Cohen-Lenstra Moments

Melanie Matchett Wood·Philip Matchett Wood
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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.

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Melanie Matchett Wood, Philip Matchett Wood. 2017-02-15. Nonabelian Cohen-Lenstra Moments. https://doi.org/10.1215/00127094-2018-0037

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