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.

arXiv commentmain article by Melanie Matchett Wood, appendix by Melanie Matchett Wood and Philip Matchett Wood, to appear in Duke Mathematical Journal

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Cited by in corpus (12)

• The Distribution of p-Torsion in Degree p Cyclic Fields
• A predicted distribution for Galois groups of maximal unramified extensions
• Statistics of the First Galois Cohomology Group: A Refinement of Malle's Conjecture
• Higher genus theory
• Number fields with prescribed norms
• The $16$-rank of $\mathbb{Q}(\sqrt{-p})$
• 4-ranks and the general model for statistics of ray class groups of imaginary quadratic number fields
• The distribution of $H_8$-extensions of quadratic fields
• Cohen-Lenstra heuristics and bilinear pairings in the presence of roots of unity
• Monogenicity and 2-torsion in the class group of number fields of odd degree
• Presentations of Galois groups of maximal extensions with restricted ramification
• On unramified solvable extensions of small number fields