On the average size of $3$-torsion in class groups of $C_2 \wr H$-extensions

TL;DR

This paper proves that the average size of 3-torsion in class groups of certain G-extensions, specifically those involving wreath products with nilpotent groups, is finite, extending previous results to more cases.

Contribution

It extends prior work by proving finiteness of the average 3-torsion size for class groups of $C_2 times H$-extensions for all nilpotent groups $H$, including cases not covered before.

Findings

Finiteness of average 3-torsion in class groups for $C_2 times H$-extensions with nilpotent $H$

Extension of previous results to include groups like $C_5$

Broader applicability of Cohen-Lenstra-Martinet heuristics

Abstract

The Cohen-Lenstra-Martinet heuristics lead one to conjecture that the average size of the $p$-torsion in class groups of $G$-extensions of a number field is finite. In a 2021 paper, Lemke Oliver, Wang, and Wood proved this conjecture in the case of $p = 3$ for permutation groups $G$ of the form $C_2 \wr H$ for a broad family of permutation groups $H$, including most nilpotent groups. However, their theorem does not apply for some nilpotent groups of interest, such as $H = C_5$. We extend their results to prove that the average size of $3$-torsion in class groups of $C_2 \wr H$-extensions is finite for any nilpotent group $H$.