The average size of $3$-torsion in class groups of $2$-extensions

TL;DR

This paper calculates the average size of 3-torsion in class groups of certain 2-group extensions of number fields, confirming predictions of Cohen–Lenstra–Martinet heuristics and extending previous results.

Contribution

It provides the first proven cases of finite average 3-torsion in class groups for a broad class of 2-group extensions, and introduces a new method applicable to other permutation groups.

Findings

Average 3-torsion matches Cohen–Lenstra–Martinet predictions

Proves new cases of heuristics for G-extensions with G as a 2-group

Method extends to many other permutation groups

Abstract

We determine the average size of the 3-torsion in class groups of $G$-extensions of a number field when $G$ is any transitive $2$-group containing a transposition, for example $D_4$. It follows from the Cohen--Lenstra--Martinet heuristics that the average size of the $p$-torsion in class groups of $G$-extensions of a number field is conjecturally finite for any $G$ and most $p$ (including $p\nmid|G|$). Previously this conjecture had only been proven in the cases of $G=S_2$ with $p=3$ and $G=S_3$ with $p=2$. We also show that the average $3$-torsion in a certain relative class group for these $G$-extensions is as predicted by Cohen and Martinet, proving new cases of the Cohen--Lenstra--Martinet heuristics. Our new method also works for many other permutation groups $G$ that are not $2$-groups.