Geodesic
The average size of 3-torsion in class groups of 2-extensions
Forum of Mathematics, Pi, Tome 13 (2025) no. 1, p. e19

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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 conjectured, 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. 
Oliver, Robert Lemke; Wang, Jiuya; Wood, Melanie Matchett. The average size of 3-torsion in class groups of 2-extensions. Forum of Mathematics, Pi, Tome 13 (2025) no. 1, p. e19. doi: 10.1017/S2050508625000009
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