Joint IAS/Princeton University Number Theory Seminar

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, $D4$.  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 $p$.  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.

This is joint work with Jiuya Wang and Melanie Matchett Wood.