The average size of 3-torsion in class groups of 2-extensions
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=S2 with p=3 and G=S3 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.