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

In 1971, Davenport and Heilbronn determined the average number of 3-torsion elements in class groups of quadratic fields. This was the prototype result on which the Cohen--Lenstra--Martinet heuristics were based, but very few other class group averages have been determined since. In joint work with Jiuya Wang and Melanie Matchett Wood, we find infinitely many more. For example, we show the average size of the 3-torsion subgroup of D_4 quartic fields is about 1.42, and in general, we determine the average size of the 3-torsion subgroup of 2-extensions of any degree over a number field.