The mean number of 3-torsion elements in ray class groups of quadratic fields

Abstract

We determine the average number of 3-torsion elements in the ray class groups of a fixed (integral) conductor c of quadratic fields ordered by their absolute discriminant, generalizing Davenport and Heilbronn’s theorem on class groups. A consequence of this result is that a positive proportion of such ray class groups of quadratic fields have trivial 3-torsion subgroup whenever the conductor c is taken to be a squarefree integer having very few prime factors none of which are congruent to 1 mod 3. Additionally, we compute the second main term for the number of 3-torsion elements in ray class groups with a fixed conductor of quadratic fields ordered by their discriminant.