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

TL;DR

This paper generalizes Davenport and Heilbronn's theorem to determine the average number of 3-torsion elements in ray class groups of quadratic fields with fixed conductor, revealing that many such groups have trivial 3-torsion under certain conditions.

Contribution

It extends the understanding of 3-torsion in ray class groups of quadratic fields and computes the second main term for their distribution with fixed conductor.

Findings

Positive proportion of ray class groups have trivial 3-torsion when conductor is squarefree with few prime factors.

Derived the average number of 3-torsion elements in ray class groups for quadratic fields.

Computed the second main term for the count of 3-torsion elements with fixed conductor.

Abstract

We determine the average number of $3$-torsion elements in the ray class groups of fixed (integral) conductor $c$ of quadratic fields ordered by 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 $1 \bmod 3$. Additionally, we compute the second main term for the number of $3$-torsion elements in ray class groups with fixed conductor of quadratic fields with bounded discriminant.