The average sizes of two-torsion subgroups in quotients of class groups of cubic fields

TL;DR

This paper generalizes Bhargava's results to compute the average sizes of certain 2-torsion subgroups in class groups and related structures of cubic fields, providing explicit formulas for these averages.

Contribution

It introduces a formula for the average size of quotients of class groups by subgroup generated by primes over a fixed set, extending previous work on class group 2-torsion.

Findings

Derived formulas for average sizes of class group quotients

Calculated average sizes of K_{2n} and Selmer groups for cubic fields

Extended Bhargava's results to more general subgroup quotients

Abstract

We prove a generalization of a result of Bhargava regarding the average size $\mathrm{Cl}(K)[2]$ as $K$ varies among cubic fields. For a fixed set of rational primes $S$, we obtain a formula for the average size of $\mathrm{Cl}(K)/\langle S \rangle[2]$ as $K$ varies among cubic fields with a fixed signature, where $\langle S \rangle$ is the subgroup of $\mathrm{Cl}(K)$ generated by the classes of primes of $K$ above primes in $S$. As a consequence, we are able to calculate the average sizes of $K_{2n}(\mathcal{O}_K)[2]$ for $n > 0$ and for the relaxed Selmer group $\mathrm{Sel}_2^S(K)$ as $K$ varies in these same families.