Title: The average sizes of two-torsion subgroups in quotients of class groups of cubic fields,
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.