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

Zev Klagsbrun

Published 2017-01-11Version 1

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 prime in $S$. We additionally obtain average sizes for the relaxed Selmer group $\mathrm{Sel}_2^S(K)$ and for $\mathcal{O}_{K,S}^\times/(\mathcal{O}_{K,S}^\times)^2$ as $K$ varies in the same families.