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arXiv · 1701.02838

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

Zev Klagsbrun
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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.

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Zev Klagsbrun. 2017-01-11. The average sizes of two-torsion subgroups in quotients of class groups of cubic fields. https://arxiv.org/abs/1701.02838

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