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 $\mathrm{Cl}(K)/⟨S⟩[2]$ as $K$ varies among cubic fields with a fixed signature, where $⟨S⟩$ 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.

13 pages. Updated to include results about the average size of K_{2n}(O_K)[2]

References in corpus (2)

• On the mean number of 2-torsion elements in the class groups, narrow class groups, and ideal groups of cubic orders and fields
• Davenport-Heilbronn Theorems for Quotients of Class Groups

Cited by in corpus (1)

• Statistics of $K$-groups modulo $p$ for the ring of integers of a varying quadratic number field