I am looking for nontrivial bounds on the sizes of the $2$-torsion subgroups of the class groups of cubic and higher degree number fields $K$. The entire class group is bounded in size by $O(|\text{disc}(K)|^{\frac 12+\epsilon})$.


The suggested answer has nothing to do with this.


Should I look for bounds of Brumer and Kramer on the sizes of $2$-Selmer groups and ranks of elliptic curves, or bounds of Helfgott and Venkatesh on the number of integral points on elliptic curves, or something else entirely?



Bounds for the $2$-torsion subgroup of the class group of a number field
Recent work of [Bhargava, Shankar, Taniguchi, Thorne, Tsimerman, and Zhao](https://arxiv.org/pdf/1701.02458.pdf) shows that if $K$ is a number field of degree $n$ then the size of the $2$-torsion subgroup of the class group of $K$ is
$$ 
O(|\text{disc}(K)|^{1/2-\delta\_n +\epsilon}),
$$
where $\delta\_n=1/(2n)$ is permissible. For cubic and quartic fields, stronger bounds are obtained. The case of quadratic fields is trivial by genus theory.


