On the $2$-torsion in class groups of number fields

Abstract

In $2020$, Bhargava, Shankar, Taniguchi, Thorne, Tsimerman, and Zhao proved that for a finite extension $K/\mathbb{Q}$ of degree $n\ge 5$, the size of the $2$-torsion class group is bounded by $\#h_2(K)=O_{n,\epsilon }(D_K^{\frac{1}{2}-\frac{1}{2n}+\epsilon })$, where $D_K$ is the absolute discriminant of $K$. In the present paper, we improve their bound by proving that $\#h_2(K)=O_{n,\epsilon }(D_K^{\frac{1}{2}-\frac{1}{2n}-{\delta }_K+\epsilon })$, for a constant ${\delta }_K\ge \frac{1}{28n}-\frac{3}{28n(n-1)}$.