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

TL;DR

This paper improves the upper bounds on the size of the 2-torsion part of class groups of number fields, refining previous results by introducing a new constant that tightens the bound based on the discriminant.

Contribution

The paper provides a sharper upper bound on 2-torsion in class groups, advancing the understanding of their size relative to the discriminant of the number field.

Findings

Improved the bound on 2-torsion class groups by a constant factor.

Established a new explicit lower bound for the constant elta_K.

Refined previous asymptotic bounds for class group torsion sizes.

Abstract

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