A new pointwise bound for $3$-torsion of class groups

TL;DR

This paper improves the upper bound on the size of the 3-torsion part of class groups of quadratic fields, reducing the exponent from 1/3 to approximately 0.3193, and also provides new average bounds for 3-torsion.

Contribution

It introduces a sharper pointwise bound for 3-torsion in class groups and combines methods to derive new average bounds for 3-torsion in real quadratic fields.

Findings

Improved the bound to h_3(d) ^{0.3193 + psilon}

Established new average bounds for 3-torsion in real quadratic fields

Extended techniques to refine torsion bounds in class groups.

Abstract

Ellenberg--Venkatesh proved in 2007 that $h_3(d) \ll_\epsilon |d|^{1/3 + \epsilon}$, where $h_3(d)$ denotes the size of the $3$-torsion of the class group of $\mathbb{Q}(\sqrt{d})$. We improve this bound to $h_3(d) \ll_\epsilon |d|^{\kappa + \epsilon}$ with $\kappa \approx 0.3193 \cdots$. We also combine our methods with work of Heath-Brown--Pierce to give new bounds for average $\ell$-torsion of real quadratic fields.