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

Ellenberg–Venkatesh proved in 2007 that $h_3(d){\ll }_{\epsilon }\mathrm{\mid }d{\mathrm{\mid }}^{1\mathrm{/}3+\epsilon }$, 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 $h_3(d){\ll }_{\epsilon }\mathrm{\mid }d{\mathrm{\mid }}^{\kappa +\epsilon }$ with $\kappa \approx 0.3193\cdots$. We also combine our methods with work of Heath-Brown–Pierce to give new bounds for average $\mathrm{\ell }$-torsion of real quadratic fields.