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

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.