Pointwise bounds for 3-torsion

For $\ell$ an odd prime number and $d$ a squarefree integer, a central question in arithmetic statistics is to give pointwise bounds for the size of the $\ell$-torsion of the class group of $\mathbb{Q}(\sqrt{d})$. This is in general a difficult problem, and unconditional pointwise bounds are only available for $\ell = 3$ due to work of Pierce, Helfgott—Venkatesh and Ellenberg—Venkatesh. The current record is $h_3(d) \ll_\epsilon d^{1/3 + \epsilon}$ due to Ellenberg—Venkatesh. We will discuss how to improve this to $h_3(d) \ll d^{0.32}$. This is joint work with Stephanie Chan.