Improving the trivial bound for $\mathrm{\ell }$-torsion in class groups

For any number field $K$ with $D_K=\mathrm{\mid }\mathrm{D}\mathrm{i}\mathrm{s}\mathrm{c}(K)\mathrm{\mid }$ and any integer $\mathrm{\ell }\ge 2$, we improve over the commonly cited trivial bound $\mathrm{\mid }{\mathrm{C}\mathrm{l}}_K[\mathrm{\ell }]\mathrm{\mid }\le \mathrm{\mid }{\mathrm{C}\mathrm{l}}_K\mathrm{\mid }{\ll }_{[K:\mathbb{Q}],\epsilon }{D}_K^{1\mathrm{/}2+\epsilon }$ on the $\mathrm{\ell }$-torsion subgroup of the class group of $K$ by showing that $\mathrm{\mid }{\mathrm{C}\mathrm{l}}_K[\mathrm{\ell }]\mathrm{\mid }=o_{[K:\mathbb{Q}],\mathrm{\ell }}(D_K^{1\mathrm{/}2})$. In fact, we obtain an explicit log-power saving. This is the first general unconditional saving over the trivial bound that holds for all $K$ and all $\mathrm{\ell }$.