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

Abstract

For any number field $K$ with $D_K=|\mathrm{Disc}(K)|$ and any integer $\ell \geq 2$, we improve over the commonly cited trivial bound $|\mathrm{Cl}_K[\ell]| \leq |\mathrm{Cl}_K| \ll_{[K:\mathbb{Q}],\varepsilon} D_K^{1/2+\varepsilon}$ on the $\ell$-torsion subgroup of the class group of $K$ by showing that $|\mathrm{Cl}_K[\ell]| = o_{[K:\mathbb{Q}],\ell}(D_K^{1/2})$. 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 $\ell$.