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 $\ell \ge 2$, we improve over the commonly cited trivial bound $|{\mathrm{Cl}}_K[\ell ]|\le |{\mathrm{Cl}}_K|{\ll }_{[K:\mathbb{Q}],\epsilon }{D}_K^{1/2+\epsilon }$ 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 $\ell$.

11 pages