Improving the trivial bound for $\ell$-torsion in class groups
Robert J. Lemke Oliver, Asif Zaman
TL;DR
The paper delivers a uniform unconditional improvement over the trivial $\ell$-torsion bound in class groups by proving $|\text{Cl}_K[\ell]| = o(D_K^{1/2})$ with an explicit log-power saving. It develops an inverse relation between $|\text{Cl}_K|$ and $|\text{Cl}_K[\ell]|$ via an Ellenberg–Venkatesh framework and refines control of the zeta-residue $\kappa_K$, using the class-number formula to link torsion size to discriminant and regulator. The results yield a bound $|\text{Cl}_K[\ell]| \ll |\text{Cl}_K| (\log D_K)^{-\delta V_K}$ under a near-extremal size assumption, and, under subconvexity-type hypotheses (or GRH-like conditions), produce effective power savings $D_K^{\Delta}$ for all number fields with $\Delta$ depending on $\ell$ and $[K:\mathbb{Q}]$. The work unifies conditional and unconditional approaches to $\ell$-torsion, clarifies the role of the zeta-residue and prime-splitting behavior, and highlights a barrier posed by quadratic subfields to fully effective unconditional results.
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 $\ell$.