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

TL;DR

This paper improves the general unconditional upper bounds on the size of the -torsion subgroup of class groups of number fields, providing explicit log-power savings over the trivial bound for all fields and all .

Contribution

It establishes the first unconditional, explicit log-power savings over the trivial bound for -torsion in class groups applicable to all number fields and all .

Findings

Proves that |-cl(K)| = o_{[K: Q],}(D_K^{1/2}) for all number fields K.

Provides explicit log-power savings over the trivial bound.

First unconditional general result improving the trivial bound for -torsion.

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 $\ell$.