Bounds for moments of \(\ell \)-torsion in class groups

Fix a number field k, integers \(\ell , n \ge 2\) $\ell ,n\ge 2$ , and a prime p. For all \(r \ge 1\) $r\ge 1$ , we prove strong unconditional upper bounds on the rth moment of \(\ell \) $\ell$ -torsion in the ideal class groups of degree p extensions of k and of degree n \(S_n\) $S_n$ -extensions of k, improving upon results of Ellenberg, Pierce and Wood as well as GRH-conditional results of Frei and Widmer. For large r, our results are comparable with work of Heath-Brown and Pierce for imaginary quadratic extensions of \(\mathbb {Q}\) $\mathbb{Q}$ . When \(r=1\) $r=1$ , our results are new even for the family of all quadratic extensions of \(\mathbb {Q}\) $\mathbb{Q}$ , leading to an improved upper bound for the count of degree p \(D_p\) $D_p$ -extensions over \(\mathbb {Q}\) $\mathbb{Q}$ (where \(D_p\) $D_p$ is the dihedral group of order 2p).