Bounds for the $\ell$-torsion in class groups

We prove for each integer $\ell\geq 1$ an unconditional upper bound for the size of the $\ell$-torsion subgroup $Cl_K[\ell]$ of the class group of $K$, which holds for all but a zero density set of number fields $K$ of degree $d\in\{4,5\}$ (with the additional restriction in the case $d = 4$ that the field be non-$D_4$). For sufficiently large $\ell$ this improves recent results of Ellenberg, Matchett Wood, and Pierce, and is also stronger than the best currently known pointwise bounds under GRH. Conditional on GRH and on a weak conjecture on the distribution of number fields our bounds also hold for arbitrary degrees $d$.