Bounds for moments of ℓ-torsion in class groups

Fix a number field k, integers ℓ,n≥2, and a prime p. For all r≥1, we prove strong unconditional upper bounds on the rth moment of ℓ-torsion in the ideal class groups of degree p extensions of k and of degree nS_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 Q. When r=1, our results are new even for the family of all quadratic extensions of Q, leading to an improved upper bound for the count of degree pD_p-extensions over Q (where D_p is the dihedral group of order 2p).