On ℓ-torsion in class groups of numberfields

For each integer ≥ 1, we prove an unconditional upper bound on the size of the -torsion subgroup of the class group, which holds for all but a zero-density set of field extensions of Q of degree d, for any fixed d ∈ \2,3,4,5\ (with the additional restriction in the case $d=4$ that the field be non-D₄). For sufficiently large (specified explicitly), these results are as strong as a previously known bound that is conditional on GRH. As part of our argument, we develop a probabilistic "Chebyshev sieve," and give uniform, power-saving error terms for the asymptotics of quartic (non-D₄) and quintic fields with chosen splitting types at a finite set of primes.