On l-torsion in class groups of number fields

For each integer l >= 1, we prove an unconditional upper bound on the size of the l-torsion subgroup of the class group, which holds for all but a zero-density set of field extensions of (sic) of degree d, for any fixed d is an element of {2, 3, 4, 5} (with the additional restriction in the case d = 4 that the field be non-D-4). For sufficiently large l (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-D-4) and quintic fields with chosen splitting types at a finite set of primes.