#### Vol. 11, No. 8, 2017

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On $\ell$-torsion in class groups of number fields

### Jordan Ellenberg, Lillian B. Pierce and Melanie Matchett Wood

Vol. 11 (2017), No. 8, 1739–1778
##### Abstract

For each integer $\ell \ge 1$, we prove an unconditional upper bound on the size of the $\ell$-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 $d\in \left\{2,3,4,5\right\}$ (with the additional restriction in the case $d=4$ that the field be non-${D}_{4}$). For sufficiently large $\ell$ (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.

##### Keywords
number fields, class groups, Cohen–Lenstra heuristics, sieves
##### Mathematical Subject Classification 2010
Primary: 11R29
Secondary: 11N36, 11R45