#### 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
##### Milestones
Revised: 10 June 2017
Accepted: 10 July 2017
Published: 15 October 2017
##### Authors
 Jordan Ellenberg Department of Mathematics University of Wisconsin Madison, WI 53706 United States Lillian B. Pierce Mathematics Department Duke University Durham, NC 27708 United States Melanie Matchett Wood Department of Mathematics University of Wisconsin Van Vleck Hall Madison, WI 53711 United States American Institute of Mathematics San Jose, CA 95112 United States