On ℓ-torsion in class groups of number fields

Ellenberg, Jordan; Pierce, Lillian; Wood, Melanie

doi:10.2140/ant.2017.11.1739

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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

DOI: 10.2140/ant.2017.11.1739

Abstract

For each integer $ℓ \geq 1$ , 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 $ℚ$ of degree $d$ , for any fixed $d \in {2, 3, 4, 5}$ (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.

Keywords

number fields, class groups, Cohen–Lenstra heuristics, sieves

Mathematical Subject Classification 2010

Primary: 11R29

Secondary: 11N36, 11R45

Milestones

Received: 1 April 2016

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

Vol. 11, No. 8, 2017