Vol. 11, No. 8, 2017

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
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 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 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.

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