Bounds for the ℓ‐torsion in class groups

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We prove for each integer ℓ⩾1 an unconditional upper bound for the size of the ℓ-torsion subgroup ClK[ℓ] of the class group of K, which holds for all but a zero density set of number fields K of degree d∈{4,5} (with the additional restriction in the case d=4 that the field be non-D4). For sufficiently large ℓ this improves recent results of Ellenberg, Matchett Wood and Pierce, and is also stronger than the best currently known pointwise bounds under GRH. Conditional on GRH and on a weak conjecture on the distribution of number fields our bounds also hold for arbitrary degrees d.
Original languageEnglish
Pages (from-to)124-131
Number of pages8
JournalBulletin of the London Mathematical Society
Issue number1
Early online date9 Nov 2017
Publication statusPublished - Feb 2018

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