Bounds for the $\ell$-torsion in class groups

Abstract

We prove for each integer $\ell \ge 1$ an unconditional upper bound for the size of the $\ell$-torsion subgroup $C{l}_K[\ell ]$ of the class group of $K$, which holds for all but a zero density set of number fields $K$ of degree $d\in \{4,5\}$ (with the additional restriction in the case $d=4$ that the field be non-$D_4$). For sufficiently large $\ell$ 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 $d$.

arXiv commentTo appear in Bull. Lond. Math. Soc

References in corpus (5)

• On $\ell$-torsion in class groups of number fields
• Geometry-of-numbers methods over global fields I: Prehomogeneous vector spaces
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• Number fields without small generators
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Cited by in corpus (6)

• Genus theory and $\epsilon$-conjectures on p-class groups
• Bounds for moments of $\ell$-torsion in class groups
• $\ell$-torsion bounds for the class group of number fields with an $\ell$-group as Galois group
• Pointwise Bound for $\ell$-torsion in Class Groups II: Nilpotent Extensions
• Pointwise Bound for $\ell$-torsion in Class Groups: Elementary Abelian Extensions
• $\ell$-torsion in class groups of certain families of $D_4$-quartic fields