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 $d$.

To appear in Bull. Lond. Math. Soc

References in corpus (1)

• Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves

Cited by in corpus (2)

• Pointwise Bound for $\ell$-torsion in Class Groups II: Nilpotent Extensions
• Pointwise Bound for $\ell$-torsion in Class Groups: Elementary Abelian Extensions