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

TL;DR

This paper establishes new unconditional upper bounds on the size of the $ ext{l}$-torsion subgroup of class groups for certain number fields, improving previous results and extending to larger degrees under conjectural assumptions.

Contribution

It provides the first unconditional bounds for $ ext{l}$-torsion in class groups for degree 4 and 5 fields, with improvements for large $ ext{l}$ and under GRH for all degrees.

Findings

Unconditional bounds for degree 4 and 5 fields for most fields.

Stronger bounds for large $ ext{l}$ compared to previous work.

Conditional bounds for arbitrary degrees under GRH and a conjecture.

Abstract

We prove for each integer $\ell\geq 1$ an unconditional upper bound for the size of the $\ell$-torsion subgroup $Cl_K[\ell]$ 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 $d$.