On $\ell$-torsion in class groups of number fields

TL;DR

This paper establishes unconditional upper bounds on the size of the -torsion subgroup of class groups for most number fields of degree 2 to 5, using probabilistic methods and asymptotic analysis.

Contribution

It introduces a probabilistic Chebyshev sieve and provides uniform error estimates for counting specific number fields, advancing understanding of class group torsion bounds.

Findings

Unconditional bounds on -torsion for most quadratic to quintic fields.

Development of a probabilistic Chebyshev sieve technique.

Asymptotic formulas with power-saving error terms for counting fields with specified splitting types.

Abstract

For each integer $\ell \geq 1$, we prove an unconditional upper bound on the size of the $\ell$-torsion subgroup of the class group, which holds for all but a zero-density set of field extensions of $\mathbb{Q}$ of degree $d$, for any fixed $d \in \{2,3,4,5\}$ (with the additional restriction in the case $d=4$ that the field be non-$D_4$). For sufficiently large $\ell$ (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-$D_4$) and quintic fields with chosen splitting types at a finite set of primes.