$\ell$-torsion in class groups of certain families of $D_4$-quartic fields

TL;DR

This paper establishes an upper bound on the -torsion in class groups for most fields within specific families of D4-quartic fields, utilizing a new Chebotarev density theorem and field count estimates.

Contribution

It introduces a novel Chebotarev density theorem tailored for D4-quartic fields and provides bounds on -torsion in their class groups.

Findings

Upper bounds for -torsion in class groups of D4-quartic fields

A new Chebotarev density theorem for these families

Quantitative estimates on the number of such fields

Abstract

We prove an upper bound for $\ell$-torsion in class groups of almost all fields in certain families of $D_4$-quartic fields. Our key tools are a new Chebotarev density theorem for these families of $D_4$-quartic fields and a lower bound for the number of fields in the families.