A short remark on the $\ell$-torsion part of class groups

TL;DR

This paper addresses Ellenberg's question on primitive elements of small height, improves bounds on the $ ext{l}$-torsion in class groups of number fields, and extends results to pure fields of odd degree.

Contribution

It answers Ellenberg's question and enhances bounds on the $ ext{l}$-torsion part of class groups, generalizing previous results to arbitrary odd degree pure fields.

Findings

Answered Ellenberg's question on primitive elements.

Improved bounds for $ ext{l}$-torsion in class groups of cubic fields.

Extended bounds to pure fields of odd degree.

Abstract

In a 2008 paper Ellenberg suggested a strategy to improve the known upper bounds for the $\ell$-torsion part of class groups of number fields of fixed degree $d$. Motivated by this he proposed a question about the number of primitive elements of small height in a number field. Here we answer Ellenberg's question. We also improve Heath-Brown's bound for the $\ell$-torsion part of class groups of purely cubic number fields, and we generalize our improvement to pure fields of arbitrary odd degree $d$.