$\ell$-Torsion in Class Groups via Dirichlet $L$-functions

TL;DR

This paper investigates upper bounds for the $ ext{ell}$-part of class numbers in quadratic and cubic fields, especially with smooth discriminants, using properties of Dirichlet $L$-functions.

Contribution

It introduces new bounds for the $ ext{ell}$-torsion in class groups of quadratic and cubic fields based on Dirichlet $L$-functions and discriminant smoothness.

Findings

Established upper bounds for $h_ ext{ell}(K)$ in quadratic fields.

Extended bounds to cubic fields with smooth discriminants.

Demonstrated the effectiveness of Dirichlet $L$-functions in bounding class group torsion.

Abstract

For a prime $\ell$, let $h_\ell(K)$ denote the $\ell$-part of the class number of the number field $K$. We investigate upper bounds for $h_\ell(K)$ when $K$ is quadratic or cubic, particularly in the case in which the discriminant of $K$ is smooth. This is achieved using properties of Dirichlet $L$-functions.