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

For a prime $\mathrm{\ell }$, let $h_{\mathrm{\ell }}(K)$ denote the $\mathrm{\ell }$-part of the class number of the number field $K$. We investigate upper bounds for $h_{\mathrm{\ell }}(K)$ 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.