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

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.

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