We study the asymptotics conjecture of Malle for dihedral groups $D_\ell$ of order $2\ell$, where $\ell$ is an odd prime. We prove the expected lower bound for those groups. For the upper bounds we show that there is a connection to class groups of quadratic number fields. The asymptotic behavior of those class groups is predicted by the Cohen--Lenstra heuristics. Under the assumption of this heuristic we are able to prove the expected upper bounds.
On $p$-class groups of relative cyclic $p$-extensions
Random matrices, the Cohen-Lenstra heuristics, and roots of unity
$\ell$-Torsion in Class Groups via Dirichlet $L$-functions
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