Asymptotics of number fields and the Cohen–Lenstra heuristics

Klüners, Jürgen

Journal de théorie des nombres de Bordeaux, Tome 18 (2006) no. 3, pp. 607-615.

Nous étudions les conjectures de Malle pour les groupes diédraux $D_{\ell }$ d’ordre $2\ell$, où $\ell$ est un nombre premier impair. Nous prouvons que les bornes inférieures sont celles attendues. Pour les bornes supérieures, nous montrons qu’il y a un lien avec les groupes de classes des corps quadratiques. Le comportement asymptotique de ces groupes de classes est prédit par les heuristiques de Cohen–Lenstra. Sous ces hypothèses, nous pouvons montrer que les bornes supérieures sont celles attendues.

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.

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