$\ell$-torsion in class groups of certain families of $D_4$-quartic fields

An, Chen

Journal de théorie des nombres de Bordeaux, Volume 32 (2020) no. 1, pp. 1-23.

We prove an upper bound for $\ell$-torsion in class groups of almost all fields in certain families of $D_4$-quartic fields. Our key tools are a new Chebotarev density theorem for these families of $D_4$-quartic fields and a lower bound for the number of fields in the families.

Nous donnons une borne supérieure pour la $\ell$-torsion des groupes de classes pour presque tous les corps de certaines familles des corps quartiques de type $D_4.$ Nos outils principaux sont une nouvelle version du théorème de densité de Chebotarev pour ces familles et une borne inférieure sur le nombre de corps dans les familles.

Classification: 11R29, 11R42, 11R45
Mots-clés : torsion, class group, Chebotarev density theorem

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