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

An, Chen

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

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.

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.

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

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