Enumerating quartic dihedral extensions of with signatures
Annales de l'Institut Fourier, Volume 53 (2003) no. 2, p. 339-377
In a previous paper, we have given asymptotic formulas for the number of isomorphism classes of D 4 -extensions with discriminant up to a given bound, both when the signature of the extensions is or is not specified. We have also given very efficient exact formulas for this number when the signature is not specified. The aim of this paper is to give such exact formulas when the signature is specified. The problem is complicated by the fact that the ray class characters which appear are not all genus characters.
Dans un précédent article, nous avons donné des formules asymptotiques pour le nombre de classes d’isomorphismes d’extensions D 4 classées par discriminant croissant, que la signature des extensions soit ou non spécifiée. Nous avons également donné des formules exactes très efficaces pour ce nombre quand on ne spécifie pas la signature. Le but du présent article est de donner de telles formules exactes quand la signature est imposée. Le problème se complique du fait de l’apparition de caractères sur les groupes de classes de rayon qui ne sont pas des caractères de genre.
DOI : https://doi.org/10.5802/aif.1946
Classification:  11R16,  11R29,  11R45,  11Y40
Keywords: discriminant counting, genus character, quartic reciprocity
@article{AIF_2003__53_2_339_0,
     author = {Cohen, Henri},
     title = {Enumerating quartic dihedral extensions of ${\mathbb {Q}}$ with signatures},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {53},
     number = {2},
     year = {2003},
     pages = {339-377},
     doi = {10.5802/aif.1946},
     mrnumber = {1990000},
     zbl = {01940698},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2003__53_2_339_0}
}
Enumerating quartic dihedral extensions of ${\mathbb {Q}}$ with signatures. Annales de l'Institut Fourier, Volume 53 (2003) no. 2, pp. 339-377. doi : 10.5802/aif.1946. https://aif.centre-mersenne.org/item/AIF_2003__53_2_339_0/

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