Crible et 3-rang des corps quadratiques
Annales de l'Institut Fourier, Volume 46 (1996) no. 4, p. 909-949
Call h 3 * (Δ) the number of cube roots of unity in the class group of ℚ(Δ), where Δ is a fundamental discriminant. Davenport and Heilbronn computed the mean value of these numbers when Δ tends to ±∞. The author gives a general geometric argument yielding an explicit bound for the error term, with the additional possibility of restricting Δ to arithmetic progressions. Sieve techniques then produce results about the 3-parts of the groups Cl (ℚ(Δ)), where P k is an almost-prime of order k. In this way, one controls simultaneously both the 2-rank and the 3-rank of the class group Cl (ℚ(Δ)). As a special case, the author gives a bound for the mean 3-rank of the ℚ(±p), where p is prime.
Considérons le cardinal h 3 * (Δ) de l’ensemble des racines cubiques de l’unité dans le groupe des classes de ℚ(Δ), où Δ est un discriminant fondamental. Un résultat de Davenport et Heilbronn calcule la valeur moyenne de ces nombres quand Δ varie. On obtient ici géométriquement une borne explicite pour le reste, avec la possibilité supplémentaire de restreindre les Δ à des progressions arithmétiques. Des techniques de crible permettent alors d’évaluer la 3-partie des ℚ(±P k ), où P k est pseudo-premier d’ordre k. On contrôle ainsi simultanément le 2-rang et le 3-rang du groupe des classes Cl (ℚ(Δ)). L’auteur donne en particulier une borne pour le 3-rang en moyenne des ℚ(±p), où p est premier. 
@article{AIF_1996__46_4_909_0,
     author = {Belabas, Karim},
     title = {Crible et 3-rang des corps quadratiques},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {46},
     number = {4},
     year = {1996},
     pages = {909-949},
     doi = {10.5802/aif.1535},
     mrnumber = {98b:11112},
     zbl = {0853.11088},
     language = {fr},
     url = {https://aif.centre-mersenne.org/item/AIF_1996__46_4_909_0}
}

Belabas, Karim. Crible et 3-rang des corps quadratiques. Annales de l'Institut Fourier, Volume 46 (1996) no. 4, pp. 909-949. doi : 10.5802/aif.1535. https://aif.centre-mersenne.org/item/AIF_1996__46_4_909_0/

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