Call the number of cube roots of unity in the class group of , where is a fundamental discriminant. Davenport and Heilbronn computed the mean value of these numbers when tends to . The author gives a general geometric argument yielding an explicit bound for the error term, with the additional possibility of restricting to arithmetic progressions. Sieve techniques then produce results about the 3-parts of the groups , where is an almost-prime of order . In this way, one controls simultaneously both the 2-rank and the 3-rank of the class group . As a special case, the author gives a bound for the mean 3-rank of the , where is prime.
Considérons le cardinal de l’ensemble des racines cubiques de l’unité dans le groupe des classes de , où est un discriminant fondamental. Un résultat de Davenport et Heilbronn calcule la valeur moyenne de ces nombres quand varie. On obtient ici géométriquement une borne explicite pour le reste, avec la possibilité supplémentaire de restreindre les à des progressions arithmétiques. Des techniques de crible permettent alors d’évaluer la 3-partie des , où est pseudo-premier d’ordre . On contrôle ainsi simultanément le 2-rang et le 3-rang du groupe des classes . L’auteur donne en particulier une borne pour le 3-rang en moyenne des , où 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}, pages = {909--949}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {46}, number = {4}, year = {1996}, doi = {10.5802/aif.1535}, mrnumber = {98b:11112}, zbl = {0853.11088}, language = {fr}, url = {http://archive.numdam.org/articles/10.5802/aif.1535/} }
TY - JOUR AU - Belabas, Karim TI - Crible et 3-rang des corps quadratiques JO - Annales de l'Institut Fourier PY - 1996 SP - 909 EP - 949 VL - 46 IS - 4 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.1535/ DO - 10.5802/aif.1535 LA - fr ID - AIF_1996__46_4_909_0 ER -
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. http://archive.numdam.org/articles/10.5802/aif.1535/
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