Crible et 3-rang des corps quadratiques

Belabas, Karim

Annales de l'Institut Fourier, Tome 46 (1996), p. 909-949 / Harvested from Numdam

Résumé

Considérons le cardinal $h^*_3(\Delta )$ de l’ensemble des racines cubiques de l’unité dans le groupe des classes de ${\Bbb Q}(\sqrt{\Delta })$, où $\Delta $ est un discriminant fondamental. Un résultat de Davenport et Heilbronn calcule la valeur moyenne de ces nombres quand $\Delta $ varie. On obtient ici géométriquement une borne explicite pour le reste, avec la possibilité supplémentaire de restreindre les $\Delta $ à des progressions arithmétiques. Des techniques de crible permettent alors d’évaluer la 3-partie des ${\Bbb Q}(\sqrt{\pm P_k})$, où $P_k$ est pseudo-premier d’ordre $k$. On contrôle ainsi simultanément le 2-rang et le 3-rang du groupe des classes ${\rm Cl}({\Bbb Q}(\sqrt{\Delta }))$. L’auteur donne en particulier une borne pour le 3-rang en moyenne des ${\Bbb Q}(\sqrt{\pm p})$, où $p$ est premier.

Abstract

Call $h^*_3(\Delta )$ the number of cube roots of unity in the class group of ${\Bbb Q}(\sqrt{\Delta })$, where $\Delta $ is a fundamental discriminant. Davenport and Heilbronn computed the mean value of these numbers when $\Delta $ tends to $\pm \infty $. The author gives a general geometric argument yielding an explicit bound for the error term, with the additional possibility of restricting $\Delta $ to arithmetic progressions. Sieve techniques then produce results about the 3-parts of the groups ${\rm Cl}({\Bbb Q}(\sqrt{\Delta }))$, 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 ${\rm Cl}({\Bbb Q}(\sqrt{\Delta }))$. As a special case, the author gives a bound for the mean 3-rank of the ${\Bbb Q} (\sqrt{\pm p})$, where $p$ is prime.

Détail

Publié le : 1996-01-01

MR 98b:11112 | Zbl 0853.11088 | 1 citation dans Numdam

DOI : https://doi.org/10.5802/aif.1535

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