SUR LE MODULE DE BERTRANDIAS–PAYAN DANS UNE p-EXTENSION – NOYAU DE CAPITULATION
Résumé
For a number field K and a prime number p we denote by BP_K the compositum of the cyclic p-extensions of K which are embeddable into a cyclic p-extension of arbitrary large degree. The extension BP_K/K is p-ramified and is a finite extension of the compositum K^ of the Z_p-extensions of K. The group bp_K := Gal(BP_K/K^) is called the Bertrandias--Payan module.
We study the transfer map j_L/K : bp_K ---> bp_L (as a capitulation morphism of ideal classes) in a p-extension L/K. In the cyclic case of degree p, we prove that j_L/K is injective except if L/K is kummerian, p-ramified, non globally cyclotomic but locally cyclotomic at p (Theorem 3.1). We give an explicit formula (Theorem 5.2) for #bp_L^G / #bp_K and we show how its entirety depends on the torsion groupT_L of the Galois group of the maximal Abelian p-ramified pro-p-extension of L, by using a suitable p-adic logarithm.
Origine : Fichiers produits par l'(les) auteur(s)
Loading...