Composantes isotypiques de pro-p-extensions de corps de nombres et p-rationalité - Archive ouverte HAL
Journal Articles Publicationes Mathematicae Debrecen Year : 2019

Composantes isotypiques de pro-p-extensions de corps de nombres et p-rationalité

Abstract

Let p be a prime number, and let K/k be a finite Galois extension of number fields with Galois group ∆ of order coprime to p. Let S be a finite set of non archimedean places of k including the set S_p of p-adic places, and let K_S be the maximal pro-p extension of K unramified outside S. Let G := G_S/H be a quotient of G_S :=Gal(K_S/K) on which ∆ acts trivially. Put X := H/[H, H]. In this paper, we study the ϕ-component X^ϕ of X for all Q_p-irreductible characters ϕ of ∆, and, in particular, by assuming Leopoldt conjecture we show that for all non-trivial characters ϕ, the Zp[[G]]-module X^ϕ is free if and only if the ϕ-component of the Z_p-torsion of G_S/[G_S, G_S] is trivial. We also make a numerical study of the freeness of X^ϕ in cyclic extensions K/Q of degree 3 and 4 (by using families of polynomials given by Balady, Lecacheux and more recently by Balady and Washington), but also in degree 6 dihedral extension over Q: the results we get support a recent conjecture of Gras.
Fichier principal
Vignette du fichier
Maire-Rougnant-08_28_2018.pdf (730.05 Ko) Télécharger le fichier
Origin Files produced by the author(s)
Loading...

Dates and versions

hal-02107052 , version 1 (23-04-2019)

Identifiers

Cite

Christian Maire, Marine Rougnant. Composantes isotypiques de pro-p-extensions de corps de nombres et p-rationalité. Publicationes Mathematicae Debrecen, 2019, 94 (1-2), pp.123-155. ⟨10.5486/PMD.2019.8281⟩. ⟨hal-02107052⟩
108 View
82 Download

Altmetric

Share

More