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

Let $p$ be a prime number, and let ${\rm K}/{\rm k}$ be a finite Galois extension of number fields with Galois group $\Delta$ of order coprime to $p$. Let $S$ be a finite set of non-Archimedean places of ${\rm k}$ including the set $S_p$ of $p$-adic places, and let ${\rm K}_S$ be the maximal pro-$p$ extension of ${\rm K}$ unramified outside $S$. Let $G:=G_S/H$ be a quotient of $G_S:=\operatorname{Gal}({\rm K}_S/{\rm K})$ on which $\Delta$ acts trivially. Put ${\rm X}:=H/[H,H]$. In this paper, we study the $\varphi$-component ${\rm X}^\varphi$ of ${\rm X}$ for all ${\rm Q}_p$-irreductible characters $\varphi$ of $\Delta$, and, in particular, by assuming the Leopoldt conjecture, we show that for all non-trivial characters $\varphi$, the ${\rm Z}_p[[G]]$-module ${\rm X}^\varphi$ is free if and only if the $\varphi$-component of the ${\rm Z}_p$-torsion of $G_S/[G_S,G_S]$ is trivial. We also make a numerical study of the freeness of ${\rm X}^\varphi$ in cyclic extensions ${\rm K}/{\rm Q}$ 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 ${\rm Q}$: the results we get support a recent conjecture of Gras.