Normes d'idéaux dans la tour cyclotomique et conjecture de Greenberg

Abstract

Pre-print of a publication in "Annales mathématiques du Qu{é}bec". Let $k$ be a totally real number field and let $k_{\infty }$ be its cyclotomic ${\mathbb{Z}}_p$-extension for $p$ totally split in $k$. This text completes our article entitled: "Approche $p$-adique de la conjecture de Greenberg pour les corps totalement réels" (Annales Mathématiques Blaise Pascal 2017), by means of heuristics on the $p$-adic behavior of the norms, in $k_n/k$, of the ideals in $k_{\infty }$ ; indeed, this conjecture (on the nullity of the invariants $\lambda$ et $\mu$ of Iwasawa) depends of images in the torsion group ${\mathcal{T}}_k$ of the Galois group of the maximal abelian $p$-ramified pro-$p$-extension of $k$, thus of Artin symbols in a finite extension $F/k$ obtained by Galois descent of ${\mathcal{T}}_k$. An assumption of distribution of these norms implies $\lambda =\mu =0$. Several statistics and numerical examples in the quadratic case confirm the probable exactness of such properties which constitute the fundamental obstruction for a proof of Greenberg's conjecture in the sole context of Iwasawa's theory.

in French, Completes with numerical computations and heuristics our previous paper arXiv:1611.09592 on Greenberg's conjecture. Annales math{é}matiques du Quebec, A para{î}tre