# Normes d'id\'eaux dans la tour cyclotomique et conjecture de Greenberg

## TL;DR

This paper investigates the $p$-adic behavior of norms in cyclotomic extensions of totally real fields, providing heuristics and numerical evidence supporting Greenberg's conjecture that certain Iwasawa invariants vanish.

## Contribution

It offers heuristics and statistical evidence on the $p$-adic norms' distribution, advancing understanding of Greenberg's conjecture in Iwasawa theory for totally real fields.

## Key findings

- Numerical examples support the conjecture that $ ext{lambda} = ext{mu} = 0$.
- Heuristics suggest a distribution assumption leading to invariants vanishing.
- Statistical data in quadratic cases confirm the proposed properties.

## Abstract

Pre-print of a publication in "Annales math\'ematiques du Qu{\'e}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 r\'eels" (Annales Math\'ematiques 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.