# Normes d'idéaux dans la tour cyclotomique et conjecture de Greenberg : Hypothèses p-adiques sur les normes d'idéaux

Abstract : Pre-print of a publication in “ Annales mathématiques du Québec ”. Let k be a totally real number field and let k_∞ be its cyclotomic 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 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_∞ ; indeed, this conjecture (on the nullity of the invariants λ et µ of Iwasawa) depends of images in the torsion group Tk 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 Tk. 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.
Keywords :
Document type :
Journal articles

Cited literature [11 references]

https://hal.archives-ouvertes.fr/hal-01546656
Contributor : Georges Gras <>
Submitted on : Thursday, October 4, 2018 - 2:01:56 PM
Last modification on : Saturday, October 6, 2018 - 1:02:10 AM
Document(s) archivé(s) le : Saturday, January 5, 2019 - 2:55:32 PM

### Files

Norms & Greenberg's conjecture...
Files produced by the author(s)

### Identifiers

• HAL Id : hal-01546656, version 3
• ARXIV : 1706.08784

### Citation

Georges Gras. Normes d'idéaux dans la tour cyclotomique et conjecture de Greenberg : Hypothèses p-adiques sur les normes d'idéaux. Annales mathématiques du Quebec, A paraître. ⟨hal-01546656v3⟩

Record views