1. Normes d'idéaux dans la tour cyclotomique et conjecture de Greenberg: Hypothèses p-adiques sur les normes d'idéaux
- Author
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Georges Gras, Laboratoire de Mathématiques de Besançon (UMR 6623) (LMB), Université de Bourgogne (UB)-Université de Franche-Comté (UFC), and Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Distribution (number theory) ,Mathematics::Number Theory ,General Mathematics ,Galois group ,010103 numerical & computational mathematics ,p-class groups ,01 natural sciences ,Combinatorics ,class field theory ,Leopoldt's conjecture ,Class field theory ,0101 mathematics ,Abelian group ,Mathematics ,p-adic regulators ,Conjecture ,Mathematics - Number Theory ,010102 general mathematics ,11R23, 11R29, 11R37 ,11Y40 ,Greenberg's conjecture ,16. Peace & justice ,Iwasawa's theory ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,Fermat quotients of algebraic numbers ,Number theory ,Totally real number field - 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., Comment: in French, Completes with numerical computations and heuristics our previous paper arXiv:1611.09592 on Greenberg's conjecture. Annales math{\'e}matiques du Quebec, A para{\^i}tre
- Published
- 2019