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.
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  • ARXIV : 1706.08784

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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⟩

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