Note sur la conjecture de Greenberg

Abstract : We use logarithmic ℓ-class groups to take a new view on the Greenberg's conjecture about Iwasawa ℓ-invariants of totally real number fields. By the way we recall and complete some classical results. We give a sufficient logarithmic criterium which is also necessary in the context of the so-called weak conjecture, when the prime ℓ splits completely in F. In the semi-simple case, we unconditionnally prove that the conjecture holds if and only if the logarithmic class group of F capitulates in the Zℓ-cyclotomic tower. We use logarithmic ℓ-class groups to take a new view on Greenberg's conjecture about Iwasawa ℓ-invariants of a totally real number field K. By the way we recall and complete some classical results. Under Leopoldt's conjecture, we prove that Greenberg's conjecture holds if and only if the logarithmic classes of K principalize in the cyclotomic Zℓ-extensions. As an illustration of our approach, in the special case where the prime ℓ splits completely in K, we prove that the sufficient condition introduced by Gras just asserts the triviality of the logarithmic class group of K.
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https://hal.archives-ouvertes.fr/hal-01407300
Contributor : Jean-François Jaulent <>
Submitted on : Wednesday, March 22, 2017 - 7:14:58 PM
Last modification on : Thursday, May 3, 2018 - 1:14:26 AM
Document(s) archivé(s) le : Friday, June 23, 2017 - 2:35:14 PM

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  • HAL Id : hal-01407300, version 2
  • ARXIV : 1612.00718

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Jean-François Jaulent. Note sur la conjecture de Greenberg. 2017. ⟨hal-01407300v2⟩

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