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Note Sur La Conjecture De Greenberg


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1 Institut de Mathematiques de Bordeaux, Universite de Bordeaux & CNRS, 351 cours de la liberation, F-33405 Talence Cedex, France
     

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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 unconditionally prove that Greenberg’s conjecture holds if and only if the logarithmic classes of K principalize in the cyclotomic ℤ𝓁-extension. 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. Last, in the abelian case, we provide an explicit description of the circular class groups in connexion with the so-called weak conjecture.
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  • Note Sur La Conjecture De Greenberg

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Authors

Jean-Francois Jaulent
Institut de Mathematiques de Bordeaux, Universite de Bordeaux & CNRS, 351 cours de la liberation, F-33405 Talence Cedex, France

Abstract


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 unconditionally prove that Greenberg’s conjecture holds if and only if the logarithmic classes of K principalize in the cyclotomic ℤ𝓁-extension. 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. Last, in the abelian case, we provide an explicit description of the circular class groups in connexion with the so-called weak conjecture.

References