Note on Greenberg conjecture

Note sur la conjecture de Greenberg

Jean-François Jaulent (1)

1 IMB - Institut de Mathématiques de Bordeaux

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.

Nous étudions la conjecture de Greenberg sur les ℓ-invariants d'Iwasawa des corps totalement réels, en termes de classes logarithmiques. Sous la conjecture de Leopoldt, et donc en particulier pour les corps abéliens, nous montrons que la conjecture de Greenberg postule tout simplement la capitulation des classes logarithmiques dans la Zℓ-tour cyclotomique. Dans le cas complètement décomposé, nous prouvons de même que le critère suffisant introduit par Gras exprime tout simplement la trivialité du ℓ-groupe des classes logarithmiques du corps de base considéré.

Mots clés: Greenberg conjecture; logarithmic class groups; Iwasawa theory

Domaines: Théorie des nombres [math.NT]

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