Abstract
Greenberg’s well known conjecture, (GC) for short, asserts that the Iwasawa invariants \(\lambda \) and \(\mu \) associated to the cyclotomic \({\mathbb {Z}}_p\)-extension of any totally real number field F should vanish. In his foundational 1976 paper, Greenberg has shown two necessary and sufficient conditions for (GC) to hold, in two seemingly opposite cases, when p is undecomposed, resp. totally decomposed in F. In this article we present an encompassing approach covering both cases and resting only on “ genus formulas ”, that is (roughly speaking) on formulas which express the order of the Galois (co-)invariants of certain modules along the cyclotomic tower. These modules are akin to class groups, and in the end we obtain several unified criteria, which naturally contain the particular conditions given by Greenberg.
Résumé
La conjecture bien connue de Greenberg, (CG) en abrégé, prédit la nullité des invariants \(\lambda \) et \(\mu \) d’Iwasawa attachés à la \({\mathbb {Z}}_p\)-extension cyclotomique de tout corps de nombres totalement réel F. Dans son article fondateur de 1976 , Greenberg a montré deux conditions nécessaires et suffisantes pour la validité de (CG) dans deux cas apparemment opposés, quand p est non décomposé, resp. totalement décomposé dans F. Dans cet article de synthèse, suivant une approche qui repose uniquement sur des “formules de genres”, i.e., sans trop de précision, des formules donnant l’ordre des (co)-invariants galoisiens de certains modules apparentés aux groupes de classes le long de la tour cyclotomique, nous produisons plusieurs critères unifiés qui contiennent naturellement les conditions particulières de Greenberg.
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Nguyen Quang Do, T. Formules de genres et conjecture de Greenberg. Ann. Math. Québec 42, 267–280 (2018). https://doi.org/10.1007/s40316-017-0093-y
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DOI: https://doi.org/10.1007/s40316-017-0093-y