Formules de genres et conjecture de Greenberg

Greenberg’s well known conjecture, (GC) for short, asserts that the Iwasawa invariants λ and μ associated to the cyclotomic 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, i...

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Bibliographic Details
Published inAnnales mathématiques du Québec Vol. 42; no. 2; pp. 267 - 280
Main Author Nguyen Quang Do, Thong
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.10.2018
Springer Nature B.V
Subjects
Algebra
Analysis
Invariants
Mathematics
Mathematics and Statistics
Modules
Number Theory
Greenberg’s conjecture
11 R 23
Genus formulas
Unified criteria
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Summary:Greenberg’s well known conjecture, (GC) for short, asserts that the Iwasawa invariants λ and μ associated to the cyclotomic 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.
ISSN:2195-4755
2195-4763
DOI:10.1007/s40316-017-0093-y