Algorithmic complexity of Greenberg’s conjecture

Let k be a totally real number field and p a prime. We show that the “complexity” of Greenberg’s conjecture ( λ = μ = 0 ) is governed (under Leopoldt’s conjecture) by the finite torsion group T k of the Galois group of the maximal abelian p -ramified pro- p -extension of  k , by means of images, in...

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Bibliographic Details
Published in:Archiv der Mathematik Vol. 117; no. 3; pp. 277 - 289
Main Author: Gras, Georges
Format: Journal Article
Language:English
Published: Cham Springer International Publishing 01.09.2021
Springer Nature B.V
Subjects:
Algorithms
Complexity
Mathematics
Mathematics and Statistics
Norms
Number theory
Theorems
Greenberg’s conjecture
11R29
Class field theory
Iwasawa’s theory
ramification theory
11R23
Class groups
11R37
adic regulators
11Y40
ISSN:0003-889X, 1420-8938
Online Access:Get full text
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Summary:Let k be a totally real number field and p a prime. We show that the “complexity” of Greenberg’s conjecture ( λ = μ = 0 ) is governed (under Leopoldt’s conjecture) by the finite torsion group T k of the Galois group of the maximal abelian p -ramified pro- p -extension of  k , by means of images, in T k , of ideal norms from the layers k n of the cyclotomic tower (Theorem 4.2 ). These images are obtained via the algorithm computing, by “unscrewing”, the p -class group of k n . Conjecture 4.3 of equidistribution of these images would show that the number of steps b n of the algorithms is bounded as n → ∞ , so that (Theorem 3.3 ) Greenberg’s conjecture, hopeless within the sole framework of Iwasawa’s theory, would hold true “with probability 1”.
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ISSN:0003-889X
1420-8938
DOI:10.1007/s00013-021-01618-9