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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| Vydáno v: | Archiv der Mathematik Ročník 117; číslo 3; s. 277 - 289 |
|---|---|
| Hlavní autor: | |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Cham
Springer International Publishing
01.09.2021
Springer Nature B.V |
| Témata: | |
| ISSN: | 0003-889X, 1420-8938 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | 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”. |
|---|---|
| Bibliografie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0003-889X 1420-8938 |
| DOI: | 10.1007/s00013-021-01618-9 |