On zeta-functions and cyclotomic \({\mathbb{Z}}_ p\)-extensions of algebraic number fields
type: Item
instance of: scholarly article
title: On zeta-functions and cyclotomic \({\mathbb{Z}}_ p\)-extensions of algebraic number fields
zbMATH Open document ID: 0557.12008
author: Keiichi Komatsu
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1984-01-01T00:00:00Z
review text: Let \(\zeta_ k\) and \(\zeta_{k'}\), be Dedekind zeta functions of function fields k and k' respectively. Then Tate and Turner gave a necessary and sufficient condition for \(\zeta_ k=\zeta_{k'}\), by using Jacobian varieties J(\({\mathcal C})\) and J(\({\mathcal C}')\) of complete non-singular curves \({\mathcal C}\) and \({\mathcal C}'\) with function fields isomorphic to k and k' respectively [cf. \textit{J. Tate}, Invent. Math. 2, 134-144 (1966; Zbl 0147.203); and \textit{S. Turner}, Bol. Soc. Bras. Mat. 9, 89-95 (1978; Zbl 0427.12011)]. In this paper, the author intends to get an analogue of this result in algebraic number fields. Namely, for Dedekind zeta functions \(\zeta_ k\) and \(\zeta_{k'}\), of algebraic number fields k and k' respectively, he gives some necessary conditions for \(\zeta_ k=\zeta_{k'}\) by using the Galois group of the maximal (unramified) abelian pro-p extension over \(k_{\infty}\) \((k'_{\infty})\) or the Pontryagin dual of the p-primary subgroup of the ideal class group of \(k_{\infty}\) \((k'_{\infty})\), where \(k_{\infty}\) is the cyclotomic \({\mathbb{Z}}_ p\)-extension of k.
reviewed by: Hideo Yokoi
Mathematics Subject Classification ID: 11R42
Mathematics Subject Classification ID: 11R18
zbMATH DE Number: 3887809
zbMATH Keywords: maximal unramified abelian pro-p-extension
zbMATH Keywords: Dedekind zeta functions
zbMATH Keywords: Galois group
zbMATH Keywords: cyclotomic \({\mathbb{Z}}_ p\)-extension
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178228762
OpenAlex ID: W2092862567
cites work: A note on class numbers of algebraic number fields
cites work: On \(p\)-adic \(L\)-functions
cites work: On \(\mathbb Z_{\ell}\)-extensions of algebraic number fields
cites work: On the class numbers of arithmetically equivalent fields
cites work: Endomorphisms of Abelian varieties over finite fields
cites work: Adele rings of global field of positive characteristc
cites work: Isomorphisms of Galois groups of solvably closed Galois extensions
cites work: On the Class Number of a Relatively Cyclic Number Field
DOI: 10.2748/TMJ/1178228762
description: scientific article; zbMATH DE number 3887809