Elliptic units and a Kummer's criterion for imaginary quadratic fields
type: Item
instance of: scholarly article
title: Elliptic units and a Kummer's criterion for imaginary quadratic fields
zbMATH Open document ID: 0617.12004
DOI: 10.1016/0022-314X(87)90015-1
author: Hiroshi Saito
published in: Journal of Number Theory
publication date: 1987-01-01T00:00:00Z
review text: Adding to the recent investigations of \textit{J. Coates} and \textit{A. Wiles} [Algebr. Number Theory, Pap. Kyoto int. Symp. 1976, 9-23 (1977; Zbl 0369.12009)] and especially \textit{G. Robert} [Bull. Soc. Math. Fr., Suppl., Mém. No.36 (1973; Zbl 0314.12006); Ann. Sci. Éc. Norm. Supér., IV. Sér. 11, 297-389 (1978; Zbl 0409.12008)], the author discusses a criterion for the existence of certain cyclic extensions of prime degree p of abelian extensions of an imaginary quadratic field K. The paper is based essentially on the author's modification of the results on elliptic units belonging to C. L. Siegel, K. Ramachandra and G. Robert. The criterion is stated in the form of the divisibility of special values of L-functions with Grössencharacters by \({\mathfrak p}\), a prime divisor of p in K, instead of Robert's independence of Hurwitz numbers modulo \({\mathfrak p}\), and what is more, this criterion gives a necessary and sufficient condition in the problem.
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11R27
Mathematics Subject Classification ID: 11R42
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 14H52
zbMATH DE Number: 4000056
zbMATH Keywords: class number
zbMATH Keywords: cyclic extensions
zbMATH Keywords: abelian extensions
zbMATH Keywords: imaginary quadratic field
zbMATH Keywords: elliptic units
zbMATH Keywords: values of L-functions
zbMATH Keywords: Grössencharacters
MaRDI profile type: Publication
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full work available at URL: 0022-314x(87)90015-1
OpenAlex ID: W2046412762
description: scientific article; zbMATH DE number 4000056