Elliptic units and a Kummer's criterion for imaginary quadratic fields
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.
scientific article; zbMATH DE number 4000056
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