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An Application of the Power Residue Theory to Some Abelian Functions

Published online by Cambridge University Press:  22 January 2016

Tomio Kubota
Tomio Kubota*
Affiliation:
Mathematical Institute, Nagoya University
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The aim of this paper is to point out a number-theoretical property of a certain product of values taken by an abelian function at points of finite order of the abelian variety on which the abelian function is considered We use the theory of power residues and the theory of complex multiplication, and the result is somewhat similar to the so-called Stickelberger’s relation in the theory of Gauss sums. A related investigation was previously made by the author in a special case of elliptic curves [2].

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 27 , Issue 1 , February 1966 , pp. 51 - 54
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1966

References

[1] Hasse, H., Zetafunktion und L-Funktion zu einem arithmetischen Funktionenkörper vom Fermatschen Typus, Abh. Deutsch. Akad. der Wiss. zu Berlin, Klasse für Math. und allg. Naturw., Jahrg. 1954, Heft 4.Google Scholar
[2] Kubota, T., Reciprocities in Gauss’ and Eisenstein’s number fields, J. reine angew. Math. 208 (1961), 35–50.CrossRefGoogle Scholar
[3] Shimura, G. and Taniyama, Y., Complex multiplication of abelian varieties and its application to number theory, Publ, of the Math. Soc. of Japan 6 (1961).Google Scholar
[4] Weil, A., Jacobi sums as Grössencharaktere, Trans. Amer. Math. Soc. 73 (1952), 487–495.Google Scholar