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On the biquadratic Gauss sum

Published online by Cambridge University Press:  24 October 2008

Andrew D. McGettrick
Andrew D. McGettrick
Affiliation:
Department of Computer Science at the University of Strathclyde, Glasgow
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Extract

1. Professor J. W. S. Cassels (in (l)) recently enunciated a conjecture concerning Kummer sums. His conjecture suggests that there is a close connexion between the Kummer sum and a certain product of Weierstrass elliptic functions.

In this paper we consider the biquadratic Gauss sum. A theory similar to that used in (1) and in (4) suggests that there is a corresponding connexion between the biquadratic sum and a product of Jacobian elliptic functions. The enunciation of this conjecture appears in section 6. But first of all we require some preliminary results.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 71 , Issue 1 , January 1972 , pp. 79 - 83
Copyright
Copyright © Cambridge Philosophical Society 1972

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References

REFERENCES

(1)Cassels, J. W. S.On Kummer sums. Proc. London Math. Soc. (3) 21 (1970), 19–27.CrossRefGoogle Scholar
(2)Eisenstein, G.Mathematische Abhandlungen (Georg Olms; Hildesheim, 1967).Google Scholar
(3)Kubota, T.Some arithmetical applications of an elliptic function. J. Reine Angew Math. 214/215 (1964), 141–145.CrossRefGoogle Scholar
(4)McGettrick, A. D. A result in the theory of Weierstrass elliptic functions. To appear in Proc. London Math. Soc.Google Scholar
(5)McGettrick, A. D. On Gaussian Sums. Ph.D. Dissertation, Cambridge (1969).Google Scholar