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On the Cubic Theta Function

Published online by Cambridge University Press:  22 January 2016

Akinori Yoshimoto
Akinori Yoshimoto*
Affiliation:
Department of Mathematics Faculty of Science Nagoya University, Japan
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The generalized theta function of a totally imaginary field including n-th roots of unity, which was defined by T. Kubota [2], was introduced in his investigation of the reciprosity law of the n-th power residue. If n = 2, it reduces to the classical theta function. In the case n = 3 for the Eisenstein field, the Fourier coefficients of the cubic theta function, which were explicitly expressed by S.J. Patterson, are essentially cubic Gauss sums [3], Furthermore in the case n = 4 for the Gaussian field those of the biquadratic theta functions, which have been investigated by T. Suzuki [4], haven’t been obtained completely yet.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 105 , March 1987 , pp. 153 - 167
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1987

References

[ 1 ] Davenport, H. and Hasse, H., Die Nullstellen der Kongruenzzetafunktionen in gewissen zyklischen Fallen, J. reine angew. Math., 172 (1934), 151–182.Google Scholar
[ 2 ] Kubota, T., Some results concerning reciprosity law and real analytic automorphic functions, Proc. Sympos. Pure Math., 20 (1971), 382–395.Google Scholar
[ 3 ] Patterson, S. J., A cubic analogue of the theta series, J. reine angew. Math., 296 (1977), 125–161.Google Scholar
[ 4 ] Suzuki, T., Some results on the biquadratic theta series, J. reine angew Math., 340 (1984), 70–117.Google Scholar
[ 5 ] Weil, A., Uber die Bestimmung Dirichletscher Reiehen durch Funktionalgleichungen, Math. Ann., 168 (1967), 149–156.Google Scholar