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Some examples of integral definite quaternary quadratic forms with prime discriminant

Published online by Cambridge University Press:  22 January 2016

Ki-Ichiro Hashimoto
Ki-Ichiro Hashimoto*
Affiliation:
Department of Mathematics, University of Tokyo
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In the theory of integral quadratic forms or the related theory such as modular forms, it is important to give examples of forms with given discriminant for various computational purposes. However, little is known about this problem even in the case where the explicit formula for the class number has been given. Among the works that deal with satisfactorily many examples or give a constructive method of them, there are [1] for ternary forms, and [5] for quaternary forms with square discriminants. As for quaternary forms with prime discriminant, P. Ponomarev [8] gave an explicit way to construct such forms from ternary forms with discriminant 2q. It was based on the result of his previous work [7] and that of Y. Kitaoka [6], but was rather complicated.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 77 , February 1980 , pp. 167 - 175
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1980

References

[1] Brandt, H. and Intrau, O., Tabellen reduzierter positiver ternärer quadratischer Formen, Akademie-Verlag, Berlin, 1958.Google Scholar
[2] Eichler, M., Theta function over Q and Lecture Notes in Math. Nr. 627, Springer-Verlag, Berlin-New York, 1977, 197–225.Google Scholar
[3] Hashimoto, K., Twisted trace formula of the Brandt matrix, Proceedings of the Japan Academy, Vol. 53, Ser A, No. 3 (1977), 98–102.Google Scholar
[4] Hashimoto, K., On the arithmetic of quadratic extension of quaternion algebras (Japanese), master thesis, University of Tokyo, 1977.Google Scholar
[5] Ibukiyama, T., A basis and maximal orders in quaternion algebras over the rational number field (Japanese), Sugaku, 24 (1972), 316–318.Google Scholar
[6] Kitaoka, Y., Quaternary even positive definite quadratic forms of prime discriminant, Nagoya Math. J., 52 (1973), 147–161.CrossRefGoogle Scholar
[7] Ponomarev, P., A correspondence between quaternary quadratic forms, Nagoya Math. J., 62 (1976), 125–140.CrossRefGoogle Scholar
[8] Ponomarev, P., Ternary quadratic forms and an explicit quaternary correspondences, Conference on Quadratic Forms, Queen’s Papers in Pure and Applied Math. No. 46 (1977), 582–594.Google Scholar