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On zeta functions associated with quadratic forms of variable coefficients

Published online by Cambridge University Press:  22 January 2016

Toshiaki Suzuki
Toshiaki Suzuki*
Affiliation:
Department of Mathematics, Nagoya University
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In 1938, C. L. Siegel studied zeta functions of indefinite quadratic forms ([6], c). On the other hand, M. Sato and T. Shintani constructed the general theory of zeta functions of one complex variable associated with prehomogeneous vector spaces in 1974 ([1]). Moreover T. Shintani studied several zeta functions of prehomogeneous vector spaces, especially, “Dirichlet series whose coefficients are class-numbers of integral binary cubic forms” ([3]) and “Zeta functions associated with the vector space of quadratic forms” ([2]).

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 73 , March 1979, pp. 117 - 147
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1979

References

[1] Sato, M. and Shintani, T., On zeta functions associated with prehomogeneous vector space, Ann. of Math. 1O0 (1974), 131–170.Google Scholar
[2] Shintani, T., On zeta functions associated with the vector space of quadratic forms, J. Fac. Science Univ. of Tokyo Sec. IA, Vol. 22, 25–65.Google Scholar
[3] Shintani, T., On Dirichlet series whose coefficients are class-numbers of integral binary cubic forms, J. Math. Soc. Japan 24 (1972), 132–188.Google Scholar
[4] Sato, M., Theory of prehomogeneous vector spaces (notes by Shintani, T. in Japanese), Sugaku no Ayumi 15 (1970), 85–157.Google Scholar
[5] Gel’fand, I. M. and Shilov, G. E., Generalized Functions, Vol. 1, Academic Press, New York, 1964.Google Scholar
[6a] Siegel, C. L., Uber die analytische Theorie der quadratischen Formen, Ann. of Math. 36 (1935), 527–606.CrossRefGoogle Scholar
[6b] Siegel, C. L., The volume of the fundamental domain for some infinite groups, Trans. Amer. Math. Soc. 39 (1936), 209–218.Google Scholar
[6c] Siegel, C. L., Uber die Zetafunctionen indefiniter quadratischer Formen, Math. Z. 43 (1938), 682–708.CrossRefGoogle Scholar
[7] Ono, T., On Tamagawa Numbers, Proc. Sympos. Pure Math., Vol. 9, Amer. Math. Soc., 1966, 122–132.Google Scholar
[8] Weil, A., Adeles and algebraic groups, Inst. Adv. Study, Princeton, 1961.Google Scholar
[9] Ono, T., A mean value theorem in adele geometry, J. Math. Soc. Japan, Vol. 20, Nos. 1-2, 1968, 275–288.Google Scholar
[10] Ono, T., On algebraic groups and discontinuous groups, Nagoya Math. J., Vol. 27, 1966, 279–322.Google Scholar
[11] Bernstein, I. N. and Gel’fand, S. I., Meromorphic property of the Functions P , Funct. anal, and its appl. 3(1) (1969), 84–86.Google Scholar
[12] Lang, S. and Weil, A., Number of points of varieties in finite fields, Amer. J. Math. 76 (1954).Google Scholar
[13] Sato, F., Eisenstein Series for Indefinite Quadratic Forms, (to appear).Google Scholar
[14] Suzuki, T., Fourier transforms of relative invariants of prehomogenous vector space, Master Thesis in Japanese, Nagoya Univ., 1975.Google Scholar