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Proceedings of the American Mathematical Society

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A zeta-function associated with zero ternary forms


Author: Min King Eie
Journal: Proc. Amer. Math. Soc. 94 (1985), 387-392
MSC: Primary 11E45; Secondary 11F99, 11M41
MathSciNet review: 787878
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Abstract: Consider the zeta-function associated with zero ternary forms defined as

$\displaystyle \tilde \xi (t) = \sum\limits_x {\frac{1} {{{{\left\vert {\det x} \right\vert}^t}}}} \quad (\operatorname{Re} t \geqslant 2),$

where $ x$ runs over all $ \operatorname{SL}_{3}({\mathbf{Z}})$-inequivalent zero ternary forms. We shall approximate $ \tilde \xi (t)$ by another zeta-function which we can compute explicitly. By the approximation, we see that $ \tilde \xi (2)$ is very close to $ 2\zeta (2)\zeta (2)$ which gives the contribution of zero ternary forms to the dimension formula of Siegel's cusp forms of degree three (computing via Selberg Trace Formula) up to a constant multiple.

References [Enhancements On Off] (What's this?)

  • [1] Takuro Shintani, On zeta-functions associated with the vector space of quadratic forms, J. Fac. Sci. Univ. Tokyo Sect. I A Math. 22 (1975), 25–65. MR 0384717
  • [2] C. L. Siegel, Lectures on quadratic forms, Notes by K. G. Ramanathan. Tata Institute of Fundamental Research Lectures on Mathematics, No. 7, Tata Institute of Fundamental Research, Bombay, 1967. MR 0271028
  • [3] Carl Ludwig Siegel, Über die Zetafunktionen indefiniter quadratischer Formen, Math. Z. 43 (1938), no. 1, 682–708 (German). MR 1545742, 10.1007/BF01181113

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DOI: https://doi.org/10.1090/S0002-9939-1985-0787878-9
Article copyright: © Copyright 1985 American Mathematical Society