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Automorphic forms and infinite matrices

Published online by Cambridge University Press:  22 January 2016

Tomio Kubota
Tomio Kubota*
Affiliation:
Department of Mathematics, Nagoya University, Chikusaku Nagoya, 464-01, Japan
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In the present paper, we show that an infinite dimensional vector whose components are Fourier coefficients of an automorphic form is characterized as an infinite dimensional vector which is annihilated by an infinite matrix constructed by the values of a Bessel function. Results and methods are all simple and concrete.

Although the idea in the present paper is applicable to more general cases, our investigation will be restricted to the case of automorphic forms of weight 0, i.e., automorphic functions, with respect to SL(2, Z) on the upper half plane, in order to explain the main idea distinctly.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 127 , September 1992 , pp. 61 - 82
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1992

References

[ 1 ] Hejhal, D. A., The Selberg trace formula and the Riemann zeta function, Duke Math. J., 43-3(1976), 441–482.Google Scholar
[ 2 ] Kubota, T., Elementary of Eisenstein series, Kodansha Scientific, 1972.Google Scholar
[ 3 ] Deshouillers, J.-M. and Iwaniec, H., Kloosterman sums and Fourier coefficients of cusp forms, Invent. Math., 70 (1982), 219–288.Google Scholar