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The inner product of an automorphic wave form with the pullback of an Eisenstein series

Published online by Cambridge University Press:  22 January 2016

Shinji Niwa
Shinji Niwa*
Affiliation:
Nagoya City College of Child Education, Owariasahi, 488, Japan
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In this paper we shall show a relation between a special value of an automorphic wave form and the inner product of the automorphic wave form with the pullback of an Eisenstein series on the upper half space. The main theorem is Theorem 3 in the end of this paper. As is shown in P. B. Garrett [13], pullbacks of Eisenstein series on Siegel upper half spaces have interesting properties as a kernel function of an integral operator. It is natural to try to investigate pullbacks of Eisenstein series of Hilbert type. We can say that Theorem 3 clarifies a property of such pullbacks in a special case. The idea of the proof is a lifting of automorphic forms by theta functions. We discuss a lifting of automorphic wave forms in 1, 2 and 3, and obtain Theorem 2 in the end of 3 as a result. We can prove Theorem 3 without much difficulty by using Theorem 2.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 108 , December 1987 , pp. 93 - 119
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1987

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