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Theta-functions and Hilbert modular forms

Published online by Cambridge University Press:  22 January 2016

Stephen S. Kudla
Stephen S. Kudla*
Affiliation:
Department of Mathematics, University of Maryland
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The purpose of this note is to show how the theta-functions attached to certain indefinite quadratic forms of signature (2, 2) can be used to produce a map from certain spaces of cusp forms of Nebentype to Hilbert modular forms. The possibility of making such a construction was suggested by Niwa [4], and the techniques are the same as his and Shintani’s [6]. The construction of Hilbert modular forms from cusp forms of one variable has been discussed by many people, and I will not attempt to give a history of the subject here. However, the map produced by the theta-function is essentially the same as that of Doi and Naganuma [2], and Zagier [7]. In particular, the integral kernel Ω(τ, z1, z2) of Zagier is essentially the ‘holomorphic part’ of the theta-function.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 69 , February 1978 , pp. 97 - 106
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1978

References

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