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On twisting operators and newforms of half-integral weight II – Complete theory of newforms for Kohnen space

Published online by Cambridge University Press:  22 January 2016

Masaru Ueda
Masaru Ueda*
Affiliation:
Department of Mathematics, Faculty of Science, Nara Women’s University, Nara 630, Japan, m-ueda@cc.nara-wu.ac.jp
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Abstract.

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The author continues his previous work with the purpose of establishing a theory of newforms in the case of half-integral weight. In this paper, the author formulates and proves a complete theory of newforms for Kohnen space. Kohnen space is an important canonical subspace in the space of cusp forms of half-integral weight k + 1/2 (k > 0). Every Hecke common eigenform in Kohnen space corresponds to a primitive form of integral weight 2k and of odd level via Shimura Correspondence.

These newforms for Kohnen space satisfy the Strong multiplicity One theorem. Moreover, we explicitly determine the corresponding primitive form (of weight 2k) to each newform for Kohnen space. The space of oldforms is also explicitly described.

In order to find all newforms for Kohnen space, the author needs a certain non-vanishing property of Fourier coefficients of cusp forms. Such property proves by using representation theory of finite linear groups. The method of proof of newform theory is mainly based on trace formulae and trace relations.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 149 , March 1998 , pp. 117 - 171
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1998

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