On the modular forms of weight 1/2 over algebraic number fields

Serre and Stark succeeded in deciding a basis of the space of modular forms of weight 1/2 over the rational number field. Achimescu and Saha generalized their result to the case of modular forms of weight 1/2 over totally real algebraic number fields. Gove also solved this problem in the case of mod...

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Bibliographic Details
Published in:Journal of number theory 2021-12, Vol.229, p.364-385
Main Authors: Kojima, Hisashi, Sakata, Hiroshi
Format: Article
Language:English
Subjects:
Automorphic forms on [formula omitted]
Forms of half-integer weight
Theta series
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Summary:Serre and Stark succeeded in deciding a basis of the space of modular forms of weight 1/2 over the rational number field. Achimescu and Saha generalized their result to the case of modular forms of weight 1/2 over totally real algebraic number fields. Gove also solved this problem in the case of modular forms of weight 1/2 over imaginary quadratic fields. In this paper, we determine an explicit basis of the space of modular forms of weight 1/2, level c and character ψ over algebraic number fields. We prove our assertion using their arguments and Shimura's transformation formula of theta series over algebraic number fields.
ISSN:0022-314X
1096-1658
DOI:10.1016/j.jnt.2021.04.009