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.

Serre 和 Stark 成功地确定了有理数域上权重为 1/2 的模形式空间的基。Achimescu 和 Saha 将他们的结果推广到完全实代数数域上权重为 1/2 的模形式的情况。Gove 在虚二次域上权重为 1/2 的模形式的情况下也解决了这个问题。

在本文中，我们确定了代数数域上权重 1/2、级别c和字符ψ的模形式的空间的显式基础。我们使用他们的论点和 Shimura 在代数数域上的 theta 级数变换公式来证明我们的断言。