On indefinite modular forms of weight one
On indefinite modular forms of weight one
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关于重量一的不定模形式
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发表时间：
1986
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通讯作者：
Y. Mimura
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作者：
T. Hiramatsu;Noburo Ishii;Y. Mimura
如我们以前的文章（[3]，[4]）所示，虚二次域上的类域与Hecke意义下的“neben型”权1的尖点形式之间有着深刻的联系。本文研究了真实的二次域上的类域满足Shintani（[13]）条件的一个类似问题。本文共分五个部分。在第一节中，我们回顾了与真实的二次域（[1]，[2]，[10]）相联系的Hecke的权为1的不定模形式的定义。在第二节中，我们总结了Shintani关于可转移到虚二次情形的真实的二次问题的某些结果（[13]）。第三节将Shintani的结果应用到我们的问题中，得到了一些权为1的二面尖形的两种表示：正定θ级数表示和不定θ级数表示。Kac和Peterson在[7]中给出了许多由Dedekind eta函数产生的权为1的尖点形式的新恒等式的例子。在第4节中，我们将从我们的观点出发，利用第3节的结果，重新构造这些例子。在最后一节中，我们建立了某些真实的二次域上射线类场的定义方程的高重性律。作者谨向N教授表示诚挚的感谢。岩堀向他们介绍了卡茨和彼得森的工作（[7]）。我们的工作特别受到这项工作的启发。
As shown in our previous papers ([3], [4]), there are deep relations between the class fields over imaginary quadratic fields and cusp forms of weight one of “neben typus” in Hecke’s sense. In this paper we study a similar problem for class fields over real quadratic fields which satisfy a condition due to Shintani ([13]). The paper consists of five sections. In Section 1 we recall the definition of Hecke’s indefinite modular forms of weight one which are associated to real quadratic fields ([1], [2], [10]). In Section 2 we summarize certain results of Shintani for the real quadratic problem which is transferable to the imaginary quadratic situation ([13]). In Section 3 we apply the result of Shintani to our problem and obtain the two representations for some dihedral cusp forms of weight one by positive definite theta series and indefinite theta series. Kac and Peterson in [7] gave many examples of new identities for cusp forms of weight one which arise from the Dedekind eta function. In Section 4 we shall reconstruct these examples from our point of view, by using the results of Section 3. In the final section we establish the higher reeiprocity law for a defining equation of ray class fields over some real quadratic fields. The authors would like to express their sincere thanks to Professor N. Iwahori for informing them of the work of Kac and Peterson ([7]). Our work has been particularly inspired by this work.