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Shimara Cerresponchence of Hilbort modular forms

希尔伯特模块化形式的 Shimara Cerresponchence

基本信息

批准号：
09640028
负责人：
TSUYUMINE Shigeaki
金额：
$ 1.79万
依托单位：
Mie University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1998
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640028/
关键词：
totally real algebraic number field Hilbert Modular form L-function quadratic form Eisenstein series 志村対応保型形式 Hilberc保型形式

项目摘要

Let K be a totally real algebraic number field. We consider the Hilbert-Eisenstein series on K.At first let .K be real and quadratic. Then we discovered some relation between the elliptic modular forms obtained by restricting the Hilbert-Eisenstein series to the diagonal, and modular forms of half integral weight which are products of theta series and Eisensteinseries. By this it is shown that all modular forms of weight at least 5/2can be lifted to modular forms of integral weight in Shimura's sense (notethat this has been known only for cusp forms). As the application, we canobtain formulas for special values of the Dirichlet L-functions by computing Fourier coefficients of the modular forms, as well as relations between some arithmetic functions.Secondly let K be a general totally real algebraic number field. Let F bea totally real algebraic number filed which is a quadratic extension of K.We consider the Hilbert modular forms on K obtained by restricting the Hilbert-Eisenstein series on F to K, and investigate how they work on number theory of a totally algebraic number fields K.When the structure of graded ring of Hilbert modular forms of K is known, this method works well as in the case of elliptic modular forms. We discovered several formulas for special values for Dede kind zeta functions or the number of representations of positive quadratic forms over K.We investigate also the Shimura correspondence of Hilbert modular forms over K.As a result it is prove that the Hilbertmodular form in the form of (theta series) x Eisenstein series, can be lifted to a Hilbert modular form of integral weight. In the elliptic case it follows from this that all modular form of half integral weight can be lifted, however in this case we need further investigation.

设K是一个全实数代数域。我们考虑k上的Hilbert-Eisenstein级数。K是实数和二次的。然后发现了将Hilbert-Eisenstein级数限定在对角线上得到的椭圆模形式与由级数与eisenstein级数乘积而成的半积分权的模形式之间的某种关系。由此证明，所有至少5/2的权的模形式都可以提升为志村意义上的积分权的模形式（注意，这只对尖形式已知）。作为应用，我们可以通过计算模形式的傅里叶系数得到狄利克雷l函数的特殊值的公式，以及一些算术函数之间的关系。其次设K为一般全实数代数域。设F为全实数域，它是K的二次扩展。我们考虑通过限制F上的Hilbert- eisenstein级数到K而得到的K上的Hilbert模形式，并研究它们如何在全代数数域K的数论上起作用。当K的Hilbert模形式的梯度环的结构已知时，这种方法与椭圆模形式的情况一样有效。我们还研究了k上Hilbert模形式的Shimura对应关系。结果证明了（级数）x爱森斯坦级数形式的Hilbert模形式可以提升为积分权的Hilbert模形式。在椭圆情况下，由此可以得出半积分权的所有模形式都可以举起，但在这种情况下，我们需要进一步研究。