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Arithmetic study of Fourier coefficients of automorphic forms and modular forms and zeta functions

自守形式和模形式的傅里叶系数以及zeta函数的算术研究

基本信息

批准号：
11640004
负责人：
KOJIMA Hisashi
金额：
$ 2.3万
依托单位：
Iwate University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2001
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640004/
关键词：
modular forms of half integral weight modular forms zeta function Jacobi forms specaial values of zeta function Jacobi形式半整数の重さのモジュラー形式モジュラー形式ゼータ関数の特殊値モジュラー形式のフーリエ係数

项目摘要

Kojima's results(1) Under the assumption of the multiplicity 2 theorem, we determined an explicit relation between the square of Fourier coefficients of modular forms f belonging to the Kohnen's space of half integral weight and of arbitrary oddlevel with arbitrary primitive character and special values of the zeta function of the modular form F which is the image of f under the Shimura correspondence.(2) We constructed the Shimura correspondence S from Maass wave forms f of half integral weight over imaginary quadratic fields to those g of integral weight. We shall determine explicitly the Fourier coefficients of g in terms of these of f. Under some assumptions about the multiplicity one theorem with respect to Hecke operators, we deduced an explicit connection between the square of Fourier coefficients of modular forms f of half integral weight over imaginary quadratic fields and the critical value of the zeta function associated with S(f). Moreover, we generalized those results in the case of Maass wave forms f of half integral weight over arbitrary number fields. This yield a generalization of Shimura's formula concerning Fourier coefficients of Hilbert modular forms f of half integral weight over imaginary quadratic fields.(3) Under the assumptions about f concerning the multiplicity one theorem with respect to Hecke operators, we deduced an explicit connection between the square of Fourier coefficients of f and the critical value of the zeta function associated with the image of Shimura correspondence, which gives a further concise improvement of the results (1)(4) We determined explicitly the trace of representations of certain linear group over finite field into the spaces of modular forms of half integral weight, Jacobi forms and automorphic forms on SU(2,1). In the some case, we can determine the multipicity of representation.

(1)在多重性2定理的假设下，我们确定了属于半积分权的Kohnen空间的任意奇级具有任意基元特征的模形式f的傅里叶系数的平方与模形式f的特殊值之间的显式关系，该函数是f在Shimura对应下的像。(2)构造了虚二次场上半积分权质量波形f到积分权质量波形g的Shimura对应S。我们将用f的傅里叶系数显式地确定g的傅里叶系数。在关于Hecke算子的多重定理的一些假设下，我们推导出虚二次域上的半积分权模形式f的傅里叶系数的平方与与S(f)相关的ζ函数的临界值之间的显式联系。此外，我们将这些结果推广到任意数场上的半积分权的质量波形f。这是Shimura关于希尔伯特模形式f在虚二次域上的半积分权的傅里叶系数公式的推广。(3)在关于f关于Hecke算子多重定理的假设下，我们推导出f的傅里叶系数的平方与与Shimura对应像相关的zeta函数的临界值之间的显式联系；(4)我们明确地确定了有限域上某线性群的表示在SU（2,1）上的半积分权模形式、Jacobi形式和自同构形式空间中的迹。在某些情况下，我们可以确定表示的多重性。