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Research on automorphic representations and L-functions

自守表示和L函数的研究

基本信息

批准号：
13640018
负责人：
IKEDA Tamotsu
金额：
$ 2.05万
依托单位：
KYOTO UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2004
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640018/
关键词：
Siegel modular forms L-functions ジーゲル保型形式 automorphic form automorphic L-function Siegel modular form hermitian modular form automorphic forms L-function

项目摘要

In late 70's, Saito and Kurokawa independently conjectured that there should be a lifting from elliptic modular forms to Siegel modular forms of degree 2. This conjecture was proved by Maass, Andrianov, Eichler, Zagier, Piatetsky-Shapiro and others in early 8O's. This is now called the Saito-Kurokawa lifts. In this research project, the author generalized the Saito-Kurokawa lifts to higher degrees, and gave an explicit Fourier coefficient formula. Moreover, the author proved an analogous lifting in hermitian modular case, and obtained a Fourier coefficient formula.The pullback of the Siegel modular form we constructed and be thought of a kernel function, and the author constructed another lifting by means of this kernel function. This lifting is now called the Miyawaki lifting.The author calculated the L-function of the Miyawaki lifting and gave a conjecture on the inner product of the Miyawaki lifting.The triple product L-function is investigated by many authors after Garrett's discovery of the integral expression.The special values of the triple product L-function has a dichotomy, the definite case and the indefinite case.The indefinite case is more difficult, and the result of Harris and Kudla seems the only result about the indefinite case.In author's joint work with Atsushi Ichino, it was shown that some indefinite special value of the triple product L-function can be expressed as the inner product of a hermitian lifting and the Saito-Kurokawa lifting.This result is compatible with the Gross-Prasad conjecture.The author is investigating a refinement of the Gross-Prasad conjecture in a joint work with Ichino.

在70年代末，Saito和Kurokawa独立地推测应该有一个从椭圆模形式到2次西格尔模形式的提升。这个猜想在80年代初被Maass、Andrianov、Eichler、Zagier、Piatetsky-Shapiro等人证明。这就是现在的齐藤-黑川电梯。在本研究项目中，作者将Saito-Kurokawa升降机推广到更高的程度，并给出了显式的傅里叶系数公式。此外，作者还证明了在厄米模情况下的类似提升，并得到了傅里叶系数公式。将西格尔模形式的回拉构造为一个核函数，并利用这个核函数构造了另一个提升。这种举法现在被称为宫崎举法。计算了宫崎举升的l函数，给出了宫崎举升内积的一个猜想。三重积l函数在加勒特发现积分表达式后被许多作者研究。三积函数的特殊值有二分类，即定值和不定值。不确定的情况更困难，哈里斯和库德拉的结果似乎是不确定情况的唯一结果。在与Atsushi Ichino的联合工作中，证明了三重积l函数的某些不定特殊值可以表示为hermite举升与Saito-Kurokawa举升的内积。这个结果与Gross-Prasad猜想是一致的。作者正在与一野共同研究Gross-Prasad猜想的改进。