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On arithmetic theory of automorphic forms and special values of automorphic L-functions

论自守形式的算术理论和自守L-函数的特殊值

基本信息

批准号：
10640028
负责人：
FURUSAWA Masaaki
金额：
$ 2.43万
依托单位：
Osaka City University (1999)Hiroshima University (1998)
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 1999
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640028/
关键词：
Siegel modular form automorphic L-function special value of L-function Deligne's conjecture relative trace formula trace formula ジーゲル保型形式保型形式の持ち上げクルスターマン和

项目摘要

We proved the fundamental lemma for the unit element in the Hecke algebra for two relative trace formulas for GSp(4). Our ultimate goal is to prove Bocherer's conjecture on the central critical values of the quadratic twists of the spinor L-functions associated to holomorphic Siegel eigen cusp forms of degree two. The announcements of the fundamental lemma have been published in C. R. Acad. Sci. Paris and the details of the proof will appear elsewhere.In the course of the proof of the fundamental lemma, we evaluated certain matrix argument Kloosterman sums explicitly in terms of the classical GL(2) Kloosterman sums. We remark that our Kloosterman sum is a special case of the generalized Kloosterman sum which appears in the Fourier coefficients of the Poincare series for the Siegel modular group. Our result on the Kloosterman sum may be of some independent interest, since it is rare that such generalized Kloosterman is evaluated explicitly.Our second conjectural trace formula is related to the quadratic base charge for GSp (4). Our result suggests that the Jacquet-Ye criterion for the quadratic base change for GL(2) generalizes to GSp(4). This clearly deserves some further investigation. Finally our result implies that it is important to study the whole L-packet when we study the special values of automorphic L-functions. It seems very interesting to clarify the relationship between the period part of the special value expected by our result and Deligne's conjecture.

证明了GSp（4）中两个相对迹公式的Hecke代数中单位元的基本引理。我们的最终目标是证明Bocherer猜想的中心临界值的旋量L-函数的二次扭曲的全纯Siegel本征尖点形式的程度。基本引理的公布已发表在C。R. Acad. Sci.巴黎，证明的细节将在其他地方出现。在基本引理的证明过程中，我们利用经典GL（2）Kloosterman和显式地计算了某些矩阵辐角Kloosterman和。我们注意到，我们的Kloosterman和是一个特殊的情况下，广义Kloosterman和出现在的Poincare级数的傅立叶系数的西格尔模群。我们关于Kloosterman和的结果可能有一些独立的意义，因为很少有这样的广义Kloosterman被显式地求值。我们的第二个几何迹公式与GSp（4）的二次基荷有关。我们的结果表明，GL（2）的二次基变的Jacquet-Ye准则推广到GSp（4）。这显然值得进一步调查。最后，我们的结果表明，在研究自守L-函数的特殊值时，研究整个L-包是很重要的。澄清我们的结果所期望的特殊值的周期部分与德利涅猜想之间的关系似乎是非常有趣的。