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.

我们证明了Hecke代数中的单位元素的基本引理，用于GSP的两个相对痕量公式（4）。我们的最终目标是证明Bocherer对与全体形态siegel eigen eigen cusp的第二型旋转曲线的焦点临界值的猜想。基本引理的公告已在C. R. Acad发表。科学。巴黎和证明的细节将出现在其他地方。在基本引理的证据过程中，我们根据经典的GL（2）Kloosterman总和明确评估了某些矩阵参数Kloosterman总和。我们指出，我们的Kloosterman总和是广义Kloosterman和的特殊情况，它出现在Siegel模块化组的Poincare系列的傅立叶系数中。我们对Kloosterman总和的结果可能具有一定的独立兴趣，因为很少对这种广义的Kloosterman进行明确评估。我们的第二个猜想痕量公式与GSP的二次基础电荷有关（4）。我们的结果表明，GL（2）的二次基础变化的jacquet-ye标准概括为GSP（4）。这显然值得进一步调查。最后，我们的结果意味着，当我们研究自动形态L功能的特殊值时，研究整个L包装很重要。澄清我们结果预期的特殊价值的时期与迪格尼的猜想之间的关系似乎很有趣。