ON AUTOMORPHIC L-FUNCTIONS OF GENERAL SYMPLECTIC AND UNITARY GROUPS OF RANK TWO

关于一般辛和二阶酉群的自同构L函数

基本信息

批准号：
16540034
负责人：
FURUSAWA Masaaki
金额：
$ 2.3万
依托单位：
OSAKA CITY UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2006
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16540034/
关键词：
Relative Trace Formula Automorphic L-function Siegel modular form Special Value of L-function

项目摘要

We have continued the projects concerning the automorphic L-functions of gereral symplectic and unitary groups of rank two. More specifically one of the main projects is to prove the generalization of Siegfried Boecherer's conjecture concerning the central critical values of the degree four L-functions for the Siegel eigen cusp forms of degree two. Our method is to establish certain relative trace formulas, which may be regarded as natural generalizations of Jacquet's relative trace formulas which have given another proof of celebrated Waldspurger's theorem on the relation between the torus period for GL(2) and the central critical values of automorphic L-functions for GL(2).In order to establish a relative trace formula, proving the fundamental lemma is the first and crucial step. We have proved the fundamental for the unit element of the Hecke algebra already and published the result as No. 782 of the Memoirs of the AMS. During the period supported by this grant, we worked on extending the fundamental lemma from the unit element to the entire Hecke algebra. We have discovered that, by applying the theory of Macdonald polynomials to the explicit formulas for the Bessel model, the evaluation of the Kloosterman orbital integral for the general element in the Hecke algebra is reduced to the computation of general Kostka numbers and that of degenerate Kloosterman orbital integrals for the unit element of the Hecke algebra. We have evaluated all of them. Now our remaining task is to compare the linear combinations of these corresponding to the both sides of the trace formula and to make sure they match.

我们继续进行了有关二等级别和统一群体的自动形态L功能的项目。更具体地说，主要项目之一是证明Siegfried Boecherer关于第二学位的siegel Eigen cusp的四个L函数的中心临界值的猜想的概括。我们的方法是建立某些相对痕量公式，这些公式可以被视为雅克相对痕量公式的自然概括，这些公式的另一个证明了著名的Waldspurger定理，waldpurger的定理是GL（2）与GL（2）的自动化l（2）的中心关键价值（2）。我们已经证明了Hecke代数的单位要素的基本要素，并将结果发布为AMS回忆录的第782号。在这笔赠款支持的期间，我们致力于将基本引理从单位元素扩展到整个Hecke代数。 We have discovered that, by applying the theory of Macdonald polynomials to the explicit formulas for the Bessel model, the evaluation of the Kloosterman orbital integral for the general element in the Hecke algebra is reduced to the computation of general Kostka numbers and that of degenerate Kloosterman orbital integrals for the unit element of the Hecke algebra.我们已经评估了所有这些。现在，我们剩下的任务是比较这些对应于跟踪公式的两侧的线性组合，并确保它们匹配。