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.

我们继续进行有关一般辛群和二阶酉群的自同构 L 函数的项目。更具体地说，主要项目之一是证明 Siegfried Boecherer 关于二阶 Siegel 本征尖点形式的四阶 L 函数的中心临界值的猜想的推广。我们的方法是建立某些相对迹公式，这可以被视为 Jacquet 相对迹公式的自然推广，它为著名的 Waldspurger 定理提供了另一个证明，该定理关于 GL(2) 的环面周期与自同构的中心临界值之间的关系GL(2)的L函数。为了建立相对迹公式，证明基本引理是第一步，也是关键的一步。我们已经证明了赫克代数单位元的基本原理，并将其结果发表为 AMS 回忆录第 782 期。在这笔资助的支持期间，我们致力于将基本引理从单位元扩展到整个赫克代数。我们发现，通过将麦克唐纳多项式理论应用于贝塞尔模型的显式公式，赫克代数中一般元素的克洛斯特曼轨道积分的计算被简化为一般科斯特卡数和简并克洛斯特曼数的计算赫克代数单位元的轨道积分。我们已经对所有这些进行了评估。现在我们剩下的任务是比较迹公式两边对应的线性组合并确保它们匹配。