SPECIAL VALUES OF AUTOMORPHIC L-FUNCTIONS BY RELATIVE TRACE FORMULAS

用相对迹公式计算自同构L函数的特殊值

• 批准号: 13640037
• 负责人: FURUSAWA Masaaki
• 金额: $ 2.24万
• 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: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640037/
• 关键词: RELATIVE TRACE FORMULA; AUTOMORPHIC L-FUNCTION; SIEGE MODULAR FORM; SPECIAL VALUES OF L-FUNCTIONS; DELIGNE CONJECTURE; TRACE FORMULA

We have continued the project concerning the conjecture of Siegfried Boecherer and its generalization on the central critical values of the degree four L-functions for the Siegel eigen cusp form of degree two. Our method is to generalize Jacquet's two relative trace formulas which have given another proof of Waldspurger's theorem on the central critical values of L-functions for GL(2). In order to establish a trace formula, the first and crucial step is to prove the fundamental lemma. In a monograph published as No. 782 of Memoirs of the AMS, we gave a proof of the fundamental lemma for the unit element in the Hecke algebra. Our proof is based on the explicit evaluation of the two by two symmetric matrix argument generalized Kloosterman sum. This evaluation might be of some independent interest because of its relationsbip with the Fourier coefficients of the Poincare series for the Siegel modular forms of degree two. Our next task is to extend the fundamental lemma to the entire Hecke algebra. Inspired by Ye's idea on the quadratic base change for GL(n) case, in a paper In Transactions of the AMS, we proved the inversion formula for the Bessel transform. Because of the inversion formula, now we can express the orbital integrals for an arbitrary element in the Hecke algebra, as a finite sum of degenerate Kloosterman orbital integrals for the unit element. We have computed all degenerate orbital integrals and now we are ready to extend the fundamental lemma to the entire Hecke algebra.

我们继续了关于齐格弗里德Boecherer猜想及其关于Siegel本征尖点形式的四次L-函数的中心临界值的推广的项目。我们的方法是推广Jacquet的两个相关迹公式，这两个公式给出了Waldspurger关于GL（2）的L-函数的中心临界值定理的另一个证明。为了建立一个迹公式，第一步也是关键的一步是证明基本引理。在《AMS回忆录》第782期中，我们给出了Hecke代数中单位元基本引理的一个证明。我们的证明是基于两个两个对称矩阵参数广义Kloosterman和的显式评估。这个评估可能是一些独立的利益，因为它的relationsbip与庞加莱级数的傅里叶系数的西格尔模形式的程度2。我们的下一个任务是将基本引理推广到整个Hecke代数。受叶文辉关于GL（n）二次基变换的思想的启发，在《AMS学报》的一篇论文中，我们证明了Bessel变换的反演公式。由于反演公式，现在我们可以将Hecke代数中任意元素的轨道积分表示为单位元的退化Kloosterman轨道积分的有限和。我们已经计算了所有的退化轨道积分，现在我们准备将基本引理推广到整个Hecke代数。

项目成果

Atsushi ICHINO: "On the local theta correspondence and R-groups"Compositio Mathematica. 140・2. 301-316 (2004)

Atsushi IHINO：“关于局部 theta 对应和 R 群”Compositio Mathematica 140・2（2004）。

Masaaki FURUSAWA, S.Boecherer, R.Schulze-Pillot: "On the global Gross-Prasad conjecture for Yoshida liftings""Contributions to Automorphic Forms, Geometry and Number Theory : Shalikafest 2002"--a supplemental volume to the American Journal of Mathematics.

Masaaki FURUSAWA、S.Boecherer、R.Schulze-Pillot：“On the global Gross-Prasad conjecture for Yoshida lifts”“Contributions to Automorphic Forms, Geometry and Number Theory : Shalikafest 2002”——美国数学杂志增刊

MASAHARU KANEDA (WITH S.NAITO): "A TWINNING DHARACTER FORMULA FOR DEMAZURE MODULES"TRANSFORMATION GROUPS. 7. 321-341 (2002)

MASAHARU KANEDA（与 S.NAITO）：“DEMAZURE 模块的孪生 DHARACTER 公式”转换组。

Masaharu Kaneda (with T.Nakashima): "On certain maximal cyclic modules for the quantized special linear algebra at a root of unity"Pacific Journal of Mathematica. 211. 273-282 (2003)

Masaharu Kaneda（与 T.Nakashima）：“论单位根处量化特殊线性代数的某些最大循环模”太平洋数学杂志。

Masaaki FURUSAWA, J.A.Shalika: "On central critical values of the degree four L-functions for GSp(4) : The fundamental lemma, Memoirs of the A.M.S., v.164, no.782"American Mathematical Society. x+139 (2003)

Masaaki FURUSAWA，J.A.Shalika：“关于 GSP(4) 的四次 L 函数的中心临界值：基本引理，A.M.S. 回忆录，v.164，no.782”美国数学会。