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.

我们继续进行了有关Siegfried Boocherer的猜想及其对第二学位的Siegel Eigen cusp形式的四个L功能的中心临界值的概括。我们的方法是概括Jacquet的两个相对痕量公式，这些公式给出了Waldspurger的另一个证据，介绍了GL（2）L功能的中心临界值。为了建立痕量公式，第一个和关键的步骤是证明基本的引理。在作为AMS回忆录的第782号的专着中，我们证明了Hecke代数中单位元素的基本引理。我们的证明是基于对两个对称矩阵参数的显式评估。该评估可能具有独立的兴趣，因为它与第二学位的Siegel模块化形式的Poincare系列的傅立叶系数具有关系。我们的下一个任务是将基本引理扩展到整个Hecke代数。受到Ye对GL（N）情况的二次基础变化的想法的启发，在AMS交易中的论文中，我们证明了Bessel变换的反转公式。由于具有倒置公式，现在我们可以表达Hecke代数中任意元素的轨道积分，作为单位元素的简并Kloosterman轨道积分的有限总和。我们已经计算出所有退化的轨道积分，现在我们准备将基本引理扩展到整个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。

Masaaki FURUSAWA, J.A.Shalika: "On central critical values of the degree four L-functions for GSp(4) : The fundamental lemma"Mem. Amer. Math. Soc.. (印刷中). (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”Mem Soc..（出版中）。

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。