SPECIAL VALUES OF AUTOMORPHIC L-FUNCTIONS BY RELATIVE TRACE FORMULAS

我们继续了关于Siegfried Boecherer猜想及其对二阶Siegel特征尖形式的四次l函数中心临界值的推广的项目。我们的方法是推广Jacquet的两个相对迹公式，这两个迹公式给出了GL(2)下关于l -函数中心临界值的Waldspurger定理的另一个证明。为了建立迹公式，第一步也是最关键的一步是证明基本引理。在发表于《AMS回忆录》第782号的专著中，我们证明了Hecke代数中单位元的基本引理。我们的证明是基于二乘二对称矩阵参数广义Kloosterman和的显式求值。由于它与二阶西格尔模形式的庞加莱级数的傅里叶系数的关系，这个评价可能会引起一些独立的兴趣。我们的下一个任务是将基本引理推广到整个赫克代数。受Ye关于GL(n)情况的二次基变换思想的启发，我们在一篇论文《AMS Transactions of the AMS》中证明了贝塞尔变换的反演公式。由于反演公式的存在，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素的简并Kloosterman轨道积分的有限和。我们已经计算了所有的简并轨道积分现在我们准备把基本引理推广到整个赫克代数。

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.