SPECIAL VALUES OF AUTOMORPHIC L-FUNCTIONS BY RELATIVE TRACE FORMULAS

我们继续研究Siegrey Boecherer的猜想及其对二次Siegel-Eigen尖点形式的四次L函数中心临界值的推广。我们的方法是推广Jacquet的两个相对迹公式，这两个公式给出了关于GL(2)的L函数的中心临界值的Waldspurger定理的另一种证明。要建立迹公式，首先要证明基本引理。在《AMS回忆录》第782期出版的专著中，我们证明了Hecke代数中单位元的基本引理。我们的证明是基于对2×2对称矩阵变元的广义Krousterman和的显式求值。这一评估可能具有一定的独立意义，因为它与二次Siegel模形式的Poincare级数的傅里叶系数BIP有关。我们的下一个任务是将基本引理推广到整个Hecke代数。受叶诗文关于GL(N)情形二次基变换思想的启发，我们在《AMS学报》上的一篇论文中证明了贝塞尔变换的反演公式。由于有了求逆公式，现在我们可以将Hecke代数中任意元素的轨道积分表示为该单位元素的退化Krousterman轨道积分的有限和。我们已经计算了所有简并的轨道积分，现在我们准备将基本引理推广到整个Hecke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.