ON AUTOMORPHIC L-FUNCTIONS OF GENERAL SYMPLECTIC AND UNITARY GROUPS OF RANK TWO

我们继续研究二阶一般辛群和酉群的自同构L函数。更具体地说，主要工作之一是证明Siegrey Boecherer关于二次Siegel Eigen尖点形式的四次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，它可以看作是Jacquet相对迹公式的自然推广，它给出了著名的Waldspurger关于GL(2)的环面周期与GL(2)的自同构L函数的中心临界值之间关系的定理的又一证明。要建立一个相对迹公式，首先要证明基本引理。我们已经证明了Hecke代数的单位元的基本原理，并将其结果发表在AMS的回忆录第782号。在这笔拨款支持的期间，我们致力于将基本引理从单位元推广到整个Hecke代数。我们发现，通过将麦克唐纳多项式理论应用于Bessel模型的显式公式，对Hecke代数中一般元素的Krousterman轨道积分的计算可归结为对Hecke代数单位元的一般Kostka数和简并Krousterman轨道积分的计算。我们已经对所有这些项目进行了评估。现在，我们剩下的任务是比较与轨迹公式的两边对应的这些线性组合，并确保它们匹配。

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.