On arithmetic theory of automorphic forms and special values of automorphic L-functions

对于GSp(4)的两个相对的迹公式，证明了Hecke代数中单位元的基本引理。我们的最终目标是证明Bocherer关于二阶全纯Siegel特征尖型旋量l函数二次旋的中心临界值的猜想。这个基本引理的公告已经发表在《学术科学》杂志上。巴黎和证据的细节将出现在其他地方。在证明基本引理的过程中，我们用经典的GL(2) Kloosterman和显式地计算了某些矩阵参数Kloosterman和。我们注意到我们的Kloosterman和是出现在西格尔模群的庞加莱级数的傅里叶系数中的广义Kloosterman和的一种特殊情况。我们关于Kloosterman和的结果可能是一些独立的兴趣，因为这种广义Kloosterman很少被明确地求值。我们的第二个推测轨迹公式与GSp(4)的二次基电荷有关。我们的结果表明GL(2)的二次基变化的Jacquet-Ye准则可以推广到GSp(4)。这显然值得进一步调查。最后，我们的结果表明，在研究自同构l函数的特殊值时，研究整个l包是很重要的。澄清我们的结果所期望的特殊值的周期部分与德列涅猜想之间的关系似乎是非常有趣的。

We proved the fundamental lemma for the unit element in the Hecke algebra for two relative trace formulas for GSp(4). Our ultimate goal is to prove Bocherer's conjecture on the central critical values of the quadratic twists of the spinor L-functions associated to holomorphic Siegel eigen cusp forms of degree two. The announcements of the fundamental lemma have been published in C. R. Acad. Sci. Paris and the details of the proof will appear elsewhere.In the course of the proof of the fundamental lemma, we evaluated certain matrix argument Kloosterman sums explicitly in terms of the classical GL(2) Kloosterman sums. We remark that our Kloosterman sum is a special case of the generalized Kloosterman sum which appears in the Fourier coefficients of the Poincare series for the Siegel modular group. Our result on the Kloosterman sum may be of some independent interest, since it is rare that such generalized Kloosterman is evaluated explicitly.Our second conjectural trace formula is related to the quadratic base charge for GSp (4). Our result suggests that the Jacquet-Ye criterion for the quadratic base change for GL(2) generalizes to GSp(4). This clearly deserves some further investigation. Finally our result implies that it is important to study the whole L-packet when we study the special values of automorphic L-functions. It seems very interesting to clarify the relationship between the period part of the special value expected by our result and Deligne's conjecture.