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

对于GSP(4)的两个相对迹公式，我们证明了Hecke代数中单位元的基本引理。我们的最终目标是证明Bocher关于旋量L函数的中心临界值的猜想，这些旋量函数与二次全纯Siegel特征尖点形式有关。基本引理的公告已发表在C.R.Acad上。SCI。在基本引理的证明过程中，我们用经典的GL(2)Krousterman和显式地评价了一类矩阵论证Krousterman和。我们指出，我们的Krousterman和是广义Krousterman和的特例，它出现在Siegel模群的Poincare级数的傅里叶系数中。我们关于Krousterman和的结果可能有一些独立的兴趣，因为这种推广的Krousterman和很少被明确地评估。我们的第二个猜想的迹公式与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判据推广到了GSP(4)。这显然值得进一步调查。最后，我们的结果表明，当我们研究自同构的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Deligne猜想之间的关系似乎非常有趣。

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.