Investigation of Dirichlet series for Siegel modular forms

（1）利用H.Katsurda [1]中的结果所定义的半整矩阵的平方Moebius函数，给出了椭圆尖点Hecke-特征形f的偶次Klingen-Eisenstein提升上的Koecher-Maass Dirichlet级数的一个显式表达式。（与T. Ibukiyama共同工作。在这个显式公式中，出现了一个与f相连的二元Dirichlet级数，它最初由W.Kohnen和D. Zagier研究。在奇次Klingen-Eisenstein提升的情形下，用同样的方法给出了一个显式公式.在这个公式中，出现了一种新的二元Dirichlet级数。（2）T.Ikeda给出了椭圆尖点Hecke特征形f的Saito-Kurokawa提升的推广，并构造了f的Siegel尖点形式，称为Ikeda极限。我们给出了f的Ikeda lif的Koecher-Maass级数的一个显式公式。（与T. Ibukiyama的部分合作）(3)We给出了两个Siegel尖点Hecke特征形重合的一个充分条件。进一步，我们对上述结果提出了一个改进的猜想，并在Koecher-Maass级数不为零的条件下证明了这个猜想. (4)We证明了Siegel-Eisenstein级数的Andrianov型zeta函数具有Euler积，并确定了它的一个“好”Euler因子.

(1)Using the squared Moebius function for half-integral matrices defined by H.Katsurda result of 1, we gave an explicit formula for the Koecher-Maass Dirichlet series attached to the Klingen-Eisenstein lifting of even degree of an elliptic cuspidal Hecke-eigenform f.(Joint work with T.Ibukiyama.)In this explicit formula, there appears a certain Dirichlet series of two variables attached to f, which was originally investigated by W.Kohnen and D.Zagier. Furthermore, by using the same method, we gave also an explicit formula in the case of the Klingen-Eisenstein lifting of odd degree. In this formula, there appears a new type of Dirichlet series of two variables attached to f.(2)T.Ikeda gave a generarization of the Saito-Kurokawa lifting of an elliptic cuspidal Hecke-eigenform f, and constructed a certain Siegel cusp form, called the Ikeda lifint of f. We gave an explicit formula for the Koecher-Maass series for the Ikeda lif of f.(Partly joint work with T.Ibukiyama.)(3)We gave a good sufficient condition for two Siegel cuspidal Hecke-eigenforms to conincide with each other. Furthermore, we proposed a certain conjecture on a refinement of the above result, and proved this conjecture under a certain condition on the non-vanishing of the Koecher-Maass series.(4)We proved that a zeta function of Andrianov type for the Siegel-Eisenstein series has Euler product, and determined a 'good' Euler factor of it.