Investigation of Dirichlet series for Siegel modular forms

(1)利用H.Katsurda[1]中定义的半整矩阵的平方Moebius函数，我们给出了椭圆尖顶Hecke-Eisenstein特征形f的偶次Klingen-Eisenstein提升所附的Koecher-Maass Dirichlet级数的一个显式公式(与T.Ibukiyama共同工作)。在这个显式公式中，出现了一个与f相关的二元Dirichlet级数，这是由W.Kohnen和D.Zagier最初研究的。此外，利用同样的方法，我们还给出了奇次Klingen-Eisenstein提升的一个显式公式。(2)T.Ikeda给出了椭圆尖顶Hecke-本征形式f的Saito-Kurokawa提升的一个推广，并构造了一种称为f的Siegel尖点形式，称为f的Ikeda Lifint。我们给出了f的Ikeda Lif的Koecher-Maass级数的一个显式公式。(3)我们给出了两个Siegel尖顶Hecke-本征形式相交的充分条件。证明了Siegel-Eisenstein级数的Andrianov型Zeta函数存在Euler积，并确定了它的一个好的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.