Investigation of Dirichlet series for Siegel modular forms

(1)利用H.Katsurda结果1所定义的半积分矩阵的平方Moebius函数，给出了椭圆尖形hecke - eigen型f的Klingen-Eisenstein偶次提升的Koecher-Maass Dirichlet级数的显式公式（与T.Ibukiyama合著）。在这个显式公式中，出现了一个附加在f上的两个变量的狄利克雷级数，它最初是由W.Kohnen和D.Zagier研究的。此外，用同样的方法，我们也给出了奇数次克林根-爱森斯坦举升的显式公式。在这个公式中，出现了一种新的双变量狄利克雷级数附于f.(2)T上。Ikeda给出了椭圆尖形Hecke-eigenform f的Saito-Kurokawa举升的推广，并构造了一个特定的Siegel尖形，称为f的Ikeda举升。我们给出了f的Ikeda举升的Koecher-Maass级数的显式公式（部分与T.Ibukiyama合作）。(3)给出了两个西格尔倒向hecke特征型相互吻合的充分条件。在此基础上，对上述结果进行了改进，提出了一个猜想，并在koecher - mass级数不灭的一定条件下证明了这个猜想。(4)证明了Siegel-Eisenstein级数的Andrianov型zeta函数具有欧拉积，并确定了它的一个“好”欧拉因子。

(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.