Investigation of Koecher-Maass series for various liftings of Siegel modular forms

1.在2000年，我们给出了椭圆尖点Hecke-特征形f与T的Ikeda提升的Koecher-Maass Dirichlet级数的一个显式公式。伊吹山本文给出了某个Dirichlet级数的特殊值出现在这样一个公式中的代数性的一些结果。2.给出了两个Siegel尖点Hecke特征形重合的充分条件。这个结果比我们在1999年给出的结果稍强。此外，我们对上述结果进行了细化，提出了一定的猜想，并在Koecher-Maass级数不为零的一定条件下证明了该猜想。3.利用Siegel Eisenstein级数的拉回公式、Ibukiyama的微分算子和Katsurada的Siegel级数的显式公式，在下列情况下，给出了模形式f被特征标χ扭曲的标准zeta函数的精确值：（1）f是neben型椭圆尖点Hecke本征形，χ是素导体的Dirichlet特征标. (2)f是水平1的2次Siegel模形式，χ是平凡特征标.我们证明了Nagaoka定义的2次p-adic Eisenstein级数是真模形式（Joint work with S. Nagaoka.）5.本文研究了Hecke算子对θ级数的作用（与R. Schulze-Pillot）

1. In 2000, we gave an explicit formula for the Koecher-Maass Dirichlet series for the Ikeda lifting of an elliptic cuspidal Hecke-eigenforms f jointly with T. Ibukiyama. We gave some result on the algebraicity of special values of a certain Dirichlet series attached to f appearing in such a formula. 2. We gave a good sufficient condition for two Siegel cuspidal Hecke-eigenforms to coincide with each other. This result is slightly stronger than the result which we gave in 1999. 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.3. By using the pullback formula for Siegel Eisenstein series, the differential operators due to Ibukiyama, and an explicit formula for Siegel series due to Katsurada, we gave an exact values of the standard zeta function of a modular form f twisted by a character χ in the following cases:(1) f is an elliptic cuspidal Hecke eigenform of neben type, and χ is a Dirichlet character of prime conductor.(2) f is a Siegel modular form of degree 2 of level 1, and χ is the trivial character.4. We proved that the p-adic Eisenstein series of degree 2 defined by Nagaoka is a true modular form (Joint work with S. Nagaoka.)5. We investigated the action of Hecke operator on the theta series (Joint work with R. Schulze-Pillot.)