Research on special values of the standard L-functions of Siegel modular forms and related fields

(1)I考虑了标准zeta函数的特殊值与任意次Siegel尖点形式的同余之间的关系。特别地，我提出了一个关于尖点Hecke本征形的Saito-Kurokawa提升关于SL（2，Z）的同余及其L-函数的特殊值的猜想，并在一定条件下得到了证明.此外，我们给出了一些数值证据，我们的猜想。2004年10月在白马举行的研讨会和2005年3月在加拉津举行的会议上，我就这些主题进行了演讲。（2）设M和N是无平方因子的正整数，φ和χ分别是模M和模N的Dirichlet特征标。对于neben型的尖点Hecke特征形f，给出了f的标准zeta函数被χ扭曲的特殊值的一个显式公式.我给出了这些值的数值例子，并检查了一些结果来自块加藤猜想。(3)I证明了二次Neu型平方自由水平的Eisenstein级数空间是亏格θ级数与R共同张成的。舒尔茨-皮洛特利用这一结果，我与S.Boecherer和R.Schulze-Pillot共同证明了二次Neumann型平方自由水平的尖点型空间是θ级数所张成的。2004年9月，他们在立教大学举行的研讨会上就这些主题发表了演讲。(4)I与T.Ibukiyama共同给出了真实的解析Eisenstein级数的Koecher-Maass级数的一个显式形式。2004年9月在立教大学举行的研讨会上，我做了关于这个主题的演讲。

(1)I considered the relation between special values of the standard zeta functions and the congruence a Siegel cusp form of arbitrary degree. In particular, I proposed a conjecture concerning the congruence of the Saito-Kurokawa lift of a cuspidal Hecke eigenform with respect to SL(2,Z) and the special value of its L-function, and proved it under certain condition. Furthermore we gave some numerical evidence for our conjecture. I gave talks on these topics at the workshop held at Hakuba on October 2004, and at the conference held at Karatsu on March 2005.(2)Let M and N be square free positive integers, and φ and χ be Dirichlet characters mod M and N, respectively. For a cuspidal Hecke eigenform f of neben type φ I gave an explicit formula for the special values of the standard zeta function of f twisted by χ. I gave numerical examples for these values and checked some results derived from the Block-Kato conjecture.(3)I proved that the space of Eisenstein series of square free level of quadratic nebentype is spanned by the genus theta series jointly with R. Schulze-Pillot. By using this result, I proved that the space of cusp forms of square free level of quadratic nebentype is spanned by the theta series jointly with S.Boecherer and R.Schulze-Pillot. They gave talks on these topics at the symposium held at Rikkyo University on September 2004.(4)I gave an explicit form of the Koecher-Maass series for the real analytic Eisenstein series jointly with T.Ibukiyama. I gave a talk on this topic at the symposium held at Rikkyo University on September 2004.