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

(1)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，我提出了关于尖顶Hecke本征形关于SL(2，Z)的Saito-Kurokawa升力与其L函数的特定值的同余的猜想，并在一定条件下证明了这一猜想。此外，我们还给出了一些数字证据来支持我们的猜想。我在2004年10月在白场举行的研讨会上和2005年3月在Karatsu举行的会议上就这些问题进行了演讲。(2)设M和N是无平方正整数，φ和χ分别是Dirichlet特征标mod M和N。对于Neben型φ的尖锥Hecke本征形式f，我给出了f被χ扭成的标准Zeta函数的特例的显式公式。我给出了这些值的数值例子，并检验了Block-Kato猜想得到的一些结果。(3)证明了二次Nebent型自由平方能级Eisenstein级数的空间是由亏格theta级数与R.Schulze-Pillot共同跨越的。利用这一结果，我证明了二次Nebent型自由平方能级的尖点形式的空间是由S.Boecherer和R.Schulze-Pillot共同建立的theta级数所跨越的。他们在2004年9月在立交大学举行的研讨会上就这些问题进行了讨论。(4)我与T.Ibukiyama共同给出了真正解析Eisenstein级数的Koecher-Maass级数的显式形式。2004年9月，我在日本立交大学举行的研讨会上就这一问题发表了演讲。

(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.