Periods and congruence of modular forms, and Selmer group

(1)我们解决了池田猜想的周期厄米Ikeda电梯。(2)本文计算了某些向量值Siegel模形式的特殊值，证实了N. Dummigan（与N. Dummigan和T. Ibukiyama）。计算了Ramanujan δ函数△的Kim-Shahidi提升的标准L-函数的特殊值，证明了它与Zagier关于△的对称第四L-函数的特殊值的猜想（与T. Ibukiyama）。(3)本文建立了一个计算椭圆模形式的三L函数的特殊值的算法（与T。Ibukiyama）。利用它，我们考虑了Ikeda-Miyawaki升力与非Ikeda-Miyawaki升力（与T. Ibukiyama角Poor和D. Yuen）。(4)给出了S_（Z）的同余子群Γ_0（N）的Poincare级数的Fourier系数的一个显式公式<p2>。(5)给出了Saito-Kurokawa升力的扭曲Koecher-Maass级数的显式表达式。作为应用，我们得到了某些半整权模形式的Rankin-Selberg级数的特殊值的线性依赖性（与Y. Mizuno）。并给出了Duke-Imamoglu-Ikeda升力的扭曲Koecher-Maass级数的显式表达式。(6)我们考虑了Klingen-Eisenstein级数与杯形之间的同余关系（与S. Mizumoto）。

(1) We solved Ikeda's conjecture on the period of the Hermitian Ikeda lift.(2) We compute the special values of ceratain vector valued Siegel modular forms and confirmed a conjecture proposed by N. Dummigan (joint work with N. Dummigan and T. Ibukiyama). Moreover we computed the special values of the standard L-function of the Kim-Shahidi lift of Ramanujan delta function △, and confirmed that it is complatible with Zagier's conjecture on the special values of symmetric forth L function of △(joint work with T. Ibukiyama).(3) We established an algorithm for computing the special values of the triple L function of elliptic modular forms (joint work with T. Ibukiyama). By using it we considered the congruence between Ikeda-Miyawaki lift and non- Ikeda-Miyawaki lift (joint work with T. Ibukiyama, C. Poor, and D. Yuen).(4) We gave an explicit formula for the Fourier coefficient of the Poincare series for the congruence subgroup Γ_0 (N) of S_<p2>(Z).(5) We gave an explicit formula of twisted Koecher-Maass series of the Saito-Kurokawa lift. As an application we obtained a linear dependence of the special values of the Rankin-Selberg series of certain half-integral weight modular forms (joint work with Y. Mizuno). Moreover we gave an explicit formula of twisted Koecher-Maass series of the Duke-Imamoglu-Ikeda lift.(6) We consider congruence between the Klingen-Eisenstein series and cups forms (joint work with S. Mizumoto).