Periods and congruence of modular forms, and Selmer group

(1)解决了Ikeda关于Hermitian Ikeda提升周期的猜想。(2)计算了向量值Ceratain向量值Siegel模形式的特定值，证实了N.DumMican(与N.DumMican和T.Ibukiyama共同工作)提出的猜想。此外，我们还计算了Ramanujan Delta函数的Kim-Shahimi提升的标准L函数的特殊值，并证实它与Zagier关于对称四次L函数的特殊值的猜想是相容的。(3)我们建立了一个计算椭圆模形式的三重L函数的特殊值的算法(与T.Ibukiyama合作)。利用它，我们考虑了池田-宫城提升和非池田-宫城提升之间的同余(与T.Ibukiyama，C.Poor和D.袁共同工作)。(4)我们给出了S的同余子群Γ_0(N)的庞加莱级数的傅立叶系数的一个显式公式。(5)我们给出了齐藤-黑川提升的扭曲科赫-马级数的显式公式。作为应用，我们得到了某些半整权模形式的Rankin-Selberg级数的特定值的线性相关性(与Y.Mizuno共同工作)。此外，我们还给出了Duke-Imamoglu-Ikeda提升的扭转Koecher-Maass级数的一个显式公式。(6)我们考虑了Klingen-Eisenstein级数与CUPS形式之间的同余(与S.Mizumoto共同工作)。

(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).