Research on special values on L-functions of automorphic forms

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(1) We gave an explicit formula for a certain Dirichlet series of Rankin-Selberg type associated with the Fourier-Jacobi coefficients of the Ikeda lifting. (Joint work with. Kawamura (Hokkaido Univ.))(2) By using (1), we rewrote a conjecture concerning the period of the Ikeda lift proposed by T Ikeda in terms of the period of a certain half-integral weight Siegel modular forms. As a result we proved Ikeda's conjecture. (Joint work with. Kawamura (Hokkaido Univ.))(3) We proposed a conjecture, which related the congruence between the Ikeda lift of an elliptic modular from f and non-Ikeda lift with the special values of L-functions of f. By using (2) we proved our conjecture.(4) We gave a talk about (1), (2) and (3), for examples, on the following meetings ^*RI MS conference (2007,1.16), ^*The 52th Algebra Symposium (2007,8.7), ^*Special lecture at the annual meetings of M. S. J (2008,3.24)(5) We gave an exact values of the standard L-function of a certain vector valued Siegel modular forms, which is related with Harder's conjecture. (Joint work with T. Ibukiyama (Osaka Unuv.).(6) We investigate the non-vanishing of the central critical values of standard L-function of Siegel modular forms of degree 2 (Joint work with K. Matsumoto (Nagoya Univ.) and M. Chida (Tohoku Univ.))