Investigation of Koecher-Maass series for various liftings of Siegel modular forms

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1. In 2000, we gave an explicit formula for the Koecher-Maass Dirichlet series for the Ikeda lifting of an elliptic cuspidal Hecke-eigenforms f jointly with T. Ibukiyama. We gave some result on the algebraicity of special values of a certain Dirichlet series attached to f appearing in such a formula. 2. We gave a good sufficient condition for two Siegel cuspidal Hecke-eigenforms to coincide with each other. This result is slightly stronger than the result which we gave in 1999. Furthermore, we proposed a certain conjecture on a refinement of the above result, and proved this conjecture under a certain condition on the non-vanishing of the Koecher-Maass series.3. By using the pullback formula for Siegel Eisenstein series, the differential operators due to Ibukiyama, and an explicit formula for Siegel series due to Katsurada, we gave an exact values of the standard zeta function of a modular form f twisted by a character χ in the following cases:(1) f is an elliptic cuspidal Hecke eigenform of neben type, and χ is a Dirichlet character of prime conductor.(2) f is a Siegel modular form of degree 2 of level 1, and χ is the trivial character.4. We proved that the p-adic Eisenstein series of degree 2 defined by Nagaoka is a true modular form (Joint work with S. Nagaoka.)5. We investigated the action of Hecke operator on the theta series (Joint work with R. Schulze-Pillot.)