Study on automorphic forms of several variables

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The purposes of our investigation are stated as follows :(1) Study of Serre's p-adic Eisenstein series.(2) Study of mod p properties of modular forms.About the object (1), we got the following result. The generalization of the notion of Serre's p-adic Eisenstein series succeeded in the case of Siegel modular forms in the previous investigation (2001-2003). We generalized the notion to the case of Hermitian modular forms, and proved the coincidence between a p-adic Hermitian Eisenstein series and a genus theta series for Hermitian forms of rank 2. Concerning this work, we wrote two papers :[1] S.Nagaoka, On p-adic Hermitian Eisenstein series, Proc.Amer.Math.Soc. 134, 2533-2540 (2006)[2] T.Munemoto and S.Nagaoka, Note on p-adic Hermitian Eisenstein series, Abh.Math.Sem.Univ.Hamburg 76, 247-260 (2006)In [1], we treated the case of Gaussian field. We generalized the result to the case of imaginary quadratic fields with class number one..About the object (2), we succeeded to solve a pending problem in this field. That is, the existence of a modular form with certain congruence property is proved. A modular form satisfying congruence one modulo p plays an important role in the theory of modular forms with positive characteristic. In the degree 2 case, the existence was proved during in the previous period (2001-2003). Our result gives a complete answer to this problem. The main part of this result was announced in[3] S.Boecherer and S.Nagaoka, On mod p properties of Siegel modular forms,and this is accepted for publication in the "Mathematische Annalen". The researcher presented a lecture on this result at the annual meeting of the Mathematical Society of Japan as an invited speaker.