The arithmetic of abelian varieties with complex multiplication and Euler systems.

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In this research, we investigated motivic objects such as the polylogarithm and the Eisenstein classes, for the case of elliptic curves, which are one dimensional abelian varieties. We have obtained the following results. Through joint research with Shinichi Kobayashi and Takeshi Tsuji, we gave an explicit description of the p-adic realization of the elliptic polylogarithm for a prime p≧5, even in the case when the elliptic curve has complex multiplication and good supersingular reduction at p, for the Frobenius lifting which is the second power of the absolute Frobenius. Also, through joint research with G. Kings, we gave an explicit description of the p-adic Eisenstein class on the modular curve when p≧5 is a prime of good ordinary reduction.