In this paper we prove the weak form of the Tamagawa number conjecture of Bloch and Kato for elliptic curves with CM by the ring of integers O_K in a quadratic number field. For this we use a new explicit description of the p-adic specialization of the elliptic polylogarithm. The work "weak" indicates that we have nothing new to say about the rank of the K-groups involved.
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