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By Li Guo


Abstract. Let E be an elliptic curve with complex multiplication by the ring of integers of an imaginary quadratic field K. The theory of complex multiplication associates E with a Hecke character ψ. The Hasse-Weil L-function of E equals the Hecke L-function of ψ, whose special value at s = 1 encodes important arithmetic information of E, as predicted by the Birch and Swinnerton-Dyer conjecture and verified by Rubin[Ru] when the special value is non-zero. For integers k, j, special values of the Hecke L-function associated to the Hecke character ψ k ¯ ψ j should encode arithmetic information of the Hecke character ψ k ¯ ψ j, as predicted by the Bloch-Kato conjecture[B-K]. When p is a prime where E has good, ordinary reduction, the p-part of the conjecture has been verified when j = 0[Ha] and when j ̸ = 0, p> k[Gu]. To verify the conjecture in other cases, it is important to have an explicit description of the exponential map of Bloch and Kato. In this paper we provide such an explicit exponential map for the case j ̸ = 0, p ∤ k. Let K be an imaginary quadratic field with ring of integers OK. Let E be a

Year: 1997
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