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Let E be an elliptic curve defined over Q. Suppose it has complex multiplication by an order of an imaginary quadratic extension K/Q and p is a prime of good ordinary reduction. Also, suppose that the sign of functional equation of L(E,s) is -1 and p splits in K. Accordingly, the anticyclotomic Katz p-adic L-function vanishes. Is there a version of p-adic Gross - Zagier formula for the two variable Katz p-adic L-function in this case? The results of Perrin - Riou seem to exclude this choice of imaginary quadratic extension.

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  • $\begingroup$ At least, is there a precise conjectural description in this case? $\endgroup$
    – jkl
    Jul 23, 2011 at 15:07

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Is there complex Gross - Zagier in this case? Their work also excludes this case. If there is, one can expect the p-adic height of that Heegner point to appear in p-adic version, as in the case of Perrin - Riou. There is a paper of Conrad on complex Gross - Zagier which includes this case. However, I do not know whether complex version follows from that paper. Probably some more work in needed.

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