## Shadow lines in the arithmetic of elliptic curves
Balakrishnan, J. S., Ciperiani, M., Lang, J., Mirza, B. and Newton, R. ORCID: https://orcid.org/0000-0003-4925-635X
(2016)
Full text not archived in this repository. It is advisable to refer to the publisher's version if you intend to cite from this work. See Guidance on citing. ## Abstract/SummaryLet E/Q be an elliptic curve and p a rational prime of good ordinary reduction. For every imaginary quadratic field K/Q satisfying the Heegner hypothesis for E we have a corresponding line in E(K)\otimes Q_p, known as a shadow line. When E/Q has analytic rank 2 and E/K has analytic rank 3, shadow lines are expected to lie in E(Q)\otimes Qp. If, in addition, p splits in K/Q, then shadow lines can be determined using the anticyclotomic p-adic height pairing. We develop an algorithm to compute anticyclotomic p-adic heights which we then use to provide an algorithm to compute shadow lines. We conclude by illustrating these algorithms in a collection of examples.
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