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) Shadow lines in the arithmetic of elliptic curves. In: Eischen, E. E., Long, L., Pries, R. and Stange, K. (eds.) Directions in number theory : Proceedings of the 2014 WIN3 Workshop. Association for Women in Mathematics series (3). Springer International Publishing. ISBN 9783319309743

Abstract/Summary

Let 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.