Abstract
Let \(E/\mathbb{Q}\) be an elliptic curve and p a rational prime of good ordinary reduction. For every imaginary quadratic field \(K/\mathbb{Q}\) satisfying the Heegner hypothesis for E we have a corresponding line in \(E(K) \otimes \mathbb{Q}_{p}\), known as a shadow line. When \(E/\mathbb{Q}\) has analytic rank 2 and E∕K has analytic rank 3, shadow lines are expected to lie in \(E(\mathbb{Q}) \otimes \mathbb{Q}_{p}\). If, in addition, p splits in \(K/\mathbb{Q}\), 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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Notes
- 1.
The formula appearing in [13, §2.9] contains a sign error which is corrected here.
- 2.
The formula appearing in [13, §2.9] contains a sign error which is corrected here.
- 3.
Note that by Gross and Zagier [8] and Kolyvagin [10] the analytic rank of \(E^{K}/\mathbb{Q}\) being 1 implies that the algebraic rank of \(E^{K}/\mathbb{Q}\) is 1 and the Tate–Shafarevich group of \(E^{K}/\mathbb{Q}\) is finite. Furthermore, in this case, computing a non-torsion point in \(E^{K}(\mathbb{Q})\) can be done by choosing an auxiliary imaginary quadratic field F satisfying the Heegner hypothesis for \(E^{K}/\mathbb{Q}\) such that the analytic rank of E K ∕F is 1 and computing the corresponding basic Heegner point in E K(F).
- 4.
Note that Steps 2 and 3 are needed to ensure that the point whose anticyclotomic p-adic height we will compute using formula (7) satisfies the required conditions.
- 5.
As a consistency check we compute the height of nP and verify that \(h_{\rho }(nP) = \frac{1} {n^{2}} h_{\rho }(P)\) for positive integers n ≤ 5.
- 6.
We compute the height of P 1 + P 2 + S as a consistency check.
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Acknowledgements
The authors are grateful to the organizers of the conference “WIN3: Women in Numbers 3” for facilitating this collaboration and acknowledge the hospitality and support provided by the Banff International Research Station. During the preparation of this manuscript: the second author was partially supported by NSA grant H98230-12-1-0208 and NSF grant DMS-1352598; the third author was partially supported by NSF grant DGE-1144087.
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Balakrishnan, J.S., Çiperiani, M., Lang, J., Mirza, B., Newton, R. (2016). Shadow Lines in the Arithmetic of Elliptic Curves. In: Eischen, E., Long, L., Pries, R., Stange, K. (eds) Directions in Number Theory. Association for Women in Mathematics Series, vol 3. Springer, Cham. https://doi.org/10.1007/978-3-319-30976-7_2
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