Shadow lines in the arithmetic of elliptic curves

TL;DR

This paper develops algorithms to compute shadow lines associated with elliptic curves over Q, using anticyclotomic p-adic height pairings, and demonstrates their application through examples.

Contribution

It introduces a new algorithm for computing shadow lines in elliptic curves using anticyclotomic p-adic heights, advancing computational methods in the arithmetic of elliptic curves.

Findings

Successfully computed shadow lines in various examples

Demonstrated the effectiveness of the anticyclotomic p-adic height pairing

Provided computational evidence supporting theoretical predictions

Abstract

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 Q_p. 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.