Shadow line distributions

TL;DR

This paper investigates the distribution of special lines called shadow lines in the Mordell--Weil groups of elliptic curves over quadratic fields, providing computational evidence and framing related conjectures.

Contribution

It introduces the concept of shadow lines in the context of elliptic curves and studies their distribution, supported by computational verification and conjectural framing.

Findings

Shadow lines are conjectured to lie in rational points when certain conditions hold.

Computational evidence supports the conjecture about the distribution of shadow lines.

The paper frames new conjectures based on the observed distribution patterns.

Abstract

Let $E$ be an elliptic curve over $\mathbb{Q}$ with Mordell--Weil rank $2$ and $p$ be an odd prime of good ordinary reduction. For every imaginary quadratic field $K$ satisfying the Heegner hypothesis, there is (subject to the Shafarevich--Tate conjecture) a line, i.e., a free $\mathbb{Z}_p$-submodule of rank $1$, in $ E(K)\otimes \mathbb{Z}_p$ given by universal norms coming from the Mordell--Weil groups of subfields of the anticyclotomic $\mathbb{Z}_p$-extension of $K$; we call it the {\it shadow line}. When the twist of $E$ by $K$ has analytic rank $1$, the shadow line is conjectured to lie in $E(\mathbb{Q})\otimes\mathbb{Z}_p$; we verify this computationally in all our examples. We study the distribution of shadow lines in $E(\mathbb{Q})\otimes\mathbb{Z}_p$ as $K$ varies, framing conjectures based on the computations we have made.