Shadow line distributions

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.

Updated following referee's comments. To appear in Math. Comp