Shadow line distributions

Let $E/\mathbb{Q}$ be an elliptic curve of analytic rank $2$, and let $p$ be an odd prime of good, ordinary reduction such that the $p$-torsion of $E(\mathbb{Q})$ is trivial. Let $K$ be an imaginary quadratic field satisfying the Heegner hypothesis for $E$ and such that the analytic rank of the twisted curve $E^K/\mathbb{Q}$ is $1$. Further suppose that $p$ splits in $\mathcal{O}_K$. Under these assumptions, there is a $1$-dimensional $\mathbb{Q}_p$-vector space attached to the triple $(E, p, K)$, known as the shadow line, and it can be computed using anticyclotomic $p$-adic heights. We describe the computation of these heights and shadow lines. Furthermore, fixing pairs $(E, p)$ and varying $K$, we present some data on the distribution of these shadow lines. This is joint work with Mirela Çiperiani, Barry Mazur, and Karl Rubin.

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