Number Theory Seminar: Shadow line distributions

SEMINARS, NUMBER THEORY

Speaker:

Jennifer Balakrishnan - Boston University

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.

For more info, see https://ashvin-swaminathan.github.io/home/NTSeminar.html