Selmer groups and anticyclotomic ${\mathbb{Z}}_p$-extensions II

Abstract

Let $E/\mathbb{Q}$ be an elliptic curve, $p$ a prime where $E$ has ordinary reduction and $K_{\infty }/K$ the anticyclotomic ${\mathbb{Z}}_p$-extension of a quadratic imaginary field $K$ satisfying the Heegner hypothesis. We give sufficient conditions on $E$ and $p$ in order to ensure that ${\text{Sel}}_{p^{\infty }}(E/K_{\infty })$ is a cofree $\Lambda$-module of rank one. We also show that these conditions imply that $\text{rank}(E(K_n))=p^n$ for all $n\ge 0$ and that the $p$-primary subgroup of the Tate-Shafarevich group of $E/K_n$ is trivial for all $n\ge 0$.

The results of this paper may be proven by a more direct method. Please see arxiv.org/abs/1808.09544