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

TL;DR

This paper investigates the structure of Selmer groups over anticyclotomic $Z_p$-extensions of quadratic imaginary fields, establishing conditions for their cofree modules and implications for ranks and Tate-Shafarevich groups.

Contribution

It provides new sufficient conditions ensuring Selmer groups are cofree of rank one and deduces consequences for ranks and Tate-Shafarevich groups in the anticyclotomic setting.

Findings

Selmer groups are cofree $Z_p$-modules of rank one under certain conditions.

The rank of $E(K_n)$ equals $p^n$ for all $n \\geq 0$.

The $p$-primary part of Tate-Shafarevich groups is trivial for all $n \\geq 0$.

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 $n \geq 0$ and that the $p$-primary subgroup of the Tate-Shafarevich group of $E/K_n$ is trivial for all $n \geq 0$.