Self-points on elliptic curves

Abstract

Let $E∕\mathbb{Q}$ be an elliptic curve of conductor $N$ and let $p$ be a prime. We consider trace-compatible towers of modular points in the noncommutative division tower $\mathbb{Q}\left(E\left[p^{\infty }\right]\right)$. Under weak assumptions, we can prove that all these points are of infinite order and determine the rank of the group they generate. Also, we use Kolyvagin’s construction of derivative classes to find explicit elements in certain Tate–Shafarevich groups.