Torsion groups of elliptic curves over some infinite abelian extensions of $\mathbb{Q}$

Abstract

We determine, for an elliptic curve $E/\mathbb{Q}$, all the possible torsion groups $E(K{)}_{tors}$, where $K$ is the compositum of all ${\mathbb{Z}}_p$-extensions of $\mathbb{Q}$. Furthermore, we prove that for an elliptic curve $E/\mathbb{Q}$ it holds that $E(\mathbb{Q}({\mu }_{p^{\infty }}){)}_{tors}=E(\mathbb{Q}({\mu }_p){)}_{tors}$, for all primes $p\ge 5$ and $E(\mathbb{Q}({\mu }_{3^{\infty }}){)}_{tors}=E(\mathbb{Q}({\mu }_{3^3}){)}_{tors}$, $E(\mathbb{Q}({\mu }_{2^{\infty }}){)}_{tors}=E(\mathbb{Q}({\mu }_{2^4}){)}_{tors}$.

Magma code used in the paper is available on Ivan Krijan's webpage (https://web.math.pmf.unizg.hr/~ikrijan/)