Torsion groups of elliptic curves over $\mathbb{Q}({\mu }_{p^{\infty }})$

Abstract

Let $E/\mathbb{Q}$ be an elliptic curve and $p\in \{5,7,11\}$ be a prime. We determine the possibilities for $E(\mathbb{Q}({\zeta }_p){)}_{tors}$. Additionally, we determine all the possibilities for $E(\mathbb{Q}({\zeta }_{16}){)}_{tors}$ and $E(\mathbb{Q}({\zeta }_{27}){)}_{tors}$. Using these results we are able to determine the possibilities for $E(\mathbb{Q}({\mu }_{p^{\infty }}){)}_{tors}$.