Torsion of Rational Elliptic Curves over the Cyclotomic Extensions of $\mathbb{Q}$

Abstract

Let $E$ be an elliptic curve defined over $\mathbb{Q}$. In this article, we classify all groups that can arise as $E(\mathbb{Q}({\zeta }_p){)}_{\text{tors}}$ up to isomorphism for any prime $p$. When $p-1$ is not divisible by small integers such as $3,4,5,7$, or $11$, we obtain a sharper classification. For any abelian number field $K$, the torsion subgroup $E(K{)}_{\text{tors}}$ is a subgroup of $E({\mathbb{Q}}^{\text{ab}}{)}_{\text{tors}}$. Our methods provide tools to eliminate non-realized torsion structures from the list of possibilities for $E(K{)}_{\text{tors}}$.

22 pages, 2 tables