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 $E(K)_{ \text{ tors}}$.
Categories: math.NT