Torsion of rational elliptic curves over the cyclotomic extensions of $\mathbb Q$

Let $E$ be an elliptic curve defined over $\mathbb Q$. We classify all groups that can arise as $E(\mathbb Q(\zeta _p))_{\mathrm{tors}}$ 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 $E(K)_{\mathrm{tors}}$ is a subgroup of $E(\mathbb Q^{\mathrm{ab}})_{\mathrm{tors}}$. Our methods provide tools to eliminate non-realized torsion structures from the list of possibilities for $E(K)_{\mathrm{tors}}$.