Torsion of Rational Elliptic Curves over the ${\mathbb{Z}}_p$-Extensions of Quadratic Fields

Let $E$ be an elliptic curve defined over $\mathbb{Q}$. For a quadratic number field $K$ and an odd prime number $p$, let $L$ be a ${\mathbb{Z}}_p$-extension of $K$. We prove that $E(L{)}_{\text{tors}}=E(K{)}_{\text{tors}}$ when $p>5$. It enables us to classify the groups that can be realized as the torsion subgroup $E(L{)}_{\text{tors}}$, by using the classification of torsion subgroups over the quadratic fields.