Torsion groups of rational elliptic curves over $\mathbb{Z}_p$-extensions of quadratic fields: the $p\le 5$ case

Let $E$ be a rational elliptic curve. We generalize a theorem due to Avcı\cite{AVCI2026153}, which asserts that for any quadratic field $K$ and prime $p>5$, the equality $E(K)_{\mathrm{tors}} = E(L)_{\mathrm{tors}}$ holds for every $\mathbb{Z}_p$-extension $L/K$. In this paper, we consider the setti