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 setting where the \(\mathbb{Z}_p\)-extension \(L\) is replaced by the compositum \(K_{\infty}\) of all \(\mathbb{Z}_p\)-extensions of \(K\).

Under this new setting, we prove the analogous statement for \(p=5\), and further provide partial results for the remaining primes \(p=3\) and \(p=2\).