Torsion of rational elliptic curves over $\Z_p$-extensions of quadratic fields for $p\geq5$, with a slope analysis for $p=3$

Let $E/\Q$ be an elliptic curve, let $K$ be a quadratic field, and let $L/K$ be a $\Z_p$-extension.

We revisit Avcı's theorem on the equality $E(L)_{\tors}=E(K)_{\tors}$ for $p>5$ and clarify a Galois-over-$\Q$ point needed in its proof: the auxiliary descent lemmas are finite Galois-over-$\Q$ statements, while finite layers of a general mixed $\Z_p$-extension of an imaginary quadratic field need not be Galois over $\Q$.

We repair this by replacing finite layers by finite Galois envelopes built from the cyclotomic and anti-cyclotomic directions.

We then prove the equality $E(L)_{\tors}=E(K)_{\tors}$ for every $p\geq5$, including the formerly exceptional prime $p=5$, by excluding the remaining cyclic $25$-torsion obstruction via quadratic twists and the theorem of Chou--Daniels--Krijan--Najman over $\Q_{\infty,5}$.

Finally, for $p=3$ and imaginary quadratic $K\neq\Q(\sqrt{-3})$, we give a slope stratification: the odd part of $E(L)_{\tors}$ comes from $L\cap K_{\cyc}$, while the $2$-primary part comes from $L\cap K(E[2])$.

In particular, the two residue classes with both slope coordinates $3$-adic units have no torsion growth, the anti-cyclotomic residue class can contribute only $2$-primary growth, and all new odd torsion is forced into the cyclotomic residue class.

The possible orders of odd cyclic subgroups are explicitly bounded, and $13$-torsion is excluded in the cases covered by the stratification.