$\ell=2$, for which the 2-torsion tower is known essentially explicitly (see [@MR3500996] for a complete classification result when $K=\mathbb{Q}$, and [@MR3426175] for a description of the 2-torsion tower of a given elliptic curve over a number field).

In Section \[sec:AdelicFailure\] we will need to bound the minimal parameter of maximal growth for the $\ell$-adic torsion representation defined over certain extensions of the base field. We will do so with the help of the following Lemma:

\[lem-param-tilde-bound\] Let $\tilde K$ be a finite extension of $K$. Let $n_\ell$ (resp. $\tilde n_\ell$) be the minimal parameter of maximal growth for the $\ell$-adic torsion representation attached to $E/K$ (resp. $E/\tilde K$). Then $\tilde n_\ell{\leqslant}n_\ell+v_\ell([\tilde K:K])$.

Let $n_0:=n_\ell+v_\ell([\tilde K:K])+1$ and consider the following diagram:

(L2tilde) at (0,1) [$\tilde K_{\ell^{n_0}}$]{}; (L1tilde) at (-2,0) [$\tilde K_{\ell^{n_\ell}}$]{}; (L2) at (2,0) [$K_{\ell^{n_0}}$]{}; (L1tildecapL2) at (0,-1) [$\tilde K_{\ell^{n_\ell}}\cap K_{\ell^{n_0}}$]{}; (Ktilde) at (-4,-1) [$\tilde K$]{}; (fantasma) at (4,-1) ; (L1) at (2,-2) [$K_{\ell^{n_\ell}}$]{}; (KtildecapL1) at (0,-3) [$\tilde K \cap K_{\ell^{n_\ell}}$]{}; (K) at (0,-4) [$K$]{}; (L2)–(L2tilde)–(L1tilde)–(Ktilde); (L1)–(L1tildecapL2)–(L1tilde); (K)–(KtildecapL1)–(Ktilde); (L2)–(L1tildecapL2); (L1)–(KtildecapL1);

Since clearly $[\tilde K_{\ell^{n_\ell}}\cap K_{\ell^{n_0}}:K_{\ell^{n_\ell}}]$ divides $[\tilde K_{\ell^{n_\ell}}:K_{\ell^{n_\ell}}]$, which in turn divides $[\tilde K:K]$, and since $[\tilde K_{\ell^{n_0}}:\tilde K_{\ell^{n_\ell}}]=[K_{\ell^{n_0}}:\tilde K_{\ell^{n_\ell}}\cap K_{\ell^{n_0}}]$, we have $$\begin{aligned}
v_\ell\left([K_{\ell^{n_0}}:K_{\ell^{n_\ell}}]\right) & = v_\ell\left([K_{\ell^{n_0}}:\tilde K_{\ell^{n_\ell}} \cap K_{\ell^{n_0}}] 