\right) + v_\ell\left( [\tilde K_{\ell^{n_\ell}}\cap K_{\ell^{n_0}}:K_{\ell^{n_\ell}}]\right) \\ &{\leqslant}v_\ell\left([\tilde K_{\ell^{n_0}}:\tilde K_{\ell^{n_\ell}}]\right)+ v_\ell\left( [\tilde K:K]\right).
\end{aligned}$$ By [@2016arXiv161202847L Theorem 12] we have $v_\ell\left([K_{\ell^{n_0}}:K_{\ell^{n_\ell}}]\right)=\delta(n_0-n_\ell)=\delta\left( v_\ell\left([\tilde K:K]\right)+1\right)$, where $\delta$ is as in Proposition \[prop-max-growth\], and we get $$\begin{aligned}
v_\ell\left([\tilde K_{\ell^{n_0}}:\tilde K_{\ell^{n_\ell}}]\right){\geqslant}\delta+ (\delta-1)v_\ell\left([\tilde K:K]\right)> (\delta-1)(n_0-n_\ell).\end{aligned}$$ Consider now the tower of extensions $$\begin{aligned}
\tilde K_{\ell^{n_\ell}}\subseteq \tilde K_{\ell^{n_\ell+1}}\subseteq \cdots \subseteq \tilde K_{\ell^{n_0}}\end{aligned}$$ and notice that by the pigeonhole principle for at least one $n\in {\left\{ n_\ell, n_\ell+1,\dots, n_0-1 \right\}}$ we must have $[\tilde K_{\ell^{n+1}}:\tilde K_{\ell^{n}}]{\geqslant}\delta$. But then by [@2016arXiv161202847L Theorem 12] we have maximal growth over $\tilde K$ from $n< n_0$. Thus we get $\tilde n_\ell{\leqslant}n_\ell+v_\ell\left([\tilde K:K]\right)$ as claimed.

Uniform growth of $\ell$-adic representations {#subsect:UniformGrowth}
---------------------------------------------

The results in this subsection and the next will be needed in Section \[sec:UniformBounds\]. We start by recalling the following result, due to Arai:

\[thm:Arai\] Let $K$ be a number field and let $\ell$ be a prime. Then there exists an integer $n {\geqslant}0$, depending only on $K$ and $\ell$, such that for any elliptic curve $E$ over $K$ with no complex multiplication over $\overline{K}$ we have $$\tau_{\ell^\infty}(\operatorname{Gal}(\overline{K} \mid K)) \supseteq \{M \in \operatorname{GL}_2({\mathbb{Z}}_\ell) : M \equiv \operatorname{Id}\pmod{\ell^n}\}.$$

For the next result we shall need a well-known Lemma about twists of 