to a pro-$\ell'$ group for $\ell \neq \ell'$ we have $W_\infty = \prod_\ell W_{\ell^\infty}$.

### The arboreal representation

Fix a sequence $\{\beta_i\}_{i\in{\mathbb{N}}}$ of points in $E(\overline{K})$ such that $\beta_1=\alpha$ and $N \beta_M=\beta_{M/N}$ for all pairs of positive integers $(N,M)$ with $N \mid M$. For every $N{\geqslant}1$ fix furthermore a ${\mathbb{Z}}/N{\mathbb{Z}}$-basis $\{T_1^N,T_2^N\}$ of $E[N]$ in such a way that $N T_1^M=T_1^{M/N}$ and $N T_2^M=T_2^{M/N}$ for every pair of positive integers $(N,M)$ with $N \mid M$. For every $N {\geqslant}1$, the map $$\begin{aligned}
\omega_N:\operatorname{Gal}(K_{N,N}\mid K)&\to \left({\mathbb{Z}}/N{\mathbb{Z}}\right)^2\rtimes\operatorname{GL}_2\left({\mathbb{Z}}/N{\mathbb{Z}}\right)\\
\sigma&\mapsto \left(\sigma(\beta_N)-\beta_N,\tau_{N}(\sigma)\right)\end{aligned}$$ is an injective homomorphism (similarly to [@JonesRouse Proposition 3.1]) and thus identifies the group $\operatorname{Gal}(K_{N,N}\mid K)$ with a subgroup of $\left({\mathbb{Z}}/N{\mathbb{Z}}\right)^2\rtimes\operatorname{GL}_2\left({\mathbb{Z}}/N{\mathbb{Z}}\right)$.

It will be important for our applications to notice that $V_N$ comes equipped with an action of $H_N$ coming from the fact that $V_N$ is the (abelian) kernel of the natural map $\operatorname{Gal}(K_{N,N}\mid K) \to H_N$. More precisely, the action of $h \in H_N$ on $v \in V_N$ is given by conjugating the element $(v, \operatorname{Id}) \in ({\mathbb{Z}}/N{\mathbb{Z}})^2 \rtimes \operatorname{GL}_2({\mathbb{Z}}/N{\mathbb{Z}})$ by $(0,h)$. Explicitly, we have $$(0,h) (v, \operatorname{Id}) (0,h)^{-1} = (hv,h)(0,h^{-1}) = (hv, \operatorname{Id}),$$ so that the action of $H_N$ on $V_N$ is induced by the natural action of $\operatorname{GL}_2({\mathbb{Z}}/N{\mathbb{Z}})$ on $\left({\mathbb{Z}}/N{\mathbb{Z}}\right)^2$. We 