\operatorname{Gal}(\overline{K} \mid K) \to \operatorname{GL}_2(\hat{{\mathbb{Z}}})$. We denote by $H_{\ell^\infty}$ (resp. $H_\infty$) the image of $\tau_{\ell^\infty}$ (resp. $\tau_\infty$). The group $H_{\ell^\infty}$ (resp. $H_{\infty}$) is isomorphic to $\operatorname{Gal}(K_{\ell^\infty} \mid K)$ (resp. $\operatorname{Gal}(K_\infty \mid K)$).

One can also pass to the limit on the torsion subgroups themselves, obtaining the *$\ell$-adic Tate module* $T_\ell E=\varprojlim_n E[\ell^n] \cong {\mathbb{Z}}_\ell^2$ and the *adelic Tate module* $TE = \varprojlim_M E[M] \cong \hat{{\mathbb{Z}}}^2 \cong \prod_{\ell} {\mathbb{Z}}_\ell^2$.

### The Kummer representation

Let $M$ and $N$ be positive integers with $N\mid M$. Let $\beta\in E({\overline{K}})$ be a point such that $N\beta=\alpha$. For any $\sigma\in\operatorname{Gal}({\overline{K}}\mid K_M)$ we have that $\sigma(\beta)-\beta$ is an $N$-torsion point, so the following map is well-defined: $$\begin{array}{cccc}
\kappa_N: & \operatorname{Gal}({\overline{K}}\mid K_M) &\to & E[N]\\
& \sigma &\mapsto & \sigma(\beta)-\beta.
\end{array}$$ Since any other $N$-division point $\beta'$ of $\alpha$ satisfies $\beta'=\beta+T$ for some $T\in E[N]$, and the coordinates of $T$ belong to $K_N \subseteq K_M$, the map $\kappa_N$ does not depend on the choice of $\beta$. It is also immediate to check that $\kappa_N$ is a group homomorphism, and that the field fixed by its kernel is exactly the $(M,N)$-Kummer extension of $K$. Fixing a basis of $E[N]$ we can identify the Galois group $\operatorname{Gal}(K_{M,N}\mid K_M)$ with a 