subgroup of $({\mathbb{Z}}/N{\mathbb{Z}})^2$. It is then clear that $K_{M,N}$ is an abelian extension of $K_M$ of degree dividing $N^2$, and the Galois group of this extension has exponent dividing $N$.

In the special case $M=N$ we denote by $V_N$ the image of $\operatorname{Gal}\left( K_{N,N} \mid K_N \right)$ in $({\mathbb{Z}}/N{\mathbb{Z}})^2$.

By passing to the limit in the previous constructions we also obtain the following:

-   There is an $\ell$-adic Kummer representation $\kappa_{\ell^\infty} : \operatorname{Gal}(\overline{K} \mid K_{\ell^\infty}) \to T_\ell E$ which factors via a map $\operatorname{Gal}(K_{\ell^\infty, \ell^\infty} \mid K_{\ell^\infty}) \to T_\ell E$ (still denoted by $\kappa_{\ell^\infty}$).

-   The image $V_{\ell^\infty}$ of $\kappa_{\ell^\infty}$ is a sub-${\mathbb{Z}}_\ell$-module of $T_\ell E \cong {\mathbb{Z}}_\ell^2$, isomorphic to $\operatorname{Gal}(K_{\ell^\infty, \ell^\infty} \mid K_{\ell^\infty})$ as a profinite group. We therefore identify $\operatorname{Gal}(K_{\ell^\infty, \ell^\infty} \mid K_{\ell^\infty})$ with $V_{\ell^\infty}$.

-   We can identify the Galois group $\operatorname{Gal}(K_{\infty, \ell^\infty} \mid K_\infty)$ with a ${\mathbb{Z}}_\ell$-submodule $W_{\ell^\infty}$ of $V_{\ell^\infty}$ (hence also of $T_\ell E$) via the representation $\kappa_{\ell^\infty}$.

-   We can identify the Galois group $\operatorname{Gal}(K_{\infty, \infty} \mid K_\infty)$ with a sub-$\hat{{\mathbb{Z}}}$-module $W_\infty$ of $TE \cong \hat{{\mathbb{Z}}}^2$.

Notice that $W_{\ell^\infty}$ is the projection of $W_{\infty}$ in ${\mathbb{Z}}_\ell^2$, and since $W_{\ell^\infty}$ is a pro-$\ell$ group and there are no nontrivial continuous morphisms from a pro-$\ell$ group 