obtain similar statements by suitably passing to the limit in $N$:

\[lemma:HnActionOnVn\] For every positive integer $N$, the group $V_N$ is an $H_N$-submodule of $({\mathbb{Z}}/N{\mathbb{Z}})^2$ for the natural action of $H_N {\leqslant}\operatorname{GL}_2({\mathbb{Z}}/N{\mathbb{Z}})$ on $V_N {\leqslant}({\mathbb{Z}}/N{\mathbb{Z}})^2$. Similarly, both $V_{\ell^\infty}$ and $W_{\ell^\infty}$ are $H_{\ell^\infty}$-modules.

\[rem:Vsubgroup\] Let $N \in \mathbb{N} \cup \{\ell^\infty\}$ and $M \in \mathbb{N} \cup \{ \ell^\infty, \infty \}$ with $N \mid M$. Then $\operatorname{Gal}(K_{M,N} \mid K_M)$ can be identified with a subgroup of $V_N$: this follows from inspection of the diagram $$\xymatrix{
& K_{M,N} \ar@{-}[dr] \ar@{-}[dl] \\
K_M \ar@{-}[dr] && K_{N,N} \ar@{-}[dl]  \\
& K_M \cap K_{N,N} \ar@{-}[d] \\
& K_N
}$$ which shows that $\operatorname{Gal}(K_{M,N} \mid K_M)$ is isomorphic to $\operatorname{Gal}( K_{N,N} \mid K_M \cap K_{N,N})$, which in turn is clearly a subgroup of $\operatorname{Gal}(K_{N,N} \mid K_N) \cong V_N$.

Curves with complex multiplication {#subsect:CMCurves}
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If $\operatorname{End}_{{\overline{K}}}(E)\neq \mathbb{Z}$ we say that *$E$ has complex multiplication*, or CM for short. In this case $\operatorname{End}_{{\overline{K}}}(E)$ is an order in an imaginary quadratic field, called *the CM-field of $E$*. The torsion representations in the CM case have been studied for example in [@Deur1] and [@Deur2].

In this case, the image of the torsion representation $\tau_{\ell^\infty}$ is closely related to the *Cartan subgroup 