*maximal* if one of the following holds:

-   $E$ does not have CM over $\overline K$ and $H_{\ell^\infty}=\operatorname{GL}_2({\mathbb{Z}}_\ell)$.

-   $E$ has CM over $K$ by an order ${\mathcal{A}}$ in the imaginary quadratic field $F$, the prime $\ell$ is unramified in $F$ and does not divide $[\mathcal{O}_F : {\mathcal{A}}]$, and $H_{\ell^\infty}$ is conjugated to the Cartan subgroup of $\operatorname{GL}_2({\mathbb{Z}}_\ell)$ corresponding to ${\mathcal{A}}$.

-   $E$ has CM over $\overline K$ (but not over $K$) by an order ${\mathcal{A}}$ in the imaginary quadratic field $F$, the prime $\ell$ is unramified in $F$ and does not divide $[\mathcal{O}_F : {\mathcal{A}}]$, and $H_{\ell^\infty}$ is conjugated to the normaliser of the Cartan subgroup of $\operatorname{GL}_2({\mathbb{Z}}_\ell)$ corresponding to ${\mathcal{A}}$.

\[thm:CMSerre\] Let $E/K$ be an elliptic curve admitting CM over $\overline{K}$. Then the $\ell$-adic representation attached to $E/K$ is maximal for all but finitely many primes $\ell$.

In the rest of this section we recall various important properties of the torsion representations: we shall need results that describe both the asymptotic behaviour of the $\bmod \,\ell^n$ torsion representation as $n \to \infty$ (§\[subsect:MaximalGrowth\] and \[subsect:UniformGrowth\]) and the possible images of the $\bmod\,\ell$ representations attached to elliptic curves defined over the rationals (§\[subsec:PossibleImagesModl\]).

Maximal growth {#subsect:MaximalGrowth}
--------------

We recall 