For a number field F, let E/F be an elliptic curve with CM by a quadratic field K. Let $\rho\_\ell: \text{Gal}\_F \to \text{Aut}(T\_{\ell}E)$ be the $\ell$-adic representation associated to E for some prime $\ell$, and let $\mathcal{G}\_{\ell}$ denote its image. According to Theorem 5 in Serre's "Groupes de Lie $\ell$-adiques attaches aux courbes elliptiques" (1966), $\mathcal{G}\_{\ell}$ is abelian if and only if K $\subset$ F.


Is it always the case that $\mathcal{G}\_{\ell}$ being abelian forces K $\subset$ F, or are there some restrictions on $\ell$ that I haven't been able to find in the paper?



Abelian image of l-adic representation
There are no restrictions on $l$. What's going on is that if the $CM$ endomorphisms are defined over $KF$ but not over $F$, there must be elements in the $\operatorname{Gal}(KF|F)$ that conjugate these endomorphisms to different endomorphisms. Since the endomorphisms act faithfully on the Galois representation, the only missing piece is to find an element of the Galois group that has the same action on the Galois-representation as one of the CM endomorphisms. One can check that the Frobenius elements of primes of $F$ that split in $KF$ act this way. 


