In "On $\ell$-adic representations attached to modular forms II", Ribet proved that the $\ell$-adic representation $\rho\_{f,\ell}$ attached to a non-CM newform form $f$ satisfies
$${\rm SL}\_2(\mathbb{F}\_\ell)\subset \overline{\rho}\_{f,\ell}(G\_\mathbb{Q}).\qquad\qquad(\*)$$


I am wondering:


(i) Why does $(\*)$ imply that $\overline{\rho}\_{f,\ell|G\_{\mathbb{Q}(\zeta\_\ell)}}$ is absolutely irreducible?


(ii) Let $n$ be a positive integer. Does $(\*)$ imply that ${\rm Sym}^n\overline{\rho}\_{f,\ell|G\_{\mathbb{Q}(\zeta\_\ell)}}$ is irreducible? If the answer is negative, under what condition does this hold?



Irreducibility of the $n$th symetric power of the reduction of the Galois representation of a non-CM newform
Here's an answer that's more work than Kevin Ventullo's, but works for $\ell=2$ and $\ell=3$, and gives you a method that will work in various more general cases.


There is a surjection $\pi\_1\colon G\_\mathbb Q \to \overline{\rho}\_{f,l} (G\_{\mathbb Q})$ and a surjection $\pi\_2 \colon G\_\mathbb Q \to \operatorname{Gal}(\mathbb Q(\zeta\_\ell) / \mathbb Q)$. The group $\overline{\rho}\_{f,l} (G\_{\mathbb Q(\zeta\_\ell)})$ is the image under $\pi\_1$ of the kernel of $\pi\_2$. In other words, it is the intersection of the image of the product map $(\pi\_1,\pi\_2) \colon G\_\mathbb Q \to \overline{\rho}\_{f,l} (G\_{\mathbb Q}) \times \operatorname{Gal}(\mathbb Q(\zeta\_\ell) / \mathbb Q)$ with $G\_{\mathbb Q}) \times 1$.


By Goursat's lemma, the image of the product map is $ \{ (a,b) \mid f\_1(a) =f\_2 (b) \}$ where $f\_1 \colon \overline{\rho}\_{f,l} (G\_{\mathbb Q}) \to Q$ and $f\_2 \colon \operatorname{Gal}(\mathbb Q(\zeta\_\ell) / \mathbb Q) \to Q$ are surjections for some finite group $Q$. Expressed in this language, $\overline{\rho}\_{f,l} (G\_{\mathbb Q(\zeta\_\ell)})$ is the kernel of $f\_1$.


Now because $f\_2$ is surjective, $Q$ is a quotient of a cyclic group of order $\ell-1$, and thus is a cyclic group of order dividing $\ell-1$.


So $\overline{\rho}\_{f,l} (G\_{\mathbb Q(\zeta\_\ell)})$ is the kernel of a map from $\overline{\rho}\_{f,l} (G\_{\mathbb Q(\zeta\_\ell)})$ to a cyclic group of order dividing $\ell-1$. Thus it contains the kernel of a map from $SL\_2(\mathbb F\_\ell)$ to the kernel of a cyclic group of order dividing $\ell-1$.


Since the abelianization of $SL\_2(\mathbb F\_\ell)$ is trivial (for $\ell>3$) or has order relatively prime to $\ell-1$ (for $\ell=2,3$, since $2$ is relatively prime to $1$ and $3$ is relatively prime to $2$), $\overline{\rho}\_{f,l} (G\_{\mathbb Q(\zeta\_\ell)})$ contains $SL\_2(\mathbb F\_\ell)$.


Since $SL\_2(\mathbb F\_\ell$ acts absolutely irreducibly on the standard representation for all $\ell$, it follows that $\overline{\rho}\_{f,l} (G\_{\mathbb Q(\zeta\_\ell)})$ acts absolutely irreducibly on the standard representation, as desired.


It immediately follows that the image of $\operatorname{Sym}^n \overline{\rho}\_{f,\ell} \mid\_{ G\_{\mathbb Q(\zeta\_\ell)}}$ containsthe image of $SL\_2(\mathbb F\_\ell)$ under the $n$th symmetric power representation. Since the $n$th symmetric power representation of $SL\_2(\mathbb F\_\ell)$ is absolutely irreducible for $n \leq \ell-1$, the representation $\operatorname{Sym}^n \overline{\rho}\_{f,\ell} \mid\_{ G\_{\mathbb Q(\zeta\_\ell)}}$ is absolutely irreducible for $n\leq \ell-1$.


