Is $H^1_{et}( \overline{C},\mathbb Q_l)$ irreducible as a Galois representation?

Question

Let $C$ be a smooth, projective, geometrically connected curve over $\mathbb Q$. We know that for any prime $\ell$ outside a finite set of primes, we have a $G_{\mathbb Q}$-representation $H^1_{et}( \overline{C},\mathbb{Q}_{\ell})$ whose dimension is twice the genus of the curve. Lets call any such $\ell$ as good. Here $\overline C$ is the curve base changed to $\overline{\mathbb Q}$.

Then a couple of idle questions:

1. Is it possible that $H^1_{et}( \overline{C},\mathbb{Q}_{\ell})$ is reducible as a $G_{\mathbb Q}$ representation for one of these good $\ell$? If so, is there any geometric consequence of this for the curve or it's Jacobian?
2. Is it possible for all the $H^1_{et}(\overline{C}, \mathbb{Q}_l)$ to be simultaneously reducible for all the good $\ell$?

Answer 1

First of all, in complete generality, if $(\rho_\ell\colon G_{\mathbb Q}\to \mathrm{GL}_n(\overline{\mathbb Q}_\ell))$ is a compatible system of $\ell$-adic Galois representations, then, conjecturally, $\rho_\ell$ is absolutely irreducible for some $\ell$ if and only if it is absolutely irreducible for all $\ell$.

This conjecture is wide open in general, but it is well-understood in your case, due to Faltings' theorem. To make things easier, I'm only going to talk about absolute irreducibility.

Let $J$ be the Jacobian of $C$. By a deep theorem of Faltings, the natural embedding $$\mathrm{End}_{\mathbb Q}(J)\otimes_{\mathbb Z}\overline{\mathbb{Q}}_\ell\hookrightarrow\mathrm{End}(T_\ell(J)\otimes\overline{\mathbb Q}_\ell)$$ is an isomorphism. By Schur's lemma, a representation is irreducible if and only if its endomorphism ring is a field. Thus,

$H^1(\overline C, \mathbb Q_\ell)$ is absolutely irreducible if and only if $\mathrm{End}_\mathbb Q(J) = \mathbb Z$.

The embedding, which is elementary, shows that $H^1(\overline C, \mathbb Q_\ell)$ is absolutely reducible if $\mathrm{End}_\mathbb Q(J) \ne \mathbb Z$. The content of Faltings' theorem is in the converse.

Note that this condition is clearly independent of $\ell$.

In particular, it's extremely common for $H^1(\overline C, \mathbb Q_\ell)$ to be absolutely reducible. @Watson has already mentioned CM curves in the comments. But another extremely important example is the modular curves $X_0(N)$. Here, the Galois representation decomposes as a direct sum of two-dimensional representations, each corresponding to a modular form in $S_2(\Gamma_0(N))$. Indeed, this process is exactly how one attaches Galois representations to weight $2$ modular forms!