See http://math.uni.lu/~wiese/galois/Boeckle-Luxemburg-Notes.pdf, Theorem 1.4(a): Is there an example of a semisimple $\mathbf{Q}_\ell$-representation $V$ of $G_F$ ($F$ a global field) ramified at a set $S$ of places where $S$ is not finite (for every $\dim{V} \geq 1$)?

Edit: It seems that for $\dim{V} = 1$, there is no such representation.

See http://math.uni.lu/~wiese/galois/Boeckle-Luxemburg-Notes.pdf, Theorem 1.4(a): Is there an example of a semisimple $\mathbf{Q}_\ell$-representation $V$ of $G_F$ ($F$ a global field) ramified at a set $S$ of places where $S$ is not finite (for every $\dim{V} \geq 0$)1$)?

Edit: It seems that for $\dim{V} = 1$, there is no such representation.

See http://math.uni.lu/~wiese/galois/Boeckle-Luxemburg-Notes.pdf, Theorem 1.4(a): Is there an example of a semisimple $\mathbf{Q}_\ell$-representation $V$ of $G_F$ ($F$ a global field) ramified at a set $S$ of places where $S$ is not finite (for every $\dim{V} \geq 0$)?