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See, 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 $\dim{V} \geq 1$)?

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@Timo: I've added a tag and would also urge you to incorporate your own "answer" into an edited question to avoid confusion and focus the question better. – Jim Humphreys Aug 19 '12 at 23:00

In the case $\dim V=2$, my advisor Ravi Ramakrishna has shown in his paper Infinitely ramified Galois representations (Ann. Math. 151) that there are surjective representations $\rho\colon G_\mathbf{Q}\to \mathrm{GL}_2(\mathbf{Z}_p)$ ramified at infinitely many primes.

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up vote 4 down vote accepted

It seems that for $\dim{V} = 1$, there is no such representation p. 10.

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