TITLE: Is there a semisimple $\mathbf{Q}_\ell$-representation of $G_F$ ramified at an infinite set of places?
QUESTION [5 upvotes]: 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 $\dim{V} \geq 1$)?

REPLY [6 votes]: 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.