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Fix a prime $l$. Let $\phi:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to GL_n(\mathbb{Q}_l)$ be a semisimple continuous representation. Assume $\phi$ has finite image when restricted to $\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$ for any $p$ (including $p=l$). Can $\phi$ have infinite image?

Since $\phi$ is de Rham at $l$, to have infinite image it must be ramified at infinitely many places. Examples of semisimple infinitely ramified Galois representations were constructed in a paper of Ramakrishna (note that the ramified primes necessarily have zero density).

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