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).



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy