Let $ \rho:G_{\mathbb{Q}}\to {\rm GL}_{2}(\mathbb{Q}_{p}) $ be a continuous $ p $-adic representation of the absolute Galois group $ G_{\mathbb{Q}} $ of the rational number field $ \mathbb{Q} $. Let $ (\rho_{i})_{i\in \mathbb{N}} $ be a sequence of $ 2 $-dimensional $ p $-adic representations of $ G_{\mathbb{Q}} $. Then we say that $ (\rho_{i}) $ (uniformly trace) converges to $ \rho $ if $ |\text{tr}(\rho_{i}(g))-\text{tr}(\rho(g))|\to 0 $ uniformly for all $ g\in G_{\mathbb{Q}} $. Now suppose that $ \rho $ is absolutely irreducible and ramified at an infinite set $ S $ of primes of $ \mathbb{Q} $. My question is the following: is there a sequence $ (\rho_{i})_{i\in \mathbb{N}} $ of $ 2 $-dimensional $ p $-adic representations of $ G_{\mathbb{Q}} $ that converges to $\rho$ such that
- each $ \rho_{i} $ is finitely ramified.
- For each $i$, if $ \rho_{i} $ is ramified at the finite set $ S_{i} $ of primes, then $ S_{i}\subset S $?
Any comments and suggestions will be appreciated.
We remark that the notion of uniformly trace convergent was studied in [Khare, Chandrashekhar. "Limits of residually irreducible p-adic Galois representations." Proceedings of the American Mathematical Society (2003): 1999-2006.] in the residually irreducible case. See also [Bellaiche, J., Chenevier, G., Khare, C., & Larsen, M. (2005). Converging sequences of p-adic Galois representations and density theorems. International Mathematics Research Notices, 2005(59), 3691-3720.]
