I have been looking at Serre's conjecture and noticed that there are two conventions in the literature for a p-adic representation $\rho:\mbox{Gal}(\bar{\mathbb Q}/\mathbb Q)\to \mbox{GL}(n,V).$ In some references (eg Serre's book on $\ell$-adic representations), $V$ is a vector space over a finite extension of $\mathbb Q_p$. However, in more recent papers (eg Buzzard, Diamond, Jarvis) $V$ is a vector space over $\bar{\mathbb Q_p}$. It is easy to show that the former definition is a special case of the latter, but I suspect, and would like to prove that they are actually the same. That is, I would like to show that the image of any any continuous Galois representation over $\bar{\mathbb{Q}_p}$ actually lies in a finite extension of $\mathbb Q_p$.
Is this the case?
I think that a proof should use the fact that $G_{\mathbb Q}$ is compact and that $\bar{\mathbb Q}_p$ is the union of finite extensions. I have tried to mimic the proof that $\bar{\mathbb Q}_p$ is not complete, but have not been able to find an appropriate Cauchy sequence in an arbitrary compact subgroup of GL($n,V$).
(This is my first question, so please feel free to edit if appropriate. Thanks!)
Best Answer
A proof of the result you're after is contained at the beginning of section two of a recent paper of Skinner here. Skinner mentions that references for this fact seem to be rare.