Does the image of a p-adic Galois representation always lie in a finite extension?

Question

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!)

Answer 1

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.

Answer 2

I tried to cut and paste here an argument from a.tex file, but it came out looking like a complete mess, so I'll give a link to a webpage link here.

Concerning the comments by Kevin and David about proofs using the Baire category theorem, I think the proof I posted above (due to Warren Sinnott) should be viewed in a different light. Consider the theorem that the alg. closure of $\mathbf Q_p$ is not complete. There are a couple of different proofs of it. (Note Jen said a proof of that noncompleteness theorem is what she was trying to adapt to prove the compactness theorem for the matrix groups, so I suspect the proof in the link above is the direction she was trying to go in, whether or not other proofs of the compactness theorem may be considered more slick.) I'll briefly describe two such proofs.

1. In the $p$-adic book by Koblitz, he explicitly constructs an infinite series $\sum c_ip^i$ with $c_i$ in $\overline{\mathbf Q}_p$ of absolute value 1 and increasing degree over $\mathbf Q_p$, and then use the increasing-degree condition on the coefficients to show the series can't converge in $\overline{\mathbf Q}_p$, although it's Cauchy since the general term tends to 0. (This is essentially what takes place in the compactness proof at the link I posted above, but in a multiplicative setting: form a product of matrices tending to the identity whose entries have higher and higher degree over $\mathbf Q_p$. The compactness hypotheses imply the product converges in $GL_n(\overline{\mathbf Q}_p)$ and then we get a contradiction. The same argument shows any compact additive subgroup of $\overline{\mathbf Q}_p$ is inside a finite extension of $\mathbf Q_p$.)

2. In the ultrametric analysis book by Schikhof, there is a proof that $\overline{\mathbf Q}_p$ is not complete which uses the Baire category theorem: the elements of $\overline{\mathbf Q}_p$ with degree up to $n$, as $n$ varies, provide a countable cover of $\overline{\mathbf Q}_p$ by closed subsets which each turn out to have no interior point, while of course their union $\overline{\mathbf Q}_p$ has many interior points. The closed set formulation of the Baire category theorem is that a countable union of closed subsets which each have no interior does not have an interior either. Thus we have a contradiction, so $\overline{\mathbf Q}_p$ is not complete.

I don't think these two strategies for proving a space is incomplete are the same, at least psychologically: in the first one you explicitly construct a non-convergent Cauchy sequence and in the second one you show a general property of complete spaces doesn't hold. For the same reason, I think the Baire and non-Baire proofs of this compactness theorem are pretty different proofs.