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

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!)

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Jen, I have a one-page note in my files of a proof that every compact subgroup of GL_n(Q_p-bar) which is different from the proof in Skinner's paper. I'll .tex it up and post it shortly. –  KConrad Mar 10 '10 at 23:41
@KConrad: I know one too that uses Baire Cat Theorem and I think is in a paper of Dickinson? Somehow I suspect that the "bottom line" is that it's the same as Skinner's... –  Kevin Buzzard Mar 11 '10 at 0:04
Keith, that would be great...Thanks! –  Johnson-Leung Mar 11 '10 at 2:00
OK, see a link I set up below. –  KConrad Mar 11 '10 at 2:21
I just want to make a cultural remark that, although this is not written down in many places, it is well-known to everyone working in the field. People switch between the two settings (finite over $\mathbb Q_p$ or $\overline{\mathbb Q}_p$) depending on which is convenient. –  Emerton Mar 11 '10 at 15:26

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.

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Another reference: it's Lemme 2.2.1.1 of Breuil-Mezard's 2002 Duke paper (which you can find on Breuil's website), they say they learned the proof from J.-B. Bost. It's a Baire Category argument, so maybe this is the reference that Kevin was thinking of? –  D. Savitt Mar 11 '10 at 0:27
An older version is in Katz-Sarnak ("Random matrices, Frobenius eigenvalues and monodromy"), Lemma 9.0.8, with a Haar-measure proof that they say they learnt from Pop in 1995, and mention of the Baire category proof by Sinnott from 1989. They indicate that the result "does not seem to appear in the literature" at that time. –  Denis Chaperon de Lauzières Jun 18 at 6:01

I tried to cut and paste here an argument from a.tex file, but it came out looking like a complete mess, so here is a webpage link I set up: http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/GLnQpbar.pdf

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.

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This is the line of argument that I was attempting to make, but I accepted jnewton's answer because it came in first. I wanted to let you know that in your write-up you switch from $K_r$ to $G_r$ midway through the proof. Thanks again! –  Johnson-Leung Mar 11 '10 at 23:51
The switch to the congruence subgroups $G_r$ was actually intentional. At the point in the proof where they start being used, what matters is the way these subgroups shrink nicely to the identity, and for that purpose I thought it was better to emphasize the containment of the terms $g_i$ in the various subgroups $G_{d_i}$ instead of the finer information that they are in the subgroups $K_{d_i}$. However, since the switch admittedly can look like a typo, I added some text to the argument to point out the switch was being (deliberately) made. –  KConrad Mar 12 '10 at 0:23
Just a comment: I think the Schikhof proof is very natural from a functional analytic perspective. Indeed, if K is any complete normed field, and V is a normed linear K-space, then it's a basic fact that any finite-dimensional subspace of V is closed (and an obvious fact that any proper subspace has empty interior). It now follows immediately from Baire Category that there is no complete normed K-linear space of countably infinite dimension. Note that, more generally, a normed field extension L/K is complete iff [L:K] is finite. This is proved in Bosch-Guntzer-Remmert. –  Pete L. Clark Mar 12 '10 at 0:43
Something is missing at the end: C_p/Q_p is a normed field extension that is complete but [C_p:Q_p] is not finite (or I could use Q_p/Q instead). Lost a countability hypothesis in the second to last sentence? –  KConrad Mar 12 '10 at 0:58
@KConrad: I lost the word "algebraic": an algebraic normed field extension $L/K$ is complete iff $[L:K]$ is finite. –  Pete L. Clark Mar 12 '10 at 4:22