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In an assignment, I have to give an example of a 2-dimensional $\ell$-adic representation of the absolute Galois group of $\mathbb{Q}$, bu I am faced with the problem that I do not a lot of these. Or not enough to find the example I am looking for.

More precisely, let $G$ be the absolute Galois group in question, and let $K$ be a (fixed) finite extension of $\mathbb{Q}_\ell$, for some prime $\ell$. I am looking for a representation $ \rho : G \to GL_2(\bar{K}).$

The examples I do know are the trivial representation and the ones arising by taking the direct sum or the tensor product of characters (that is, 1-dimensional representations) of $G$. The problem is that they do not seem to give what I am looking for.

Added: Most $\ell$-adic representations that arise from geometry have image lying in $GL_2(\mathcal{O}_K)$. Although these are very interesting representations, and that (in some sense) you can always reduce it to this case (see BR comments below), what I am looking for are representations that cannot be conjugated (by any element of $GL_2(\bar{K})$) in such a way that the image lies in $GL_2(\mathcal{O}_K)$.

The reason the original phrasing was ambiguous and unclear is that I was hoping to build a better repertoire of Galois representations, in the hope of eventually finding an example with the desired properties.

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    @MattE I see now that the problem, as stated, is much easier than I first thought. Although a representation such as the one you propose would not answer the particular question I am being asked in my assignment, I feel like my question has confused too many people as it is, and so I will not reformulate the question another time to include the extra hypotheses. I will just go back to my homework. Thank you all for your time, and sorry for the confusion.2011-11-27

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The question remains unclear. As BR and Keenan Kidwell have pointed out, any continuous Galois representations with values in $GL_2(K)$ (with $K$ a finite extension of $\mathbb Q_{\ell}$) can always be conjugated by an element of $GL_2(K)$ into $GL_2(\mathcal O_K)$.

What exactly is is that you want?

Added in response to the comment below: If a continuous Galois rep. takes values in $GL_2(\overline{K})$, then it lies in $GL_2(L)$ for some finite extension $L$ of $K$, and so lies in $GL_2(\mathcal O_L)$ after conjugating by an element of $GL_2(L)$. So this changes nothing, except for replacing the label $K$ by the label $L$ (and since $K$ was arbitrary, this is not a real change).

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    @M Turgeon: De$a$r M Turgeon, See my additional remarks. Regards,2011-11-27