Extending systems of l-adic representations to other l

I'm asking this not because I have an idea how one might approach it, but because it seems natural and inherently interesting.

Let $K$ be a number field, $G_K$ its absolute Galois group, and $\ell\neq\ell'$ prime numbers. An $\ell$-adic representation of $K$, i.e. a continuous homomorphism $\rho_\ell$ from $G_K$ to $\mathrm{GL}(n,\mathbb Q_{\ell})$ is said to be compatible with an $\ell'$-adic representation $\rho_{\ell'}$ of $K$ if for almost all (i.e. all but finitely many) places $v$ of $K$, the representations $\rho_\ell,\rho_{\ell'}$ are unramified, and the characteristic polynomials of $\rho_\ell(F_v)$ and $\rho_{\ell'}(F_v)$ are equal and have rational coefficients, where $F_v \in D_v/I_v \subseteq G_K$ denotes the Frobenius at $v$.

Note that by the Chebotarev density, this uniquely determines the trace of $\rho_{\ell'}$, which, in turn, uniquely determines it up to semi-simplification.

If $\mathcal{L}$ is a collection of prime numbers, then a strictly compatible system of $\ell$-adic representations consists of an $\ell$-adic representation $\rho_\ell$ of $K$ for each $\ell \in \mathcal{L}$ such that there is a finite set $S$ of places of $K$ (called the exceptional set) such that for $\ell\neq\ell'$, $v \notin S$, and $v \nmid \ell\ell'$, the representations $\rho_\ell$ and $\rho_{\ell'}$ are unramified at $v$, and the characteristic polynomials of $\rho_\ell(F_v)$ and $\rho_{\ell'}(F_v)$ are equal and have rational coefficients.

Can we find an $\ell$-adic representation unramified almost everywhere with integral traces of Frobenius and a prime $\ell'$ such that there is no $\ell'$-adic representation compatible with it? The Fontaine-Mazur conjecture would predict that the representation cannot be geometric.

But what's more, I'm wondering the following: suppose we have a strictly compatible system of $\ell$-adic representations for almost all $\ell$. Then can we necessarily extend this to a strictly compatible system for all $\ell$?

• I think your definition is non-standard. See Taylor's ICM paper for how you might want to do it (you don't want equality of traces but of char polys, and you don't want traces to be integral but instead to land in a coefficient field, a number field). If you assume that your $\ell$-adic representation is potentially semistable at all primes above ell as well then it should conjecturally live in a family because it should be attached to an automorphic representation. But without any hypothesis at $\ell$ I'm not sure what to expect. – user30035 Dec 31 '12 at 12:58
• I should probably also add that the recent preprint on potential automorphy by Barnet-Lamb, Gee, Geraghty and Taylor (see Taylor's web page) contains what I believe is the state of the art on such questions. With some extra added assumptions one can prove that the rep is potentially automorophic, which is strong enough to go a long way. But the theory isn't mature enough yet to answer your question positively, I don't think. I wouldn't rule out weird counterexamples either. – user30035 Dec 31 '12 at 13:12
• Yes, I don't know of how this might be answered. I asked it because it seemed natural and interesting, not because I thought it might be approachable. – David Corwin Dec 31 '12 at 13:29
• In some sense it's very unnatural, because I cannot really imagine how you're going to find an ell-adic representation in any way other than via etale cohomology in some guise, and then potential semistability, a crucial condition if you actually want to prove anything, will come for free :-) Here's another point of view -- what could you possibly prove about e.g. L-functions of varieties, or the Langlands Programme, or ??, could you prove if you could answer your question positively? What applications does your question have that the more natural question (addressed by Taylor et al) does not? – user30035 Dec 31 '12 at 13:42
• I'm not saying it has applications. It just seems like a natural question to ask. It says something about the structure of the Galois group of the number field and convergence $\ell$-adically for different primes $\ell$. – David Corwin Dec 31 '12 at 22:21