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Let $\rho_p : \mbox{Gal}(\overline{\mathbb{Q}}_p / {\mathbb{Q}_p}) \to \mbox{GL}_n(\mathbb{Q}_p)$ be a de Rham $p$-adic representation. Can one find a representation $\rho : \mbox{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \mbox{GL}_n(\mathbb{Q}_p)$ such that $\rho$ is geometric (in the sense of Fontaine-Mazur) and such that the restriction of $\rho$ to $\mbox{Gal}(\overline{\mathbb{Q}}_p / {\mathbb{Q}_p})$ is $\rho_p$ ?

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up vote 5 down vote accepted

No. There are uncountably many unramified representations from the local Galois group to $\mathbb{Q}^{\times}_p$, since Frobenius can be sent to anything in $\mathbb{Z}^{\times}_p$. However, there are only countably many global representations of this form, since, by class field theory, they all factor through the Galois group of a (finite) cyclotomic field.

The general picture is pretty much the same --- the local Galois representations form large $p$-adic analytic families, yet, assuming the Fontaine-Mazur conjecture, there are only countably many geometric representations.

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We can also see this in another way, bypassing cardinality arguments, as follows. For a number field $K$ and a (continuous) character $G_K \rightarrow \overline{\mathbf{Q}}_p^{\times}$ ramified at only finitely many places, the image on inertia at the places away from $p$ is finite. So if unramified at all places dividing $p$ then must have finite image (since inertia subgroups of $G_K^{\rm{ab}}$ at the non-archimedean places topologically generate a finite-index subgroup, due to finiteness of class groups). So that gives a "concrete" obstruction to globalizing. – BCnrd Oct 12 '10 at 20:35
Dear Boris the Orange: Sure, I agree it's pretty much the same argument. I was mainly just commenting that we can relax the coefficients to be $\overline{\mathbf{Q}}_p$ and use any number field without harming the argument. I don't know what AM's background is, so it seemed worthwhile to point that out. – BCnrd Oct 12 '10 at 22:34
Dear Boris the Orange, The "Weil number argument", although only a pseudo-argument (in that it requires motivic rather than just geometric), is still interesting, I think, since it shows that the local analogue of Fontaine--Mazur is false (and so adds to the mystery of the --- presumably correct global (i.e. actual) FM conjecture). On the other hand, your cardinality argument also sheds important light on the situation. (And, since A M presumably knows the characterization of e.g. crystalline representations in terms of weakly admissible filtered $\varphi$-modules, it's easy for them to ... – Emerton Oct 13 '10 at 0:48
... see analytic families of local crystalline Galois reps. just by varying the data on the filtered $\varphi$-module side.) (Easier than by deformation theory, I mean.) – Emerton Oct 13 '10 at 0:49

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