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 $\rho\_p$ ?



local to global Galois representation
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.


