In the language of Richard Taylor's 2004 (extended) ICM article (''Galois Representations'', Annales de la faculté des sciences de Toulouse (2004) Tome XIII, no. 1, 73-119), the conjecture is the following
Conjecture: (Fontaine-Mazur) Suppose that
$$R\colon G_{\mathbf{Q}}\rightarrow \mathrm{GL}(V),$$
is an irreducible $\ell$-adic representation which is unramified at all but finitely many primes and with the $R|_{G_{\mathbf{Q}_\ell}}$ de Rham. Then there is a smooth projective variety $X/\mathbf{Q}$ and integers $i\ge 0$ and $j$ such that $V$ is a subquotient of $H^i(X(\mathbf{C}),\overline{\mathbf{Q}}_\ell(j))$. In particular $R$ is pure of some weight $w\in \mathbf{Z}$.
(here $G_K$ means absolute galois group of $K$ etc.) The notion of a de Rham representation is rather long – it may be found e.g. in the lecture notes of O. Brinon and B. Conrad here, and see loc. cit. for explanations of the other conditions. The references for the conjecture are Fontaine (J.M. Fontaine talk at ''Mathematische Arbeitstagung 1988'', Max-Planck-Institut für mathematik preprint no. 30 of 1988) and Fontaine-Mazur
Question: What is the current status of this conjecture? What results are known in its direction?
There seems to be a lot of research papers published in the last number of years on this topic. I would be grateful if anyone would be able to present some kind of rough panorama of results related to the conjecture.
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Edit: In a Feb. 2015 survey article of C.M. Sorensen here on the Breuil-Schneider conjecture, there is a description given (in §1) of the contribution of M. Emerton to F.-M. conjecture together with a brief sketch of the method of proof.
Best Answer
It is true if $V$ is of dimension 1, essentially by class field theory (as you are considering only representations of $G_{\mathbb Q}$). Otherwise, it is still largely open for the following reasons.
If $n\geq 2$, the best tool we have to study $G_{\mathbb Q}$-representations are so called modularity lifting theorems. These theorems take as hypotheses the hypotheses of the Fontaine-Mazur conjecture plus supplementary conditions (typically, assumptions on the residual irreducibility of $\rho$ and stronger conditions on $\rho|G_{\mathbb Q_{\ell}}$ than just being de Rham) and prove that $\rho$ then comes from an automorphic representation $\pi$ of $\mathbf{G}(\mathbb A_{\mathbb Q}^{(\infty)})$ (the finite adelic points of a reductive group $\mathbf{G}$). However, except in very rare cases, we don't know that such Galois representations (automorphic galois representations for short) occur in the cohomology of smooth projective varieties, so even when they apply, the full conclusion of the Fontaine-Mazur conjecture is largely out of reach.
Worse, notice how above I mentioned that modularity lifting theorems required some supplementary local assumptions at $\ell$. One of them is that the Hodge-Tate weights of $\rho|G_{\mathbb Q_{\ell}}$ are distinct. This happens for a very good reason: even when $n=2$, there are some automorphic Galois representations (conjecturally satisfying the hypotheses of the Fontaine-Mazur conjecture) we believe exist but don't know how to construct or study, namely the automorphic Galois representation attached to algebraic Maas forms (those with eigenvalue $1/4$). A proof of the full Fontaine-Mazur conjecture even for $n=2$ would need to deal with those, and this is currently seemingly out of reach.
A bit of good news after this gloom assessment. If $n=2$ and if you are fine with disregarding automorphic Galois representations attached to Maas forms (in particular if you assume that the Hodge-Tate weights are distinct), then the supplementary hypotheses required to establish the full Fontaine-Mazur conjecture are rather mild. I don't pretend to know the cutting edge, but the following theorem should certainly be true.
This is the culmination of the work of many people; the most recent being Pierre Colmez, Matthew Emerton and Mark Kisin.