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][2], 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][3]

>**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](http://www.math.ucsd.edu/~csorense/Sorensen_Carleton.pdf) 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.

  [2]: http://math.stanford.edu/~conrad/papers/notes.pdf
  [3]: http://www.ams.org/mathscinet-getitem?mr=1363495