Status of Fontaine-Mazur conjecture

Question

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.

Answer 1

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.

Assume $\rho:G_{\mathbb Q}\longrightarrow\operatorname{GL}_{2}(\bar{\mathbb Q}_{\ell})$ satisfies the following hypotheses. 1) $\rho|G_{\mathbb Q_{\ell}}$ is de Rham and its Hodge-Tate weights are distinct. 2) $\bar{\rho}$ is absolutely irreducible and odd. 3) $\bar{\rho}|G_{\mathbb Q_{\ell}}$ is neither the sum of two identical characters nor a twist of an extension of $\bar{\chi}_{cyc}$ by 1. Then $\rho$ occurs in the cohomology of smooth projective variety (namely the desingularization of some Kuga-Sato variety).

This is the culmination of the work of many people; the most recent being Pierre Colmez, Matthew Emerton and Mark Kisin.

Answer 2

I'm far from an expert on this, and there are certainly users here more qualified to answer than me, but for now here's a short answer.

This is now known in the case of two-dimensional representations $R:G_Q\rightarrow GL_2(\bar{Q}_\ell)$, due to Emerton and Kisin.

Kisin proves the result under the additional hypothesis that $R$ is odd and has distinct Hodge--Tate weights at $\ell$ (the same $\ell$ as the residue characteristic of the coefficient ring of $R$). By "having distinct Hodge--Tate weights at $\ell$", I mean that the restriction of $R$ to the decomposition group at $\ell$ $D_\ell\simeq Gal(\bar{Q}_\ell/Q_\ell)$ has distinct Hodge--Tate weights. I'm not completely sure on this part, but I believe that it's since been proved (by Calegari if I recall correctly; maybe under some small, almost always satisfied hypotheses) that there are no even representations with distinct Hodge--Tate weights at $\ell$, removing the requirement that $R$ be odd in almost all cases.

I'm less familiar with Emerton's contribution. Certainly, Kisin's paper requires on the local-global compatibility result of Emerton and some subsequent result by Colmez; otherwise one is forced to require further that $R$ is potentially semistable over some abelian extension. I have a vague idea that perhaps (in the same local-global compatibility paper) Emerton gives an independent proof of the same 2-dimensional result, but I'm even less familiar with this.

I guess I should say that the two main papers that I've mentioned are The Fontaine--Mazur conjecture for $GL_2$ by Mark Kisin, and Local-global compatibility in the $p$-adic Langlands program for $GL_2$ by Matt Emerton.

To the best of my knowledge, there hasn't been any substantial progress on the result for higher-dimensional representations. This seems to me to be due to the fact that there are essentially no results towards a $p$-adic local Langlands correspondence for groups other than $GL_2$, but that isn't based on anything other than guesswork.