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The Fontaine-Mazur conjecture predicts that an $l$-adic Galois representation of a number field is 'geometric' if it is unramified outside a finite set of primes and is De Rham for primes lying over $l$. Now, what happens if one forgets about the latter restriction; are there any counterexamples, and is there any (general?) way to understand that those are not geometric without using the De Rham restriction?

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Dear Mikhail, Note that any $\ell$-adic rep'n of geometric origin is de Rham locally at $\ell$, by $\ell$-adic Hodge theory (what is usually called "$p$-adic Hodge theory), and so certainly the de Rham condition in the FM conjecture is necessary. The force of the conjecture is that it is also sufficient (along with the other obvious necessary condition of being unram. outside finitely many primes). Best wishes, Matthew –  Emerton Oct 22 '12 at 2:05
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up vote 10 down vote accepted

To complete Kevin's good answer: the number of $\ell$-adic representations (up to isomorphism) of a number field $K$ is countable, since so are varieties over a $K$. On the other hand, we know by Mazur's theory of deformations that representations of the type you consider that is, of the Galois group of the maximal extensions of a number field $K$ unramified outside a finite set of places $S$ containing places above $\ell$ and $\infty$) form multi-dimensional $\ell$-adic family (e.g. parametrized by spaces like $\mathbb{Z}_\ell^n$ for some $n>0$, hence are uncountable. Thus not only are there counter-examples to Fontaine-Mazur's conjecture without the de Rham hypothesis, but most examples of such representations are counter-examples.

Among all representations, the de Rham (or geometric) representations are expected to be dense in certain cases (e.g. representation of dimension 2 of $\mathbb Q$) but not in general (e.g. representations of dimension $2$ of a non-totally real number field)

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Dear Joël, Just to clarify, let me note here that "dense" means "Zariski dense" (or at least, I'm not sure whether any stronger sense of dense would be correct). Best wishes, Matthew –  Emerton Oct 22 '12 at 2:02
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P.S. Also, in your first sentence, I guess you mean $\ell$-adic reps. of geometric origin. –  Emerton Oct 22 '12 at 2:03
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In fact there are one-dimensional counterexamples: if $\chi$ is the $l$-adic cyclotomic character, and $k\in \mathbb{Z}_l \backslash \mathbb{Z}$, then $\chi^{(l-1)k}$ is unramified outside $l$, but does not arise from geometry (in particular it is not de Rham).

A general way of understanding 'all' Galois representations is via Mazur's universal deformation rings. These parametrize all $l$-adic Galois representations (unramified outside a prescribed set of primes and perhaps with extra conditions) whose reduction mod $l$ is isomorphic to a given mod $l$ representation. The $R=T$ theorems you may have heard of state that certain universal deformation rings are isomorphic to rings coming from the theory of modular forms.

I don't know if it's expected in general whether all Galois representations are limits of deRham representations.

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