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?



Can one ignore primes lying over $l$ in the Fontaine-Mazur conjecture? Counterexamples?
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)


