Fix $p>0$ a rational prime, and $K$ an algebraic number field with Galois group $G\_K:=Gal(\bar{\mathbb{Q}}/K) $. The Fontaine-Mazur conjecture predicts that if $\rho:G\_K\rightarrow GL(V)$ is a finite dimensional $\mathbb{Q}\_p$-representation, then it comes from a motive over $K$ (like a subquotient of $H^i\_c(X\times\_K\bar{\mathbb{Q}}, \mathbb{Q}\_p)$) exactly when it is unramified over almost every finite place, and potentially semi-stable over those finite prime dividing $p$.


My question is the contrary: how many examples do we have for $p$-adic Galois representations having infinite images but for which the conditions of Fontaine-Mazur fails? Maybe it should be difficult to construct them when the dimension of the representation is large? Could one get continuous $p$-adic representations that ramifies at infinitely many places?



how irregular can a $p$-adic Galois representation be?
Here are two things that can occur:


1) If V is the representation attached to an overconvergent modular form f, then V will be unramified at almost every prime but will not be de Rham at p (unless f is classical).


2) Ramakrishna has written an article "Infinitely ramified Galois representations". Here's part of the introduction: "In this paper we show how to construct [...] representations [...] that are ramified at an infinite number of primes." Under GRH, these repns are crystalline at p.


So both conditions in FM can fail independently.


