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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?

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up vote 18 down vote accepted

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.

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On the contrary, it is not so hard by deformation theoretic arguments to create $p$-adic Galois representations which are not even Hodge--Tate at $p$ (and hence not de Rham (equivalently, pst) at $p$). So a reasonable intuition is that "most" $p$-adic Galois representations are not de Rham at $p$.

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