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?


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.


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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