This question unfortunately has a very similar name to this one, but I what want to ask here is different.

Let $K$ be a finite extension of $\mathbb{Q}_p$. It seems to be well known that the local version of the Fontaine-Mazur conjecture is false, in the sense that if $\rho:Gal(\overline{K}/K)\rightarrow GL_d(\mathbb{Q}_p)$ is some de Rham representation, it may very well be that $\rho$ does not come from a variety defined over a $p$-adic field. For instance, this is claimed by Mark Kisin in his survery paper "What is... a Galois representation?".

Maybe this question is naive, but can anyone give me a concrete example of such a representation?

Also, a slightly unrelated question: does one expect the answer to this version of the local Fontaine-Mazur conjecture to change if we allow representations coming from the etale cohomology of analytic spaces?

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    $\begingroup$ Take a crystalline representation. If it comes from a global object, the eigenvalues of Frobenius should be algebraic numbers (over the rationals). So just choose a crystalline representations not having this last property. $\endgroup$ Sep 12, 2019 at 9:12
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    $\begingroup$ Dear @LaurentBerger, thank you for your comment. In the case of a variety defined over $\mathbb{Q}_p$, but not necessarily over $\mathbb{Q}$, would it still be true that the eigenvalues are algebraic over $\mathbb{Q}$ (and not just over $\mathbb{Q}_p$)? (Note in my question above I asked also about representations coming from varieties defined over $\mathbb{Q}_p$, as in Kisin's article alluded to above) $\endgroup$
    – xlord
    Sep 13, 2019 at 11:45
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    $\begingroup$ Good point! I don't know what happens for varieties over $Q_p$. $\endgroup$ Sep 19, 2019 at 7:16
  • $\begingroup$ What if we consider representations coming from rigid analytic varieties? Does that have more of a chance of making some version of local Fontaine-Mazur true? $\endgroup$ Oct 22, 2020 at 18:44
  • $\begingroup$ What if we consider crystalline representations with integral characteristic polynomial of Frobenius with Weil number roots? $\endgroup$ Oct 25, 2020 at 19:03


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