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In Berger's "An introduction to the Theory of $p$-adic Representations", he mentions that due the the equivalence of categories between etale $(\varphi,\Gamma)$-modules and $p$-adic representations, it should be possible to recover all properties of $p$-adic representations in terms of $(\varphi,\Gamma)$-modules.

For example, in Herr's 1998 PhD thesis, he shows that the Galois cohomology $H^{i}(-,V)$ of a $p$-adic representation $V$ can be obtained from a complex involving the fixed elements under torsion of the Dieudonne module $\mathbb{D}(V)$.

Since Berger wrote in 2002, which properties do people think can (easily?) be recovered and which have so far?

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  • $\begingroup$ This question is really hard to interpret. Berger shows in his articles how to recover things like $D_{cris}(V)$ and $D_{st}(V)$ and so on directly from the $(\phi,\Gamma)$-module. You've said yourself how to recover the Galois cohomology. What is left? I am finding it hard to think of a property that is natural but which is not dealt with in the paper you reference. Can you give one specific example of a property that you don't know how to recover from the $(\phi,\Gamma)$-module? $\endgroup$
    – Kevin Buzzard
    Commented Oct 4, 2011 at 20:22
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    $\begingroup$ I've heard Colmez say that he doesn't know how to extract the $\varepsilon$-factor of a p-adic representation from its $(\varphi, \Gamma)$-module. $\endgroup$
    – David Loeffler
    Commented Oct 4, 2011 at 21:31
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    $\begingroup$ Well that sounds like a good answer to James' question! I'm a little confused though. Can't one read the $\epsilon$-factor off from $D_{cris}$? Or is this some fancier $\epsilon$-factor? <p> Regardless of all this I still find the question hard to interpret in the following sense: Colmez does know how to extract the $\epsilon$-factor -- he just applies the functor which sends the $(\phi,\Gamma)$-module to the corresponding representation and then gets the factor from there. But somehow the underlying premise of the question is that this isn't allowed. $\endgroup$
    – Kevin Buzzard
    Commented Oct 5, 2011 at 10:17

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