3

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?

flag
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? – Kevin Buzzard Oct 4 2011 at 20:22
6 
 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. – David Loeffler Oct 4 2011 at 21:31
1 
 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. – Kevin Buzzard Oct 5 2011 at 10:17

Your Answer

Get an OpenID
or

Browse other questions tagged or ask your own question.