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I am currently studying $p$-adic Hodge theory and searching for help.

If $X$ is a variety over a p-adic field $K$ (should we take it global ? $p$-adic ?), then the étale cohomology $H^i(X_{\bar K}, \mathbb{Q}_p)$ is equipped with the Galois action of $G_K = Gal(\bar K \mid K)$ incuded functiorially by the action on $X_{\bar K} = X \underset{Spec(K)}{\times} {Spec(\bar K)}$. This gives a $p$-adic representation.

It seems to be common knowledge that this representation should be de Rham. In fact, people somehow go as far as saying things like "any Galois representation that comes from geometry is de Rham", which seems too vague to me to be a provable statement. What do people mean by that ? Where can I find a complete proof/reference for these facts.

The mentions of such facts I've found so far :

  • Berger says (in "An introduction to the theory of $p$-adic representations", page 19-20) the following :

[Tsuji] showed that if $X$ has semi-stable reduction, then $V=H^i_ {ét}(X_{\bar K},\mathbb{Q}_p)$ is $B_{st}$-admissible. A different proof was given by Niziol (in the good reduction case) and also by Faltings (who proved that $V$ is crystalline if $X$ has good reduction and that $V$ is de Rham otherwise).

The article of Tsuji which it refers to was a bit unhelpful, but I might have missed something. The fact that interests me seems to be the one Faltings proved, but I can't get a reference.

Another weird thing is that some people talk about the case where $K$ is a global field (like in the mathoverflow question I linked), while some others take $K$ to be a $p$-adic local field (like the thesis I referred to). Are both cases true ? Also, why do people seem to be handwavy about being de Rham being always true as soon as representations arise from geometry, when the only precise statement I found requires precise hypotheses like "being a smooth projective variety".

This seems to be the starting point of the Fontaine-Mazur conjecture, which is a kind of converse, so this should probably be an easy to find result, but I can't see it proven anywhere...

Has anyone got a reference which would answer all my questions ?

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  • $\begingroup$ what do you mean specifically when you say "Are both cases true?"? $\endgroup$
    – user691994
    Aug 1 '19 at 11:28
  • $\begingroup$ Can we state the theorem for global fields AND for $p$-adic local fields ? The statement of the question doesn't seem to require any condition on the field K. $\endgroup$ Aug 1 '19 at 11:30
  • $\begingroup$ if you are over a global field, then you also have to choose an embedding of the Galois group of the local field to the Galois group of the global field, right? How are you making this choice? $\endgroup$
    – user691994
    Aug 1 '19 at 11:31
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    $\begingroup$ If I have an element $g$ of the Galois group $G_K = Gal(\bar K \mid K)$, it is a map $\bar K \rightarrow \bar K$, so I functorially get a map $Spec(\bar K) \rightarrow Spec(\bar K)$ and then a map $X_{\bar K} \rightarrow X_{\bar K}$ (because the action on $Spec(K)$ is trivial so I can apply the fiber product universal property), and then functorially a map $H^i(X_{\bar K}, \mathbb{Q}_p) \rightarrow H^i(X_{\bar K}, \mathbb{Q}_p)$, which I may call $\rho(g)$, and thus $\rho$ is a $p$-adic representation of $G_K$. At which point don't you agree ? I didn't have to make any choice. $\endgroup$ Aug 1 '19 at 11:38
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    $\begingroup$ what is a de Rham representation of the Galois group of a global field? I think there is no standard definition (not involving a choice that does not matter). $\endgroup$
    – user691994
    Aug 1 '19 at 11:42
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I am not an expert on this topic but then, there are few experts on this topic active on MSE. I hope Alex Youcis can write a detailed answer (and criticize mine if I make factually or philosophically wrong claims).

Where can you find a proof that for a smooth proper variety over a finite extension of $\mathbb{Q}_p$ the representation of the absolute Galois group on the $p$-adic etale cohomology is de Rham? Some options:

There are much earlier works but I know close to nothing about them (typically when a theorem has both a modern proof and an old proof, I find the modern proof more enjoyable but tastes differ).

It should be noted that the first one is relying in part on the perfectoid ideas which are not necessary for the proof that the cohomology of algebraic varieties is de Rham (but is apparently necessary for analytic varieties). Whether that is a pedagogically a good or a bad thing, you judge for yourself.

I personally think that Scholze's writing style is tidier than Beilinson's, maybe you will find it easier to understand his proof.

Why are people being imprecise about what exactly is a representation coming from geometry? I do not know. I think certain conjectures about etale cohomology (if you take compact support) should extend to non-smooth non-proper varieties but I am not sure what exactly are the expectations. Some posts by David Loeffler on MathOverflow may be helpful.

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