Let $X$ be a smooth, projective variety defined over some $p$-adic field $K$. Is it true that if the etale cohomology $H^i_{et}(X_{\overline{K}},\mathbb{Q}_\ell)$ is crystalline at $\ell=p$, then $H^i_{et}(X_{\overline{K}},\mathbb{Q}_\ell)$ is unramified for $\ell\neq p$? If it is true, can someone provide a reference for this fact?
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1$\begingroup$ I suspect this is unknown without additional hypotheses. (It's OK for $i=1$ and should be OK when the cohomology is known to be automorphic). $\endgroup$– Will SawinCommented Aug 16 at 13:36
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$\begingroup$ @WillSawin Thanks for the comment! Can you give a reference, or a reason, why it should be OK for automorphic representations? $\endgroup$– T.Ch.Commented Aug 19 at 16:57
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1$\begingroup$ I think more precisely this is known for Galois representations arising from regular algebraic conjugate self-dual automorphic forms on $GL_n$ over CM fields (and thus, I guess, also over totally real fields). The most difficult step is compatibility of the local Langlands correspondence and global Langlands correspondence when $\ell=p$ which was done in this level of generality by Caraiani in her paper Monodromy and local-global compatibility for $l=p$. $\endgroup$– Will SawinCommented Aug 19 at 18:14
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2$\begingroup$ This says that the Weil-Deligne representation associated by $p$-adic Hodge theory to the $\ell$-adic Galois representation arising from the automorphic form for $\ell=p$ agrees with the Weil-Deligne associated by the local Langlands correspondence to the local representation at $p$, which, by the easier $\ell\neq p$ case of local-global compatiblity, agrees with the Weil-Deligne representation associated to the $\ell$-adic Galois representation for $\ell \neq p$. $\endgroup$– Will SawinCommented Aug 19 at 18:15
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1$\begingroup$ An $\ell$-adic Galois representation for $\ell\neq p$ is unramified if and only if the associated Weil-Deligne representation is unramified (inertia and monodromy acts trivially). If $\ell=p$, an $\ell$-adic Galois representation is crystalline if and only if the associated Weil-Deligne representation is unramified. It looks to me like recent results in the non-conjugate self-dual case are more suited to proving the converse of the statement you want then the statement itself. See e.g. Ordinary parts and local-global compatibility at $\ell=p$ by Bence Hevesi. $\endgroup$– Will SawinCommented Aug 19 at 18:18
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