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I know very little about the conjecture, beyond Grothendieck's monodromy theorem perhaps (a dense open subgroup of inertia acting unipotently on pure motives). But I heard that it was completely solved in characteristic $p$, and using Scholze's perfectoid spaces, also for hypersurfaces of toric varieties.

What is it about pure $\mathbb{Q}_p$-motives that makes the conjecture that much harder?

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  • $\begingroup$ The problem is with motives over $\mathbb{Z}_p$ and not over $\mathbb{Q}_p$. $\endgroup$
    – Mikhail Bondarko
    Commented Aug 28 at 10:22
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    $\begingroup$ Could you elaborate? Isn't it a conjecture that for $X$ a smooth proper over $\mathbb{Q}_p$, (so that its neron model is not necessarily proper over $\mathbb{Z}_p$), and $l\neq p$ a prime of bad reduction, $H^i_{et}(\overline{X},\mathbb{Q}_l)$ becomes unipotent upon restriction to a dense open subgroup of $I_p$, and that there is a certain monodromy operator, and a certain filtration $M_{\bullet}$ associated to it, such that $Gr_jH^i_{et}(\overline{X},\mathbb{Q}_l)$ w.r.t. the monodromy filtration is pure of weight $j+i$? Are you saying this is a theorem? Do you have a reference? $\endgroup$
    – kindasorta
    Commented Aug 28 at 11:27
  • $\begingroup$ I am not saying anything like this. Yet $I_p$ doesn't make any "motivic sense" if you consider $\mathbb{Q}_p$ "just as a field" that has no relation to $\mathbb{Z}_p$. $\endgroup$
    – Mikhail Bondarko
    Commented Aug 28 at 11:37
  • $\begingroup$ As about motives: any Chow motif over $\mathbb{Q}_p$ is a retract of a motif of a regular scheme that is proper over $\mathbb{Z}_p$. Hence it suffices to prove the conjecture assuming that a lift of this sort exists. $\endgroup$
    – Mikhail Bondarko
    Commented Aug 28 at 20:10
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    $\begingroup$ Weight-monodromy allows any prime $\ell \neq p$, and you cannot say the words "Neron model" unless $X$ is an abelian variety. If you don't understand the formulation of the conjecture, how could you expect this question to help you? $\endgroup$
    – Satan's Minion
    Commented Sep 7 at 9:10

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