Timeline for Why is the weight monodromy hard in mixed characteristics?
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| Sep 7 at 12:20 | comment | added | kindasorta | Fair point, indeed I am caught again for not reading the book on neron models, knowing how they are constructed, or that their construction is limited to abelian varieties. I do feel however that I have the ability to understand an experts answer to a level which might quench some of my curiosity, or perhaps help me frame this space of knowledge in terms of related problems/key references on the subject. I do not aim to represent knowledge I don't have, but am very eager to learn nonetheless. And interactions such as these are helpful, albeit your skepticism. | |
| Sep 7 at 9:10 | comment | added | Satan's Minion | 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? | |
| Aug 28 at 20:10 | comment | added | Mikhail Bondarko | 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. | |
| Aug 28 at 11:37 | comment | added | Mikhail Bondarko | 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 $\mathbb{Z}_p$. | |
| Aug 28 at 11:27 | comment | added | kindasorta | 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? | |
| Aug 28 at 10:22 | comment | added | Mikhail Bondarko | The problem is with motives over $\mathbb{Z}_p$ and not over $\mathbb{Q}_p$. | |
| Aug 20 at 22:58 | history | edited | YCor |
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| Aug 20 at 19:27 | history | asked | kindasorta | CC BY-SA 4.0 |