Timeline for $p$-adic lifts of varieties over finite fields
Current License: CC BY-SA 3.0
15 events
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| Apr 13, 2018 at 11:38 | history | undeleted | S. Carnahan♦ | ||
| Apr 11, 2018 at 22:52 | history | deleted | user122285 | via Vote | |
| Mar 27, 2018 at 23:23 | history | edited | user122285 | CC BY-SA 3.0 |
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| Mar 27, 2018 at 22:54 | history | edited | user122285 | CC BY-SA 3.0 |
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| Mar 27, 2018 at 7:46 | history | edited | user122285 | CC BY-SA 3.0 |
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| Mar 27, 2018 at 0:49 | comment | added | R. van Dobben de Bruyn | My first comment says that the assumption that the cohomology lifts as a Galois representation is not really an assumption. Since most varieties don't lift, this is not sufficient to conclude that $X_p$ lifts to $X_0$. | |
| Mar 27, 2018 at 0:45 | comment | added | user122285 | @R. van Dobben de Bruyn, (contd) In other words, does the knowledge of its reduction and etale cohomology suffice to reconstruct a variety over the $p$-adics? | |
| Mar 27, 2018 at 0:43 | comment | added | user122285 | @R. van Dobben de Bruyn, sorry for not fully understanding the implications of your comments to answering my question. Perhaps I am being specially obtuse; if so, please bear with me. My question was: Given an $X_p$, a priori we don't know if it'd lift to any $X_0$ (unless it's a curve, K3 or ppav, etc.). Suppose all we know is that its cohomology spaces do lift as Galois representations from $G_{k_p}$ to $G_{k_0}$. Is this sufficient for $X_p$ to lift to an appropriate $X_0$? My instinct would be to say, "Not in general", and then ask what more do I need to ensure a lift exists. | |
| Mar 27, 2018 at 0:23 | comment | added | R. van Dobben de Bruyn | Oh and this also trivialises my other comment: the map $G_{k_0} \to G_{k_p}$ is surjective, so any $G_{k_p}$-representation gives rise to a $G_{k_0}$-representation... | |
| Mar 27, 2018 at 0:22 | comment | added | R. van Dobben de Bruyn | If $X_0$ is a lift of a smooth proper variety $X_p$ over $k_p$, then the smooth and proper base change theorems give isomorphisms $H^i(X_{\bar k_p},\mathbb Q_\ell) \cong H^i(X_{\bar k_0},\mathbb Q_\ell)$. Thus, lifting the Galois action here only means extending it from $G_{k_p}$ to $G_{k_0}$; the module stays the same. This quite different from the Mazur/Boston story. | |
| Mar 26, 2018 at 22:50 | comment | added | user122285 | @R. van Dobben de Bruyn, thanks! What I am wondering about is: if the Galois representation on the cohomology of $X_p$ does lift, what are further obstructions to lifting $X_p$ to an $X_0$ in char. 0 such that cohomology of $X_0$ is a lift of the cohomology of $X_p$. There is a lot of work on lifting Galois representations, e..g, by Mazur, Boston et al., and I imagine it partly goes towards answering this question. | |
| Mar 26, 2018 at 22:39 | comment | added | R. van Dobben de Bruyn | Conjecturally (e.g. on the Tate + semisimplicity conjectures), the $\ell$-adic cohomology of $X_p$ is semisimple, and all simple parts occur inside the cohomology of abelian varieties (see e.g. this post). Since abelian varieties always lift, in particular the Galois action lifts. I don't know if the last statement is also known unconditionally. | |
| Mar 26, 2018 at 22:06 | history | edited | user122285 | CC BY-SA 3.0 |
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| Mar 26, 2018 at 21:14 | history | edited | user122285 | CC BY-SA 3.0 |
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| Mar 26, 2018 at 21:09 | history | asked | user122285 | CC BY-SA 3.0 |