Timeline for Enriques surfaces over $\mathbb Z$
Current License: CC BY-SA 3.0
17 events
| when toggle format | what | by | license | comment | |
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| Jun 21 at 18:34 | comment | added | Will Sawin | @JackYo For example stacks.math.columbia.edu/tag/0GKD | |
| Jun 21 at 18:32 | comment | added | Will Sawin | @JackYo It is not obvious but it is a standard result in introductions to étale cohomology. Holds for $\mathcal A_l$ locally constant. | |
| Jun 21 at 12:53 | comment | added | JackYo | is it obvious from proper& sm base change tm that $\ell$-adic cohomology sheaf $R^i f_* \mathcal{A}_l$ with $\mathcal{A}_l$ $\ell$-adic sheaf on $E$ is locally constant on $\mathbb Z[1/ \ell]$? ($f$ struct map $E$ to spec of $\mathbb Z[1/ l]$ after restr from $\mathbb Z$) Holds it for any $A_l$ or are you adressing specifically cohom sheaf $R^i f_* \mathbb Z_{\ell}$ only? by defin of $\ell$-adic sheaves this is equivalent to $R^i f_* \mathbb{Z}/\ell^n$ for all $n$ locl constant, so that there exist étale $u: U \to \mathbb Z[1/ \ell]$ trivializing it. I not see how you get it from proper bs? | |
| Jun 20 at 11:07 | comment | added | Will Sawin | @JackYo I just mean the usual cohomology classes that are invariant under the inertia group, not unramified cohomology, which is something else. The inertia action is trivial because the proper and smooth base change theorem implies the $\ell$-adic cohomology is a locally constant sheaf on $\operatorname {Spec} \mathbb Z[1/\ell]$ so the Galois action factors through the étale fundamental group of that space, and the inertia group is sent to the trivial group under the map from the Galois group to the étale fundamental group. | |
| Jun 20 at 8:20 | comment | added | JackYo | What you mean by unramified cohomology classes? There is notion of unramified cohomology,e.g. Definition 4.3 in arXiv:2106.01057 but I'm not sure if this is the same thing as what you use above. Moreover, could you elaborate why the Galois action on Picard group of $E$ must be unramified at each prime and why this would imply that the action must be trivial? | |
| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| Apr 17, 2020 at 0:15 | vote | accept | Will Sawin | ||
| Apr 16, 2020 at 7:20 | answer | added | Davide Cesare Veniani | timeline score: 15 | |
| Apr 1, 2015 at 17:41 | comment | added | Will Sawin | @ulrich The Picard scheme is a group scheme over $\mathbb Z$. But the group scheme $\alpha_2$ does not lift to $\mathbb Z$ or even $\mathbb Z_2$. I'll try to figure out what the Galois action mod $2$ is in my case. | |
| Apr 1, 2015 at 6:27 | comment | added | naf | If the action is non-trivial, then I think one gets an obstruction to good reduction if the fibre over 2 is classical (since the Picard scheme is smooth in this case). However, I don't see why the special fibre cannot be an $\alpha_2$ surface. | |
| Mar 29, 2015 at 14:06 | comment | added | Will Sawin | @ulrich I'm not sure. I think the exceptional classes are invariant even integrally, but the classes coming from $H^1(E)$ and $H^2(E)$ might not be. One may be able to figure this out by studying the two elliptic vibrations over $\mathbb P^1$. If one did get a nontrivial action, because it only shows up in $2$-adic cohomology, you might need to use $p$-adic Hodge theory to see if it's compatible with good reduction. | |
| Mar 29, 2015 at 14:01 | history | edited | Will Sawin | CC BY-SA 3.0 |
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| Mar 29, 2015 at 7:01 | comment | added | naf | Is the Galois action on the (geometric) Picard group of the surface in your example trivial? From what you write this is clear rationally but it seems possible to me that it could be non-trivial integrally (though I am not sure whether knowing this is useful). Also, a couple of typos: in the second last paragraph Kunneth should presumably be Kummer and $H^2(E)$ should be $H^1(E_2)$. | |
| Mar 28, 2015 at 16:55 | history | edited | Will Sawin | CC BY-SA 3.0 |
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| Mar 28, 2015 at 16:34 | history | edited | Will Sawin | CC BY-SA 3.0 |
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| Mar 28, 2015 at 6:16 | comment | added | naf | Ekedahl and Shepherd-Barron (arxiv.org/abs/math/0405510) have given examples of classical Enriques surfaces with non-trivial vector fields and these need not have unramified lifts. You might also find the paper by Liedtke "Arithmetic moduli and lifting of Enriques surfaces" useful. | |
| Mar 27, 2015 at 22:19 | history | asked | Will Sawin | CC BY-SA 3.0 |