Timeline for Moduli interpretation for integral models of PEL Shimura variety at parahoric level?
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| Sep 26, 2022 at 9:16 | comment | added | Suzet | In my original question, I should also mention the cases studied by Rapoport and Zink in their book, namely the PEL Shimura varieties with a parahoric structure level at p which can be described as the stabilizer of some self-dual period lattice chain. If I understand correctly, this integral model has good properties (eg. flatness) when the $G$ only involves type A and C and splits over an unramified $p$-adic field. In such a case, I understand from Kisin and Pappas' paper that their integral model coincide with Rapoport-Zink's one. Thus, we have a moduli description here | |
| Sep 24, 2022 at 11:21 | comment | added | Suzet | Thank you all for the comments and references, which I was not aware of. I have some reading to do! | |
| Sep 24, 2022 at 3:19 | comment | added | naf | This is known in many PEL cases, though perhaps not all. See Section 8 of the paper of Pappas and Zhu "Local models of Shimura varieties and a conjecture of Kottwitz." (The moduli interpretation goes back to the work of Rapoport and Zink in their book "Period Spaces for p-Divisible Groups", but it is not quite correct in some cases.) | |
| Sep 23, 2022 at 21:38 | comment | added | David Loeffler | You might also enjoy Tilouine's 2006 paper in the Coates 60th proceedings, math.uni-bielefeld.de/documenta/vol-coates/tilouine.html, where he makes a careful study of integral models for GSp4 Shimura varieties for each of the possible parahoric levels. I'd be extremely surprised if the integral models arising from these older moduli-space constructions didn't coincide with the relevant special cases of Kisin and Pappas' general results, but I don't know if this compatibility has been carefully written down anywhere. | |
| Sep 23, 2022 at 13:22 | comment | added | Jef | The case of the modular curve should be enlightening, as in this case parahoric level at $p$ corresponds to $\Gamma_0(p)$ and a moduli interpretation indeed exists (Deligne--Rapoport). | |
| Sep 23, 2022 at 13:00 | comment | added | Will Sawin | I would guess there is no type of level structure where a moduli interpretation is not expected to exist, because you're only trying to explain a finite cover of the moduli space, and there is a potentially infinite amount of extra data or conditions on that extra data you could attach for abelian varieties to try to match that cover. | |
| Sep 23, 2022 at 11:10 | history | asked | Suzet | CC BY-SA 4.0 |