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Kottwitz has built canonical integral models for a large family of PEL Shimura varieties, associated to a certain reductive group $G$ over $\mathbb Q$, when the structure level has the form $K = K_pK^p$ with $K^p \subset G(\mathbb A_f^p)$ small enough and $K_p \subset G(\mathbb Q_p)$ hyperspecial. This construction admits a moduli interpretation, classifying abelian schemes with integral additional structures.

Using the notion of full set of sections, Mantovan has built integral models for the same Shimura varieties with structure level of the form $K_m = K_{p,m}K^p$ where, for $m\geq 0$, we have $$K_{p,m} := \{g \in K_p \,|\, g \cong 1 \mod p^m\Lambda \}.$$ To explain the notations, the group $G$ arises as the group of symplectic similitudes of some space $V$ equipped with an alternate form. One hypothesis to ensure the existence of hyperspecial $K_p$ is that $V\otimes_{\mathbb Q} \mathbb Q_p$ must contain a self-dual lattice $\Lambda$, so that we can take $K_p = \mathrm{Stab}_{G}(\Lambda)$. Note that $K_p = K_{p,0}$.
The integral models defined by Mantovan also admits a moduli interpretation, refining the problem considered by Kottwitz.

Nowadays, for more general levels $K = K_{\mathrm{par}}K^p$ where $K_{\mathrm{par}} \subset G(\mathbb Q_p)$ is parahoric, we can consider the canonical integral model built by Kisin and Pappas. Their construction works for Shimura varieties of abelian type, a much broader class than those of PEL type, and as such Kisin and Pappas did not rely on moduli problems to build their integral models.

Are there known cases of PEL Shimura variety with general parahoric level structure, whose integral model built by Kisin and Pappas admits a moduli interpretation ? As a rule of thumb, is a moduli interpretation even expected to exist at such parahoric level structure ?

I am not aware of any such case in the litterature, but if it does exist I'd be very interested in learning more about it.

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  • $\begingroup$ 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. $\endgroup$
    – Will Sawin
    Sep 23, 2022 at 13:00
  • $\begingroup$ 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). $\endgroup$
    – Jef
    Sep 23, 2022 at 13:22
  • $\begingroup$ 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. $\endgroup$ Sep 23, 2022 at 21:38
  • $\begingroup$ 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.) $\endgroup$
    – naf
    Sep 24, 2022 at 3:19
  • $\begingroup$ Thank you all for the comments and references, which I was not aware of. I have some reading to do! $\endgroup$
    – Suzet
    Sep 24, 2022 at 11:21

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