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1. Parahoric groups

The current (syntomic) theory of Gardner–Madapusi [arXiv:2412.10226] and Lee–Madapusi only works for reductive group schemes $\mathcal{G}$ over $\mathbb{Z}_p$. Can we find a version of this theory for parahoric groups?

    1. Dilatated groups

          The theory for general parahoric $(\mathcal{G}, \mu)$ seems to be currently out of reach. But we may hope that if $\mathcal{G}$ is obtained by dilatating a reductive group scheme over $\mathbb{Z}_p$, the situation simplifies and becomes managable.

      Add remark

      Problem 1.1.

      [G. Pappas] Let $G$ be a connected reductive group over $\mathbb{Q}_p$ that is unramified. For $\mathcal{G}$ a parahoric model of $G$ and $\mu$ a minuscule cocharacter of $G$, can we develop a theory of $\mathrm{BT}^{\mathcal{G},\mu}$ that has the expected properties? For example, when $\mathcal{G}$ is the Iwahori of $\mathrm{GL}_2$, we want it to parametrize a $p$-divisible group of height $2$ together with an order $p$ subgroup.

        • Integral models

              Let $(\mathsf{G}, X)$ be a Shimura datum where $G = \mathsf{G}_{E_v}$ is unramified. Let $\mathcal{G}$ be a hyperspecial model with $\mathcal{G}^\prime \to \mathcal{G}$ a dilatated parahoric model. Takaya [arXiv:2505.10000] proves that if there is a Pappas–Rapoport canonical integral model $\mathscr{S}_{\mathcal{G}(\mathbb{Z}_p)}$ then there exists a Pappas–Rapoport canonical integral model $\mathscr{S}_{\mathcal{G}^\prime(\mathbb{Z}_p)}$ for $\mathcal{G}^\prime$, which represents the fiber product $\mathscr{S}_{\mathcal{G}(\mathbb{Z}_p)}^\diamond \times_{\mathrm{Sht}_{\mathcal{G},\mu}} \mathrm{Sht}_{\mathcal{G}^\prime,\mu}$.

          Add remark

          Problem 1.2.

          [K. Madapusi, A. Youcis] Can we prove a similar statement for the syntomic theory? In other words, assuming that there is a Madapusi–Youcis integral model $\mathscr{S}_{\mathcal{G}(\mathbb{Z}_p)} \to \mathrm{BT}^{\mathcal{G},\mu}$, can we construct an integral model $\mathscr{S}_{\mathcal{G}^\prime(\mathbb{Z}_p)}$ together with a formally étale map $\mathscr{S}_{\mathcal{G}^\prime(\mathbb{Z}_p)} \to \mathrm{BT}^{\mathcal{G}^\prime,\mu}$?

            • Sheared displays

                  Recently Hoff–Lau [arXiv:2601.12565] and Bhatt–Mathew–Vologodsky–Zhang has developed a theory of sheared display. As Hoff–Lau proves, this has the pleasant property that the category of sheared displays is equivalent to $p$-divisible groups over any ring $R$ in which $p$ is nilpotent.

              Add remark

              Problem 1.3.

              [Y. Luo, G. Pappas] Let $(\mathcal{G}, \mu)$ be a parahoric group together with a geometric minuscule cocharacter on the generic fiber. Can we develop a theory of sheared $(\mathcal{G},\mu)$-displays via a sheared $(\mathcal{G},\mu)$-affine Grassmannian?
                  When $(\mathcal{G},\mu)$ is of local Hodge type, i.e., when it has a nice embedding $\mathcal{G} \hookrightarrow \mathrm{GL}_{n,\mathbb{Z}_p}$, it seems like one can follow Pappas and develop a reasonable theory.

                  Cite this as: AimPL: New p-adic perspectives on canonical integral models for Shimura varieties, available at http://aimpl.org/canonicalshimura.