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4. Crystalline local systems

The notion of a crystalline local system on a smooth rigid analytic variety is subtle. In particular, it involves some control of its integral model.

    1. Local system on the Shimura variety

          Given a $p$-adic Shimura variety $\mathbf{Sh}_K(\mathsf{G}, X)_E$, there is a canonical $K_p$-local system. One expects that this is always a crystalline local system over the good reudction locus.

      Add remark

      Problem 4.1.

      [H. Guo] Let $\mathscr{S}_K(\mathsf{G},X)$ be a smooth integral model at hyperspecial level $K_p$. Let $\mathbb{L}$ be a $\mathbb{Z}_p$-local system on the generic fiber $\mathbf{Sh}_K(\mathsf{G},X)_E^\circ$. If $\mathbb{L}$ is crystalline at all CM points, does it follow that $\mathbb{L}$ is crystalline with respect to the smooth integral model $\mathscr{S}_K$? Is it perhaps the case that the collection of CM points is effective in the sense of Guo–Yang [arXiv:2409.19742]?

        • Syntomic realization for normal models

              Work of Guo–Reinecke [arXiv:2203.09490] and Du–Liu–Moon–Shimizu [arXiv:2203.03444] show that a crystalline local system on the generic fiber of a smooth $p$-adic formal scheme corresponds lifts to a prismatic F-crystal.

          Add remark

          Problem 4.2.

          [A. Youcis] Let $\mathfrak{X}/\mathbb{Z}_p$ be a normal $p$-adic formal scheme. What is the essential image of the $p$-adic realization functor $\mathsf{Refl}(\mathfrak{X}^\mathrm{syn}) \to \mathsf{Loc}(\mathfrak{X}_\eta)$, where $\mathsf{Refl}$ denotes the category of reflexive sheaves?

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