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3. Cyclic base change

The work of Madapusi–Lee also gives an Igusa stack for exceptional Shimura varieties. One may hope that this can be used to prove local-global compatibility results with the Fargues–Scholze [arXiv:2102.13459] local Langlands correspondence, and unltimately prove compatibility of cyclic base change with Fargues–Scholze.

    1. Igusa stacks at all unramified primes

          Fix an exceptional Shimura datum $(\mathsf{G}, X)$. The current methods of Madapusi–Youcis and Madapusi–Lee only yields Igusa stacks $\mathrm{Igs}(\mathsf{G}, X)$ for $p$ large enough.

      Add remark

      Problem 3.1.

      Can we relax this assumption to construct Igusa stacks for all $p$ that is unramified in $\mathsf{G}$?

        • Relatively unramified groups


          Add remark

          Problem 3.2.

          Let $\mathcal{O}$ be complete discrete valuation ring, finite over $\mathbb{Z}_p$, but possibly ramified. Let $\mathcal{G} / \mathcal{O}$ be a reductive group scheme with minuscule cocharacter $\mu \colon \mathbb{G}_{m,\mathcal{O}} \to \mathcal{G}$. Can we run Madapusi–Lee for $\operatorname{Res}_{\mathcal{O}/\mathbb{Z}_p} \mathcal{G}$?
              Gardner has upcoming work on studying the moduli of $(\mathcal{G},\mu)$-apertures in this context.

            • Local-global compatibility

                  Madapusi–Lee’s Igusa stack is of slightly different flavor than the Igusa stack of Zhang [arXiv:2309.05152] and Daniels–van Hoften–Kim–Zhang [arXiv:2408.01348].

              Add remark

              Problem 3.3.

              Can we bridge the gap and obtain local-global compatibility between the cohomology of Shimura varieties and the Fargues–Scholze parametrization? Do we have the Mantovan product formula for such exceptional Shimura varieties?

                • Cyclic base change


                  Add remark

                  Problem 3.4.

                  [M. Harris] Can we use the above ingredients to prove compatibility of cyclic base change with Fargues–Scholze semisimple L-parameters?

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