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6. Miscellany

Here are some other questions that came up.

    1. Comparison of integral models

          There are roughly two types of canonical integral models. Milne [MR1155229] and Bakker–Shankar–Tsimerman [arXiv:2405.12392] uses an extension property based on $\ell$-adic local systems, while Madapusi–Youcis uses an extension property based on the $p$-adic local system. It is not obvious how they are related to each other.

      Add remark

      Problem 6.1.

      [J. Yang] Given an integral model that is canonical in the sense of Madapusi–Youcis, it is canonical in the sense of Milne or Bakker–Shankar–Tsimerman? If the $p$-adic local system at a closed point is crystalline, then are all the $\ell$-adic local systems unramified?

        • Mod-$p$ Mumford–Tate


          Add remark

          Problem 6.2.

          [R. Jiang] Consider a subvariety $X \subseteq \mathcal{A}_{g,\mathbb{F}_p}$ and the universal isocrystal over $X$. This corresponds to a module with filtration and connection on the rigid tube $]X[$, and this has $p$-adic monodromy $G \subseteq \mathrm{GSp}_{2g}$ cut out by crystalline cycles that are horizontal sections. We now consider the analytic locus where cycles lie in $\mathrm{Fil}^0$. What is the geometry of its connected components? The fact that this is algebraic is equivalent to the Mumford–Tate conjecture in characteristic $p$.

            • Special cycles


              Add remark

              Problem 6.3.

              [Y. Shi] Is there a way to use this syntomic language to define special cycles on Shimura varieties or their compactifications? For example, are there special cycles on $\mathrm{BT}^{\mathcal{G},\mu}$?

                • $p$-adic Green’s currents


                  Add remark

                  Problem 6.4.

                  [K. Madapusi] Can we explicit complexes to compute syntomic cohomology? Using them, can we write down $p$-adic Green currents?
                      There is some prior work due to Antieau–Mathew–Morrow–Nikolaus [arXiv:2003.12541] for computing syntomic cohomology.

                    • $p$-adic modular forms


                      Add remark

                      Problem 6.5.

                      [M. Harris] Is there a uniform definition of a $p$-adic modular form that does not rely on having a universal $p$-divisible group?

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