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5. Miscellaneous


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      Problem 5.1.

      [Xinwen Zhu] Can one write down an explicit correspondence on a Shimura variety whose induced endomorphism on cohomology doesn’t come from a Hecke correspondence?
        1. Remark. [Zavosh Amir-Khosravi] More precisely: Suppose $\pi_1, \pi_2$ are nonisomorphic cohomological automorphic forms which have the same Hecke eigenvalues. Can one construct a cycle which induces a correspondence that distinguish $\pi_1$ and $\pi_2$?


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              Problem 5.3.

              [Xinwen Zhu] In characteristic 0, the complex uniformization gives a classification of all Hecke-equivariant automorphic bundles. Can you classify all Hecke-equivariant automorphic bundles on the special fiber? (There are more such, e.g. partial Hasse invariants come from them.)
                1. Remark. [Xinwen Zhu] This will illuminate what Tate classes can be constructed by Chern classes of such bundles.


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                      Problem 5.5.

                      [Yifeng Liu] Can you construct classes in motivic cohomology or higher Chow groups of the generic fiber of Shimura varieties? e.g do Rong Zhou’s classes on the special fiber of Shimura varieties lift to the generic fiber (as a residue)?


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                          Problem 5.6.

                          [org.aimpl.user:fengt@mit.edu] To what extent are the integral Hodge conjecture and integral Tate conjectures true for ($\pi$-isotypic components of) Shimura varieties?


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                              Problem 5.7.

                              [Julia Gordon] Can we make a table with expected analogies and differences between the characteristic $0$ setting of Ichino-Prasanna and the characteristic $p$ setting of Xiao-Zhu?


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                                  Problem 5.8.

                                  [Charlotte Chan] What are the irreducible components of affine Deligne-Lusztig varieties outside the fully Hodge-Newton decomposable setting? e.g. for $G = U(3,2)$?

                                      Cite this as: AimPL: Geometric realizations of Jacquet-Langlands correspondences, available at http://aimpl.org/geomjacqlang.