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5. Exotic Hecke correspondences

The work of Xiao–Zhu [arXiv:1707.05700] constructs exotic Hecke correspondences between special fibers of PEL Shimura varieties. In general, such a phenomenon is expected to occur for pairs of Shimura data $(\mathsf{G}, X)$ and $(\mathsf{G}^\prime, X^\prime)$ where $\mathsf{G}^\prime$ is an extended pure inner form of $\mathsf{G}$ (in a way that is compatible with $X, X^\prime$) and $B(G, \mu^{-1}) \cap B(G^\prime, \mu^{\prime-1}) \neq \emptyset$.

    1. Classifying pairs


      Add remark

      Problem 5.1.

      Can we classify pairs for which we expect there to be an exotic Hecke correspondence? For example, can be classify local pairs $(G, \mu)$ and $(G^\prime, \mu^\prime)$ together with an isomorphism $G^\prime \cong G_b$ under which we have $B(G, \mu^{-1}) = B(G^\prime, \mu^{\prime-1})$ as subsets of $B(G) \cong B(G^\prime)$?

        • Beyond Xiao–Zhu

              A limitation of Xiao–Zhu’s method is that one needs a common Hodge embedding of both Shimura data. Can we construct exotic Hecke correspondences beyond this setup?

          Add remark

          Problem 5.2.

          Let $F/\mathbb{Q}$ be a totally real cubic field and let $D/F$ be a quaternion algebra that is split at places over $p$ and $\infty$. Can we produce an exotic Hecke correspondence between Shimura varieties for $\operatorname{Res}_{F/\mathbb{Q}} \mathrm{GL}_2$ and $\operatorname{Res}_{F/\mathbb{Q}} D^\times$?

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