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Compatibility of canonical $\ell$-adic local systems on Shimura varieties, II
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abstract
Let $(G, X)$ be a Shimura datum. In previous work with Christian Klevdal, we showed that the canonical $G(\mathbb{Q}_{\ell})$-valued local systems on Shimura varieties for $G$ form compatible systems after projection to the adjoint group of $G$. In this note, we strengthen this result to prove compatibility for the $G(\mathbb{Q}_{\ell})$-local systems themselves. We also include the crystalline compatibility, extending the adjoint case established in our joint work Jake Huryn, Kiran Kedlaya, and Christian Klevdal.
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Compatibility of $F$-isocrystals on adjoint Shimura varieties
For adjoint Shimura varieties in the superrigid regime, the canonical overconvergent F-isocrystal has Frobenius conjugacy classes with rational, ℓ-independent characteristic polynomials matching the canonical ℓ-adic l...
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