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 local systems.
Thorne, G -local systems on smooth projective curves are potentially automorphic , Acta Math
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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 local systems.