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An irreducible integrable connection (E,∇) on a smooth projective complex variety X is called rigid if it gives rise to an isolated point of the corresponding moduli space M_{dR}(X). According to Simpson’s motivicity conjecture, irreducible rigid flat connections are of geometric origin, that is, arise as subquotients of a Gauss-Manin connection of a family of smooth projective varieties defined on an open dense subvariety of X. In this article we study mod-p reductions of irreducible rigid connections and establish results which confirm Simpson’s prediction. In particular, for large p, we prove that p-curvatures of mod-p reductions of irreducible rigid flat connections are nilpotent, and building on this result, we construct an F-isocrystalline realization for irreducible rigid flat connections. More precisely, we prove that there exist smooth models X_R and (E_R,∇_R) of X and (E,∇), over a finite-type ring R, such that for every Witt ring W(k) of a finite field k and every homomorphism R→W(k), the p-adic completion of the base change (\hat E_{W(k)}, \hat ∇_{W(k)}) on \hat X_{W(k)} represents an F-isocrystal. Subsequently, we show that irreducible rigid flat connections with vanishing p-curvatures are unitary. This allows us to prove new cases of the Grothendieck–Katz p-curvature conjecture. We also prove the existence of a complete companion correspondence for F-isocrystals stemming from irreducible cohomologically rigid connections.

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