Ogus's conjecture on F-isocrystals

Abstract

In 1984, Ogus conjectured the existence of a canonical F-isocrystal that enhances the Gauss--Manin connection, for a proper relative rigid space with analytically good reduction. We give a positive answer to this conjecture in full generality, through p-adic local systems and prismatic methods. Along the way, we introduce a prismatic refinement of the p-adic Riemann--Hilbert functor and prove a primitive purity theorem for Frobenius modules.