Pointwise criteria of p-adic local systems

Given a Z_p-linear local system over a smooth rigid space, we show that it is crystalline (resp. semi-stable) with respect to any smooth (resp. semi-stable) integral model if and only if its restrictions at many classical points are crystalline (resp. semi-stable) representations. To this end, we introduce a crystalline Riemann–Hilbert functor, and give several applications, including a semi-stable comparison theorem in the relative setting.