Monodromy and rigidity of crystalline local systems

We study several rigidity properties of $p$-adic local systems on a smooth rigid analytic space $X$ over a $p$-adic field. We prove that the monodromy of the log isocrystal attached to a $p$-adic local system is ''rigid'' along irreducible components of the special fiber. Then we give several applications. First, suppose that $X$ has good reduction. We show that if a family of semistable representations is crystalline at one classical point on $X$, then it is crystalline everywhere. Second, combining with the $p$-adic monodromy theorem recently studied by the authors and their collaborators, we prove the following surprising rigidity result conjectured by Shankar: for any $p$-adic local system on a smooth projective variety with good reduction, if it is potentially crystalline at one classical point, then it is potentially crystalline everywhere. Finally, we show that if a $p$-adic local system on the complement of a reduced normal crossing divisor on a smooth rigid analytic space is crystalline at all classical points, then it extends uniquely to a $p$-adic local system on the entire space. In other words, such a local system cannot have geometric monodromy if it has no arithmetic monodromy everywhere on the complement of a reduced normal crossing divisor.