Finite local monodromy of overconvergent unit-root F-isocrystals on a curve

We prove that the category of p-adic continuous representations of a fundamental group of a curve over a perfect field of a positive characteristic p with finite local monodromy is equivalent to that of overconvergent unit-root F-isocrystals on the curve. To show the equivalence of categories, we study the local theory and apply it to the global case.