We discuss Tate-type problems for F-isocrystals, that is, the full faithfulness of the natural restriction functors between categories of overconvergent F-isocrystals on schemes of positive characteristic. We prove it in the cases of unit-root F-isocrystals. Using this result, we prove that an overconvergent unit-root F-isocrystal has a finite monodromy.
"Morphisms of F-isocrystals and the finite monodromy theorem for unit-root F-isocrystals." Duke Math. J. 111 (3) 385 - 418, 15 February 2002. https://doi.org/10.1215/S0012-7094-02-11131-4