Purity of the startification by Newton polygons and Frobenius-periodic vector bundles

This thesis includes two parts. In the first part, we show a purity theorem for stratifications by Newton polygons coming from crystalline cohomology, which says that the family of Newton polygons over a noetherian scheme have a common break point if this is true outside a subscheme of codimension bigger than 1. The proof is similar to the proof of [dJO99, Theorem 4.1].;In the second part, we prove that for every ordinary genus-2 curve X over a finite field κ of characteristic 2 with automorphism group $Z$/2$Z$ × S₃, there exist SL(2,κ[[ s]])-representations of π₁(X) such that the image of π₁(X) is infinite. This result produces a family of examples similar to Laszlo's counterexample [Las01] to a question regarding the finiteness of the geometric monodromy of representations of the fundamental group [dJ01].