p-adic vector bundles and parallel transport
We will sketch the construction of canonical parallel transport isomorphisms along etale paths between the fibres of degree zero vector bundles with "pss = potentially strongly semistable reduction" on p-adic curves. In particular this gives representations of the etale fundamental group on the fibres of such bundles. For pss-bundles of arbitrary degree one can construct representations of a certain central extension of the fundamental group on the fibres. If we are given a model of the vector bundle over a valuation ring the above p-adic representation is integral and hence can be reduced. We will show that its reduction agrees with a representation coming from the reduced bundle via Nori's fundamental group scheme. Some Tannakian properties of the category of pss-bundles will also be explained. It is known that pss-bundles are semistable. The main open question is whether the converse holds.