Principal Bundles on p-Adic Curves and Parallel Transport

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We define functorial isomorphisms of parallel transport along étale paths for a class of principal G-bundles on a p-adic curve. Here G is a connected reductive algebraic group of finite presentation over the ring of integers of ℂp and the considered principal bundles are those with potential strongly semistable reduction of degree zero. The constructed isomorphisms yield continuous functors from the étale fundamental groupoid of the curve to the category of topological spaces with a simply transitive continuous right G(ℂp)-action. This generalizes a recent construction for vector bundles on a p-adic curve by Deninger and Werner. Our result can be viewed as a partial p-adic analogue of the classical theory by Ramanathan of principal bundles on compact Riemann surfaces, which generalizes the classical Narasimhan-Seshadri theory of vector bundles on compact Riemann surfaces.