Principal bundles on $p$-adic curves and parallel transport

TL;DR

This paper constructs functorial parallel transport isomorphisms for certain principal G-bundles on p-adic curves, extending classical theories and providing a p-adic analogue of known results for vector bundles.

Contribution

It introduces a new functorial framework for parallel transport of principal bundles on p-adic curves with potentially strongly semistable reduction, generalizing previous vector bundle results.

Findings

Defines functorial isomorphisms of parallel transport for principal G-bundles

Establishes continuous functors from the étale fundamental groupoid to topological G-spaces

Generalizes Deninger and Werner's construction for vector bundles to principal bundles

Abstract

We define functorial isomorphisms of parallel transport along \'etale paths for a class of principal $G$-bundles on a $p$-adic curve. Here $G$ is a connected reductive algebraic group of finite presentation and the considered principal bundles are just those with potentially strongly semistable reduction of degree zero. The constructed isomorphisms yield continous functors from the \'etale fundamental groupoid of the given curve to the category of topological spaces with a simply transitive continous right $G(\mathbb{C}_{p})$-action. This generalizes a construction in the case of vector bundles on a $p$-adic curve by Deninger and Werner. It may 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.