Parallel transport and the p-adic Simpson correspondence

TL;DR

This paper investigates the p-adic Simpson correspondence for curves, establishing properties of a functor linking vector bundles and p-adic representations, and addressing questions about faithfulness, Hodge-Tate structures, and compatibility.

Contribution

It proves the functor's full faithfulness, analyzes the Hodge-Tate filtration of associated cohomology, and confirms compatibility with Faltings' p-adic Simpson correspondence.

Findings

The functor is fully faithful.

Cohomology admits a Hodge-Tate filtration.

Construction is compatible with Faltings' correspondence.

Abstract

Deninger and Werner developed an analogue for p-adic curves of the classical correspondence of Narasimhan and Seshadri between stable bundles of degree zero and unitary representations of the topological fundamental group for a complex smooth proper curve. Using parallel transport, they associated functorially to every vector bundle on a p-adic curve whose reduction is strongly semi-stable of degree 0 a p-adic representation of the fundamental group of the curve. They asked several questions: whether their functor is fully faithful; whether the cohomology of the local systems produced by this functor admits a Hodge-Tate filtration; and whether their construction is compatible with the p-adic Simpson correspondence developed by Faltings. We answer these questions in this article.