Parallel transport for Higgs bundles over p-adic curves

TL;DR

This paper establishes an equivalence between p-adic representations and semi-stable Higgs bundles over p-adic curves, providing evidence for Faltings' conjecture in the context of higher genus curves.

Contribution

It proves an equivalence between p-adic representations and certain Higgs bundles with semi-stable reduction over p-adic curves of genus at least 2.

Findings

Higgs bundles are semi-stable of degree zero

Evidence supports Faltings' conjecture in this setting

Established equivalence over genus g ≥ 2 curves

Abstract

Faltings conjectured that under the p-adic Simpson correspondence, finite dimensional p-adic representations of the geometric \'etale fundamental group of a smooth proper p-adic curve X are equivalent to semi-stable Higgs bundles of degree zero over X. In this article, we establish, over a p-adic curve of genus $g\ge 2$, an equivalence between these representations and Higgs bundles, whose underlying bundles potentially admit a strongly semi-stable reduction of degree zero. We show that these Higgs bundles are semi-stable of degree zero and investigate some evidence for the aforementioned conjecture.