Parallel transport for Higgs bundles over p-adic curves

Abstract

Faltings conjectured that under the p-adic Simpson correspondence, finite dimensional p-adic representations of the geometric étale 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.

49 pages, with an appendix joint with Tongmu He. We improve the appendix and add the assumption that genus $g\ge 2$