Appears in collection : Franco-Asian Summer School on Arithmetic Geometry

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. We will talk about an equivalence between these representations and Higgs bundles whose underlying vector bundle admits potentially a strongly semi-stable reduction of degree zero. These Higgs bundles are semi-stable of degree zero and we will investigate some evidence for Faltings' conjecture.