Appears in collection : Conférence de mi-parcours du programme ANRThéorie de Hodge p-adique et Développements (ThéHopaD)

The p-adic Simpson correspondence of Faltings between small Higgs bundles and small generalized representations, and another approach for it by A. Abbes and M. Gros both depend on the choice of a certain smooth lifting of the relevant variety. I will give a way to elimite this dependence by working with a kind of (iso)crystals, which I call Higgs (iso)crystals, instead of bundles. I will also discuss cohomology of Higgs isocrystals and comparison of cohomologies.