A p-adic Simpson correspondence

For curves over a p-adic field we construct an equivalence between the category of Higgs-bundles and that of “generalised representation” of the etale fundamental group. The definition of “generalised representations” uses p-adic Hodge theory and almost etale coverings, and it includes usual representations which form a full subcategory. The equivalence depends on the choice of an exponential function for the multiplicative group.