𝑝-Adic non-abelian Hodge theory for curves via moduli stacks

For a smooth projective curve X X over C p C_p and any reductive group G G , we show that the moduli stack of G G -Higgs bundles on X X is a twist of the moduli stack of v-topological G G -bundles on X v X_v in a canonical way. We explain how a choice of an exponential trivialises this twist on points. This yields a geometrisation of Faltings’ p p -adic Simpson correspondence for X X , which we recover as a homeomorphism between the points of moduli spaces. We also show that our twisted isomorphism sends the stack of p p -adic representations of π 1 ( X ) π _1(X) to an open substack of the stack of semi-stable Higgs bundles of degree 0 0 .