Moduli spaces in 𝑝-adic non-abelian Hodge theory

We propose a new moduli-theoretic approach to the p p -adic Simpson correspondence for any smooth proper rigid space over C p Cₚ with coefficients in any rigid analytic group G G. For its formulation, we introduce the class of “smoothoid spaces” which are perfectoid families of smooth rigid spaces. We then prove a generalisation of Faltings’ local p p -adic Simpson correspondence to v v -topological G G -bundles on smoothoid spaces. We use this to show that there are small moduli v-stacks for both sides of the correspondence. Second, we use it to construct an analogue of the Hitchin morphism on either side. This allows us to give a conjectural reformulation of the p p -adic Simpson correspondence in a more geometric and canonical way: The moduli stack of v v -topological G G -bundles is a twist of the moduli stack of G G -Higgs bundles over the Hitchin base.