We explore generalizations of the p-adic Simpson correspondence on smooth proper rigid spaces to principal bundles under rigid group varieties G. For commutative G, we prove that such a correspondence exists if and only if the Lie group logarithm is surjective. Second, we treat the case of general G on ordinary abelian varieties, in which case we prove a generalization of Faltings’ “small” correspondence to general rigid groups. On abeloid varieties, we also prove an analog of the classical Corlette–Simpson correspondence for principal bundles under linear algebraic groups.

For any smooth proper rigid space $X$ over a complete algebraically closed extension $K$ of $\mathbb {Q}_p$ we give a geometrisation of the $p$-adic Simpson correspondence of rank one in terms of analytic moduli spaces: the $p$-adic character variety is canonically an étale twist of the moduli space of topologically torsion Higgs line bundles over the Hitchin base. This also eliminates the choice of an exponential. The key idea is to relate both sides to moduli spaces of $v$-line bundles. As an application, we study a major open question in $p$-adic non-abelian Hodge theory raised by Faltings, namely which Higgs bundles correspond to continuous representations under the $p$-adic Simpson correspondence. We answer this question in rank one by describing the essential image of the continuous characters $\pi ^{{\mathrm {\acute {e}t}}}_1(X)\to K^\times$ in terms of moduli spaces: for projective $X$ over $K=\mathbb {C}_p$, it is given by Higgs line bundles with vanishing Chern classes like in complex geometry. However, in general, the correct condition is the strictly stronger assumption that the underlying line bundle is a topologically torsion element in the topological group $\operatorname {Pic}(X)$.