A geometric $p$-adic Simpson correspondence in rank one

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 \'etale twist of the moduli space of topological 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: We develop a theory of topological torsion subsheaves of $v$-sheaves and apply this to the diamantine $v$-Picard functor of arXiv:2103.16557. As an application of this geometric correspondence, we study a major open question in $p$-adic non-abelian Hodge theory raised by Faltings, namely which Higgs bundles will 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 $\pi^{\acute{e}t}_1(X)\to K^\times$ 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, but in general we show that the correct condition is the strictly stronger assumption that the underlying line bundle is a topological torsion element in the topological group $\mathrm{Pic}(X)$.