For a connected smooth proper rigid space X over a perfectoid field extension of Qp, we show that the Picard functor of X ♦ defined on perfectoid test objects is the diamondification of the rigid analytic Picard functor. In particular, it is represented by a rigid group variety if and only if the rigid analytic Picard functor is.As an application, we determine which line bundles are trivialized by pro-finiteétale covers, and prove unconditionally that the associated "topological torsion Picard functor" is represented by a divisible analytic group. We use this to generalize and geometrize a construction of Deninger-Werner in the p-adic Simpson correspondence: There is an isomorphism of rigid analytic group varieties between the moduli space of continuous characters of π1(X, x) and that of pro-finite-étale Higgs line bundles on X.This article is part II of a series about line bundles on rigid spaces as diamonds.1. The natural map (Pic X,ét ) ♦ → Pic ♦ X,ét is an isomorphism of sheaves on Perf K,ét .2. If moreover K is algebraically closed, then the v-Picard functor Pic ♦ X,v fits into a split exact sequence of abelian sheaves on Perf K,ét 0 → Pic ♦ X,ét → Pic ♦ X,v → H 0 (X, Ω 1 )(−1) ⊗ K G a → 0.

For a smooth rigid space X over a perfectoid field extension K of $\mathbb {Q}_p$ , we investigate how the v-Picard group of the associated diamond $X^{\diamondsuit }$ differs from the analytic Picard group of X. To this end, we construct a left-exact ‘Hodge–Tate logarithm’ sequence $$\begin{align*}0\to \operatorname{Pic}_{\mathrm{an}}(X)\to \operatorname{Pic}_v(X^{\diamondsuit})\to H^0(X,\Omega_X^1)\{-1\}. \end{align*}$$ We deduce some analyticity criteria which have applications to p-adic modular forms. For algebraically closed K, we show that the sequence is also right-exact if X is proper or one-dimensional. In contrast, we show that, for the affine space $\mathbb {A}^n$ , the image of the Hodge–Tate logarithm consists precisely of the closed differentials. It follows that, up to a splitting, v-line bundles may be interpreted as Higgs bundles. For proper X, we use this to construct the p-adic Simpson correspondence of rank one.

For a smooth rigid space X over a complete algebraically closed extension of Qp, we investigate how the v-Picard group of the associated diamond differs from the analytic Picard group of X. To this end, we construct an injective "Hodge-Tate logarithm"and show that this is an isomorphism if X is proper or one-dimensional. In contrast, we show that for the affine space A n , the image consists precisely of the closed differentials. It follows that up to a splitting, v-line bundles may be interpreted as Higgs bundles. For proper X, we show that this realises the p-adic Simpson correspondence in rank 1.

We study Picard groups in profinite perfectoid limits of rigid spaces in the case of good reduction. Our main result is that for sufficiently large covers, these can be described in terms of the special fibre: For example, our description implies that for a supersingular variety X of good reduction X over a perfectoid field of characteristic p,Via Raynaud uniformisation, it moreover allows us to describe Picard groups of proétale covers of abeloid varieties, which has applications to pro-étale uniformisation. We use our results to answer some general questions about Picard groups and Picard functors of perfectoid spaces, e.g. we show that these are not always p-divisible. Along the way we construct a "multiplicative Hodge-Tate spectral sequence" for O × . This is part III of a series on line bundles on diamonds.

We propose a new moduli theoretic approach to the p-adic Simpson correspondence for a smooth proper rigid space X with coefficients in any rigid analytic group G, in terms of a comparison of moduli stacks. For its formulation, we introduce a class of "smoothoid spaces" which are perfectoid families of smooth rigid spaces, well-suited for studying relative p-adic Hodge theory. For any smoothoid space Y , we then construct a "sheafified non-abelian Hodge correspondence", namely a canonical isomorphismwhere ν : Yv → Y ét is the natural morphism of sites, and where Higgs G is the sheaf of isomorphism classes of G-Higgs bundles on Y ét . We also prove a generalisation of Faltings' local p-adic Simpson correspondence to G-bundles and to perfectoid families. We apply these results to deduce v-descent criteria for étale G-bundles which show that G-Higgs bundles on X form a small v-stack H iggs G . As a second application, we construct an analogue of the Hitchin morphism on the Betti side: a morphism BunG,v → AG from the small v-stack of v-topological G-bundles on X to the Hitchin base. This allows us to reformulate the conjectural p-adic Simpson correspondence for X in a more geometric and more canonical way as a comparison of Hitchin morphisms.