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 $\operatorname {Pic}(X)$.

We introduce the notion of completed $F$-crystals on the absolute prismatic site of a smooth $p$-adic formal scheme. We define a functor from the category of completed prismatic $F$-crystals to that of crystalline étale $\mathbf {Z}_p$-local systems on the generic fiber of the formal scheme and show that it gives an equivalence of categories. This generalizes the work of Bhatt and Scholze, which treats the case of a mixed characteristic complete discrete valuation ring with perfect residue field.

We prove that the Hodge–Tate spectral sequence of a proper smooth rigid analytic variety can be reconstructed from its infinitesimal $\mathbb{B}_{\text{dR}}^+$-cohomology through the Bialynicki–Birula map. We also give a new proof of the torsion-freeness of the infinitesimal $\mathbb{B}_{\text{dR}}^+$-cohomology independent of Conrad–Gabber spreading theorem, and a conceptual explanation that the degeneration of Hodge–Tate spectral sequences is equivalent to that of Hodge–de Rham spectral sequences.

We show that perfectoidization can be (almost) calculated by using p-root closure in certain cases, including the semiperfectoid case. To do this, we focus on the universality of perfectoidizations and uniform completions, as well as the p-root closed property of integral perfectoid rings. Through this calculation, we establish a connection between a classical closure operation “p-root closure” used by Roberts in mixed characteristic commutative algebra and a more recent concept of “perfectoidization” introduced by Bhatt and Scholze in their theory of prismatic cohomology.

Valuation rings and perfectoid rings are examples of (usually non-Noetherian) rings that behave in some sense like regular rings. We give and study an extension of the concept of regular local rings to non-Noetherian rings so that it includes valuation and perfectoid rings and it is related to Grothendieck’s definition of formal smoothness as in the Noetherian case. For that, we have to take into account the topologies. We prove a descent theorem for regularity along flat homomorphisms (in fact for homomorphisms of finite flat dimension), extending some known results from the Noetherian to the non-Noetherian case, as well as generalizing some recent results in the non-Noetherian case, such as the descent of regularity from perfectoid rings by B. Bhatt, S. Iyengar and L. Ma.

We study the étale cohomology of Hilbert modular varieties, building on the methods introduced by Caraiani and Scholze for unitary Shimura varieties. We obtain the analogous vanishing theorem: in the ‘generic’ case, the cohomology with torsion coefficients is concentrated in the middle degree. We also probe the structure of the cohomology beyond the generic case, obtaining bounds on the range of degrees where cohomology with torsion coefficients can be non-zero. The proof is based on the geometric Jacquet–Langlands functoriality established by Tian and Xiao and avoids trace formula computations for the cohomology of Igusa varieties. As an application, we show that, when $p$ splits completely in the totally real field and under certain technical assumptions, the $p$-adic local Langlands correspondence for $\mathrm {GL}_2(\mathbb {Q}_p)$ occurs in the completed homology of Hilbert modular varieties.

In this article, we prove that a complete Noetherian local domain of mixed characteristic $p>0$ with perfect residue field has an integral extension that is an integrally closed, almost Cohen–Macaulay domain such that the Frobenius map is surjective modulo p. This result is seen as a mixed characteristic analog of the fact that the perfect closure of a complete local domain in positive characteristic is almost Cohen–Macaulay. To this aim, we carry out a detailed study of decompletion of perfectoid rings and establish the Witt-perfect (decompleted) version of André’s perfectoid Abhyankar’s lemma and Riemann’s extension theorem.

We study horizontal semistable and horizontal de Rham representations of the absolute Galois group of a certain smooth affinoid over a $p$-adic field. In particular, we prove that a horizontal de Rham representation becomes horizontal semistable after a finite extension of the base field. As an application, we show that every de Rham local system on a smooth rigid analytic variety becomes horizontal semistable étale locally around every classical point. We also discuss potentially crystalline loci of de Rham local systems and cohomologically potentially good reduction loci of smooth proper morphisms.

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.

Kottwitz’s conjecture describes the contribution of a supercuspidal representation to the cohomology of a local Shimura variety in terms of the local Langlands correspondence. A natural extension of this conjecture concerns Scholze’s more general spaces of local shtukas. Using a new Lefschetz–Verdier trace formula for v-stacks, we prove the extended conjecture, disregarding the action of the Weil group, and modulo a virtual representation whose character vanishes on the locus of elliptic elements. As an application, we show that, for an irreducible smooth representation of an inner form of $\operatorname {\mathrm {GL}}_n$ , the L-parameter constructed by Fargues–Scholze agrees with the usual semisimplified parameter arising from local Langlands.

In this paper we give an interpretation, in terms of derived de Rham complexes, of Scholze's de Rham period sheaf and Tan and Tong's crystalline period sheaf.

We completely classify the possible extensions between semistable vector bundles on the Fargues–Fontaine curve (over an algebraically closed perfectoid field), in terms of a simple condition on Harder–Narasimhan (HN) polygons. Our arguments rely on a careful study of various moduli spaces of bundle maps, which we define and analyze using Scholze’s language of diamonds. This analysis reduces our main results to a somewhat involved combinatorial problem, which we then solve via a reinterpretation in terms of the Euclidean geometry of HN polygons.