Fargues-Rapoport conjecture for p-adic period domains in the non-basic case

Noetherian properties of Fargues-Fontaine curves

We establish that the extended Robba rings associated to a perfect nonarchimedean field of characteristic p, which arise in p-adic Hodge theory as certain completed localizations of the ring of Witt vectors, are strongly noetherian Banach rings; that is, the completed polynomial ring in any number of variables over such a Banach ring is noetherian. This enables Huber's theory of adic spaces to be applied to such rings. We also establish that rational localizations of these rings are principal ideal domains and that etale covers of these rings (in the sense of Huber) are Dedekind domains.

Sur la cohomologie étale de la courbe de Fargues-Fontaine

In this article the étale cohomology of constructible torsion sheaves on the étale site of the algebraic resp. adic Fargues-Fontaine curve is analyzed. In the $\ell\neq p$-torsion case, two conjectures of Fargues are verified: vanishing in degrees greater than two and the comparison between the étale cohomology of the adic and the algebraic curve. In the $p$-torsion case, under a certain assumption, the vanishing of the étale cohomology in degrees greater than two of those Zariski-constructible sheaves on the adic curve that come via pullback from the algebraic curve is proven.

Rapoport-Zink uniformization for the moduli space of polarized K3 surfaces

We compute the image of the $p$-adic period map for polarized K3 surfaces with supersingular reduction. This gives rise to a Rapoport-Zink type uniformization of their moduli space by an explicit open rigid analytic subvariety of a local Shimura variety of orthogonal type. In contrast to the case of Rapoport-Zink uniformization of Shimura varieties and in analogy to the complex case, the uniformizing domain does not carry an action of a $p$-adic Lie group, but only of a discrete subgroup. We briefly sketch how the same arguments can be applied to obtain a uniformization for the moduli space of…

法格斯-福恩塔因曲线：对 $p$-adic 伽罗华表示理论研究的激励

La curva de Fargues--Fontaine: Una motivación al estudio de la teoría de representaciones de Galois $p$-ádicas

本文综述了法格斯-福恩塔因曲线在 $p$-adic Hodge 理论中的核心地位及其在分类 $p$-adic 伽罗华表示中的作用，分析了 Fontaine 时期环的性质及其对曲线构造的贡献。

Harris-Viehmann conjecture for Hodge-Newton reducible Rapoport-Zink spaces

Rapoport-Zink spaces, or more generally local Shimura varieties, are expected to provide geometric realization of the local Langlands correspondence via their $l$-adic cohomology. Along this line is a conjecture by Harris and Viehmann, which roughly says that when the underlying local Shimura datum is not basic, the $l$-adic cohomology of the local Shimura variety is parabolically induced. We verify this conjecture for Rapoport-Zink spaces which are Hodge type and Hodge-Newton reducible. The main strategy is to embed such a Rapoport-Zink space into an appropriate space of EL type, for which th…

On the structure of some p-adic period domains

We prove the Fargues-Rapoport conjecture for p-adic period domains: for a reductive group G over a p-adic field and a minuscule cocharacter μ of G, the weakly admissible locus coincides with the admissible one if and only if the Kottwitz set B(G,μ) is fully Hodge-Newton decomposable.

(G, μ)-displays and Rapoport-Zink spaces

Let (G, μ) be a pair of a reductive group G over the p-adic integers and a minuscule cocharacter μ of G defined over an unramified extension. We introduce and study "(G, μ)-displays" which generalize Zink's Witt vector displays. We use these to define certain Rapoport-Zink formal schemes purely group theoretically, i.e. without p-divisible groups.

On irreducible components of Rapoport-Zink spaces

Under a mild condition, we prove that the action of the group of self-quasi-isogenies on the set of irreducible components of a Rapoport-Zink space has finite orbits. Our method allows both ramified and non-basic cases. As a consequence, we obtain some finiteness results on the representation obtained from the l-adic cohomology of a Rapoport-Zink tower.

Cell decomposition of some unitary group Rapoport-Zink spaces

In this paper we study the $p$-adic analytic geometry of the basic unitary group Rapoport-Zink spaces $\M_K$ with signature $(1,n-1)$. Using the theory of Harder-Narasimhan filtration of finite flat groups developed by Fargues in \cite{F2},\cite{F3}, and the Bruhat-Tits stratification of the reduced special fiber $\M_{red}$ defined by Vollaard-Wedhorn in \cite{VW}, we find some relatively compact fundamental domain $\D_K$ in $\M_K$ for the action of $G(\Q_p)\times J_b(\Q_p)$, the product of the associated $p$-adic reductive groups, and prove that $\M_K$ admits a locally finite cell decompositi…

Construction of a Rapoport-Zink space for $\mathrm{GU}(1,1)$ in the ramified $2$-adic case

Let $F|\mathbb{Q}_2$ be a finite extension. In this paper, we construct an RZ-space $\mathcal{N}_E$ for split $\mathrm{GU}(1,1)$ over a ramified quadratic extension $E|F$. For this, we first introduce the naive moduli problem $\mathcal{N}_E^{\mathrm{naive}}$ and then define $\mathcal{N}_E \subseteq \mathcal{N}_E^{\mathrm{naive}}$ as a canonical closed formal subscheme, using the so-called straightening condition. We establish an isomorphism between $\mathcal{N}_E$ and the Drinfeld moduli problem, proving the $2$-adic analogue of a theorem of Kudla and Rapoport. The formulation of the straighte…

Correspondances de Langlands locales dans la cohomologie des espaces de Rapoport-Zink

We generalize the work of M. Harris and R. Taylor on the local Langlands correspondence for the linear group over $\mathbb{Q}_p$. We prove some cases of the Kottwitz conjectures for the supercuspidal part of the compactly supported $\ell$-adic cohomology of Rapoport-Zink rigid-analytic spaces. This means that we decompose this part in terms of local Langlands correspondences. In particular, we prove the first case of a non-abelian reciprocity law constructed geometrically and associated to groups other than inner forms of the linear group. More precisely, we do this for the unramified unitary …

The Carlitz Logarithm as a Period Morphism for Local $G$-Shtukas

Local shtukas are the function field analogs for $p$-divisible groups. Similar to the $p$-adic theory, one defines Rapoport-Zink functors and Rapoport-Zink spaces for these local shtukas. The associated Hodge-Pink structures are described uniquely by a morphism, called the period morphism of the moduli problem. We will prove the ind-representability of the Rapoport-Zink functor in a particular case and compute the corresponding Rapoport-Zink space as well as the corresponding period morphism. In this case, the period morphism is given by the Carlitz logarithm.

关于GL_n的范畴局部朗兰兹纲领

On categorical local langlands program for $GL_n$

研究GL_n的范畴局部朗兰兹纲领，通过在法尔格斯纲领下研究局部shtukas模空间，明确谱作用，构造赫克本征层，证明法尔格斯猜想部分内容，还能证明哈里斯 - 维赫曼猜想新情况及计算井草variety部分上同调。

On certain extensions of vector bundles in p-adic geometry

Given two arbitrary vector bundles on the Fargues-Fontaine curve, we give an explicit criterion in terms of Harder-Narasimhan polygons on whether they realize a semistable vector bundle as their extensions. Our argument is largely combinatorial and builds upon the dimension analysis of certain moduli spaces of bundle maps developed in [BFH+22].

On the coherence conjecture of Pappas and Rapoport

We prove the (generalized) coherence conjecture of Pappas and Rapoport. As a corollary, one theorem of Pappas an Rapoport, which describes the geometry of the special fibers of the local models for ramified unitary groups, holds unconditionally. Our proof is based on the study of the geometry (in particular certain line bundles and $\ell$-adic sheaves) of the global Schubert varieties, which are the (generalized) equal characteristic counterparts of the local models.

A cohomological smoothness conjecture for moduli of mixed characteristic local shtukas with one leg

We give a simple geometric characterization of the locus where the inscribed Banach--Colmez Tangent Spaces of moduli of mixed characteristic local shtukas with one leg and fixed determinant are connected. We conjecture that the structure morphism for the underlying diamond is cohomologically smooth over this locus and, applying the Fargues--Scholze Jacobian criterion, we prove this conjecture in the case of EL infinite level Rapoport--Zink spaces, generalizing a result of Ivanov--Weinstein in the basic case.

志村簇与p进Shtuka模空间的S=T猜想

S=T for Shimura Varieties and Moduli Spaces of p-adic Shtukas

本研究借助Scholze的钻石、v-栈理论及Fargues–Scholze几何Satake等价，证明了Xiao–Zhu提出的S=T猜想，并推导出Hodge型志村簇的Eichler–Shimura关系。

解析空间的算术$p$-进前平展上同调的对偶性

Duality for Arithmetic $p$-adic Pro-étale Cohomology of Analytic Spaces

本研究验证了Colmez等人关于$p$-进数有限扩张上光滑部分紧合空间的算术$p$-进前平展上同调满足对偶性的猜想，借助Fontaine伽罗瓦下降技术由Fargues-Fontaine曲线的几何对偶性推导得出。

任意抛物子群水平下基本非分歧 $GU(1,n-1)$ Rapoport-Zink 空间的 Bruhat-Tits 分层

The Bruhat-Tits stratification for basic unramified $GU(1,n-1)$ Rapoport-Zink spaces at arbitrary parahoric level

本文对签名 $(1,n-1)$ 的基本非分歧酉 Rapoport-Zink 空间在任意抛物子群水平下的约化特殊纤维进行分层，证明了闭分层的光滑性、不可约性并计算了维数，同时利用 Bruhat-Tits 指标描述了分层间的关联关系。

扩展的Fargues–Scholze谱作用

The extended Fargues--Scholze spectral action

本文证明Fargues的猜想，将Fargues与Scholze构造的谱作用扩展到Kaletha挠子群的G挠子栈，为非拟分裂群的内形式提供范畴化局部朗兰兹猜想，还对环面证明该猜想并归约中心条件下的扩展版本。

范畴局部Langlands对应与anabelomorphy

The Categorical Local Langlands Correspondence and Anabelomorphy

本文证明若K和L是anabelomorphic p-adic域，则其Langlands参数栈同构，并提出联系Fargues-Scholze主猜想与p-adic域anabelomorphy的猜想，对分裂环面证明了该猜想。

p-进shtukas的堆栈与空间kimberlites

Stacks of p-adic shtukas and spatial kimberlites

本文通过研究p-进shtukas的堆栈与空间kimberlites，证明了特殊牛顿多边形映射在钻石中的可表示性，并探讨了其在同调考虑中的性质，为构造局部朗兰兹范畴等价提供了关键工具。

Weight-monodromy conjecture for p-adically uniformized varieties

The aim of this paper is to prove the weight-monodromy conjecture (Deligne's conjecture on the purity of monodromy filtration) for varieties p-adically uniformized by the Drinfeld upper half spaces of any dimension. The ingredients of the proof are to prove a special case of the Hodge standard conjecture, and apply a positivity argument of Steenbrink, M. Saito to the weight spectral sequence of Rapoport-Zink. As an application, by combining our results with the results of Schneider-Stuhler, we compute the local zeta functions of p-adically uniformized varieties in terms of representation theor…

Moduli of p-divisible groups

We prove several results about p-divisible groups and Rapoport-Zink spaces. Our main goal is to prove that Rapoport-Zink spaces at infinite level are naturally perfectoid spaces, and to give a description of these spaces purely in terms of p-adic Hodge theory. This allows us to formulate and prove duality isomorphisms between basic Rapoport-Zink spaces at infinite level in general. Moreover, we identify the image of the period morphism, reproving results of Faltings. For this, we give a general classification of p-divisible groups over the ring of integers of a complete algebraically closed fi…

Harder-Narasimhan strata and $p$-adic period domains

We revisit the Harder-Narasimhan stratification on a minuscule $p$-adic flag variety, by the theory of modifications of $G$-bundles on the Fargues-Fontaine curve. We compare the Harder-Narasimhan strata with the Newton strata introduced by Caraiani-Scholze. As a consequence, we get further equivalent conditions in terms of $p$-adic Hodge-Tate period domains for fully Hodge-Newton decomposable pairs. Moreover, we generalize these results to arbitrary cocharacters case by considering the associated $B_{dR}^+$-affine Schubert varieties. Applying Hodge-Tate period maps, our constructions give appl…

Weakly admissible locus and Newton stratification in p-adic Hodge theory

The basic admissible locus $\mathcal{F}(G, μ, b)^a$ inside the flag variety $\mathcal{F}(G, μ)$, attached to a reductive group $G$ with a minuscule cocharacter $μ$ of $G$, is a $p$-adic analogue of the complex analytic period spaces. It has an algebraic approximation $\mathcal{F}(G, μ, b)^{wa}$ inside the flag variety, called the weakly admissible locus. On the flag variety $\mathcal{F}(G, μ)$, we have the Newton stratification which has the admissible locus as its unique open stratum. In this paper, we study the relation between the Newton strata and the weakly admissible locus. We show that …

超收敛 de Rham 周期环的 Galois 和 Pro-étale 上同调

Galois and Pro-étale Cohomology of Overconvergent de Rham Period Rings

受 p 进微分方程和 p 进几何表示论启发，引入 Fontaine 经典周期环的超收敛变体，计算其 Galois 上同调及相关周期层和周期结构层的上同调。

Weight-monodromy conjecture for certain threefolds in mixed characteristic

The weight-monodromy conjecture claims the coincidence of the weight filtration and the monodromy filtration, up to shift, on the $l$-adic étale cohomology of a proper smooth variety over a complete discrete valuation field. Although it has been proved in some cases, the case of dimension $\geq 3$ in mixed characteristic is still open so far. The aim of this paper is to give a proof of the weight-monodromy conjecture for a threefold which has a projective strictly semistable model such that, for each irreducible component of the special fiber, the Picard number is equal to the second $l$-adic …

Topological flatness of local models for ramified unitary groups. II. The even dimensional case

Local models are schemes, defined in terms of linear-algebraic moduli problems, which give étale-local neighborhoods of integral models of certain p-adic PEL Shimura varieties defined by Rapoport and Zink. In the case of a unitary similitude group whose localization at Q_p is ramified, quasi-split GU_n, Pappas has observed that the original local models are typically not flat, and he and Rapoport have introduced new conditions to the original moduli problem which they conjecture to yield a flat scheme. In a previous paper we proved that their new local models are topologically flat when n is o…

可接受对和p进Hodge结构 I：de Rham格的超越性

Admissible pairs and $p$-adic Hodge structures I: Transcendence of the de Rham lattice

本文定义了p进Hodge结构的Tannakian范畴，并展示了其与可接受对之间的等价关系，通过研究p进周期的超越性来推广一维形式群的结果。

阿贝尔动机模空间的$p$进双曲性

$p$-adic hyperbolicity for moduli spaces of abelian motives

本研究证明阿贝尔型Shimura簇在离散赋值域上满足$p$进Borel延拓性质，推导了其在多领域的应用，并证明了Rapoport-Zink空间的延拓结果。

刻痕、扭量子和$p$-adic周期

Inscription, twistors, and $p$-adic periods

本文提出刻痕$v$-sheaves理论，用于研究$p$-adic周期，构建了Hodge和Hodge-Tate周期映射的刻痕版本，并引入了新的张量 functor 和几何Sen morphism 结构。

Weight-monodromy conjecture over equal characteristic local fields

The aim of this paper is to study certain properties of the weight spectral sequences of Rapoport-Zink by a specialization argument. By reducing to the case over finite fields previously treated by Deligne, we prove that the weight filtration and the monodromy filtration defined on the $l$-adic étale cohomology coincide, up to shift, for proper smooth varieties over equal characteristic local fields. We also prove that the weight spectral sequences degenerate at $E_2$ in any characteristic without using log geometry. Moreover, as an application, we give a modulo $p>0$ reduction proof of a Hodg…

The generic fiber of moduli spaces of bounded local $G$-shtukas

Moduli spaces of bounded local $G$-shtukas are a group-theoretic generalization of the function field analog of Rapoport and Zink's moduli spaces of $p$-divisible groups. In this article we generalize some very prominent concepts in the theory of Rapoport-Zink spaces to our setting. More precisely, we define period spaces, as well as the period map from a moduli space of bounded local $G$-shtukas to the corresponding period space, and we determine the image of the period map. Furthermore, we define a tower of coverings of the generic fiber of the moduli space which is equipped with a Hecke act…

Period Spaces for Hodge Structures in Equal Characteristic

We develop the analog in equal positive characteristic of Fontaine's theory for crystalline Galois representations of a p-adic field. In particular we describe the analog of Fontaine's functor which assigns to a crystalline Galois representation an isocrystal with a Hodge filtration. In equal characteristic the role of isocrystals and Hodge filtrations is played by z-isocrystals and Hodge-Pink structures. The latter were invented by Pink. Our first main result in this article is the analog of the Colmez-Fontaine Theorem that "weakly admissible implies admissible". Next we construct period spac…

范畴局部Langlands猜想

The categorical local Langlands conjecture

本文提出一个程序来证明Fargues-Scholze的范畴局部Langlands猜想，通过建立增强Whittaker系数函子与Eisenstein级数的兼容性，并利用归纳原理将一般群约化到真Levi子群。

关于紧支撑上同调的猜想 $C_{\rm st}$

Une conjecture $C_{\rm st}$ pour la cohomologie à support compact

本文提出针对 $p$-adic 解析簇紧支撑上同调的 $C_{\rm st}$ 类猜想，通过分析 $\mathbf{B}$ 中加入特定元素对 Galois 上同调的影响。

环面型 Shimura 簇的典范整模型

Canonical integral models for Shimura varieties of toral type

本文在环面型 Shimura 簇情形下证明了 Pappas-Rapoport 猜想，并利用棱镜 F-晶体理论建立了 G-值晶体表示与 G-shtuka 之间的全忠实函子。

Topological flatness of orthogonal local models in the split, even case. I

Local models are schemes, defined in terms of linear algebra, that were introduced by Rapoport and Zink to study the étale-local structure of integral models of certain PEL Shimura varieties over p-adic fields. A basic requirement for the integral models, or equivalently for the local models, is that they be flat. In the case of local models for even orthogonal groups, Genestier observed that the original definition of the local model does not yield a flat scheme. In a recent article, Pappas and Rapoport introduced a new condition to the moduli problem defining the local model, the so-called s…