Bhargav Bhatt, Jacob Lurie
In this note, we introduce and study the Cartier--Witt stack $\mathrm{WCart}_X$ attached to a $p$-adic formal scheme $X$ as well as some variants. In particular, we reinterpret the notion of prismatic crystals on $X$ and their cohomology in terms of quasicoherent sheaf theory on $\mathrm{WCart}_X$ in favorable situations.
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Prismatization
本文构造并研究p进整数环形式谱的三种棱镜化函子，旨在为棱镜上同调提供类似F-规范的系数范畴。
The aim of this article is to given an extension of the prismatization functor for $p$-adic formal schemes (whose construction was first sketched by Drinfeld and then given by Bhatt-Lurie) to all schemes over $\mathrm{Spec}(\mathbf{Z})$. We then prove some basic properties of this extension (algebraicity, flatness for syntomic morphisms, perfectness of cohomology) and show that for smooth schemes over $\mathbf{Q}$ this construction recovers (a version of) the filtered de Rham stack.
A 1-dimensional formal group over the prismatization of Spf Z_p
本文研究 Spf(Z_p) 棱镜化上的乘法群映射的核，证明其为某一定义在其上的一维形式群的 Cartier 对偶，并给出其李代数及在 q-de Rham 棱镜商上的拉回的显式描述。
Prismatization via spherical loop spaces
本文遵循Manam方法引入拓扑循环同调变体的Frobenius-逆扭，借此构造球谱自由环路空间的修改，为Bhatt-Lurie和Drinfeld的棱镜化栈提供偶周期谱增强，并将该增强编码的额外结构与Breuil-Kisin扭转及Drinfeld形式群等同。
On the Drinfeld formal group
本文证明 Drinfeld 在 Spf Z_p 棱柱化上的形式群等同于拓扑周期循环同调去完备化变体的 Quillen 形式群，并建立其向句法化的延拓及 Drinfeld 猜想的代数化。
We investigate a certain class of (geometric) finite (Galois) coverings of formal fibres of $p$-adic curves and the corresponding quotient of the (geometric) étale fundamental group. A key result in our investigation is that these (Galois) coverings can be compactified to finite (Galois) coverings of proper $p$-adic curves. We also prove that the maximal prime-to-$p$ quotient of the geometric étale fundamental group of a (geometrically connected) formal fibre of a $p$-adic curve is (pro-)prime-to-$p$ free of finite computable rank.
Formal models for relative adic spaces
本文将 Raynaud 的形式模型理论从刚性解析空间扩展到任意 Tate 仿元基上的一致 qcqs adic 空间，并引入规范形式爆破作为关键新概念。
We investigate generalizations along the lines of the Mordell--Lang conjecture of the author's $p$-adic formal Manin--Mumford results for $n$-dimensional $p$-divisible formal groups $\mathcal{F}$. In particular, given a finitely generated subgroup $Γ$ of $\mathcal{F}(\overline{\mathbb{Q}}_p)$ and a closed subscheme $X\hookrightarrow \mathcal{F}$, we show under suitable assumptions that for any points $P\in X(\mathbb{C}_p)$ satisfying $nP\inΓ$ for some $n\in\mathbb{N}$, the minimal such orders $n$ are uniformly bounded whenever $X$ does not contain a formal subgroup translate of positive dimens…
We show the non-existence of sections of arithmetic fundamental groups of open p-adic annuli of small radii, this implies the non-existence of sections of arithmetic fundamental groups of formal boundaries of formal germs of p-adic curves.
We prove that for any proper smooth formal scheme $\frak X$ over $\mathcal O_K$, where $\mathcal O_K$ is the ring of integers in a complete discretely valued nonarchimedean extension $K$ of $\mathbb Q_p$ with perfect residue field $k$ and ramification degree $e$, the $i$-th Breuil-Kisin cohomology group and its Hodge-Tate specialization admit nice decompositions when $ie<p-1$. Thanks to the comparison theorems in the recent works of Bhatt, Morrow and Scholze, we can then get an integral comparison theorem for formal schemes when the cohomological degree $i$ satisfies $ie<p-1$, which generalize…
Given a smooth formal scheme over the ring of integers of a mixed-characteristic perfectoid field, we study its $p$-adic vanishing cycles via de Rham--Witt and $q$-de Rham complexes.
The goal of this paper is to study a $p$-adic analog of the joint of the conjectures of André--Oort and André--Pink. More precisely, on a product of ordinary Siegel formal moduli schemes, we study the distribution of points whose components are either CM points or points in Hecke orbits. We use linearity of formal subschemes of the product as the $p$-adic analog of geodesicness over complex numbers. Moreover, we relax the usual incidence relations by using $p$-adic distance. We also study a $p$-adic formal scheme theoretic analog of the Ax--Lindemann theorem.
The p-adic Cauchy Theorem and Overconvergent Period Sheaves
本文建立了任意光滑刚性解析流形上p-adic Cauchy定理的几何类比，证明超收敛de Rham周期结构层的水平截面函子与Scholze的OBdR水平截面函子一致，并应用于识别D-cap-模的de Rham函子。
Quotients Of Admissible Formal Schemes and Adic Spaces by Finite Groups
本文自洽地处理了局部强Noetherian adic空间上局部拓扑有限型的容许（形式）概形与adic空间的有限群商问题。
Non-adic formal schemes
本文重新定义形式概型为 proringed 空间，研究其基本性质，并给出若干非 adic 形式概型的例子，以补充非诺特或非 adic 情形下研究较少的理论基础。
We establish several new properties of the $p$-adic Jacquet-Langlands functor defined by Scholze in terms of the cohomology of the Lubin-Tate tower. In particular, we reprove Scholze's basic finiteness theorems, prove a duality theorem, and show a kind of partial Künneth formula. Using these results, we deduce bounds on Gelfand-Kirillov dimension, together with some new vanishing and nonvanishing results. Our key new tool is the six functor formalism with solid almost $\mathcal{O}^+/p$-coefficients developed recently by the second author [Man22]. One major point of this paper is to extend the …
Topics in Adic Spaces
本文为海德堡春季学校迷你课程的扩展讲义，系统介绍 adic 空间中的凝聚层、态射性质及其与形式几何和 Berkovich 空间的联系。
In this sequel of arXiv:1211.5294 and arXiv:1211.5948, we develop an adic formalism for étale cohomology of Artin stacks and prove several desired properties including the base change theorem. In addition, we define perverse t-structures on Artin stacks for general perversity, extending Gabber's work on schemes. Our results generalize results of Laszlo and Olsson on adic formalism and middle perversity. We continue to work in the world of $\infty$-categories in the sense of Lurie, by enhancing all the derived categories, functors, and natural transformations to the level of $\infty$-categories…
We show that the Deligne formal model of the Drinfeld p-adic halfplane relative to a non-archimedean local field F represents a moduli problem of polarized O_F-modules with an action of the ring of integers O_E in a quadratic extension E of F. The proof proceeds by establishing a comparison isomorphism with the Drinfeld moduli problem. This isomorphism reflects the accidental isomorphism of SL_2(F) and SU(C)(F) for a two-dimensional split hermitian space C for E/F.
Our results concern analytic functions on the open unit $p$-adic poly-disc in $\mathbb{C}^n_p$ centered at the multiplicative unit and we prove that such functions only vanish at finitely many $n$-tuples of roots of unity $(ζ_1-1,\ldots,ζ_n-1)$ unless they vanish along a translate of the formal multiplicative group. For polynomial functions, this follows from the multiplicative Manin-Mumford conjecture. However we allow for a much wider class of analytic functions; in particular we establish a rigidity result for formal tori. Moreover, our methods apply to Lubin-Tate formal groups beyond just …
The aim of the present paper is to provide a new aspect of the $p$-adic Teichmüller theory established by S. Mochizuki. We study the symplectic geometry of the $p$-adic formal stacks $\widehat{\mathcal{M}}_{g, \mathbb{Z}_p}$ (= the moduli classifying $p$-adic formal curves of fixed genus $g>1$) and $\widehat{\mathcal{S}}_{g, \mathbb{Z}_p}$ (= the moduli classifying $p$-adic formal curves of genus $g$ equipped with an indigenous bundle). A major achievement in the (classical) $p$-adic Teichmüller theory is the construction of the locus $\widehat{\mathcal{N}}_{g, \mathbb{Z}_p}^{\mathrm{ord}}$ in…
A p-adic Cartier isomorphism between the A_inf-cohomology and de Rham-Witt complexes for semistable formal schemes
本文证明了半稳定形式概型上 A_inf-上同调与 de Rham-Witt 复形之间的 p-adic Cartier 同构，从而将 Bhatt-Morrow-Scholze 在光滑情形下的结果进行了推广。
The goal of the present text is to state and prove a generalization of Raynaud localization theorem in the setting of derived geometry. More explicitly, we show that the $\infty$-category of quasi-paracompact and quasi-separated derived $k$-analytic spaces can be realized as a localization of the $\infty$-category of admissible derived formal schemes. We construct a derived rigidification functor generalizing Raynaud rigidification functor. In order to construct the latter we will need to formalize derived formal $\mathcal{O}_k$-adic formal geometry via a structured spaces approach. We prove t…
Let $\mathcal{V}$ be a complete discrete valued ring of mixed characteristic $(0,p)$, $K$ its field of fractions, $k$ its residue field which is supposed to be perfect. Let $X$ be a separated $k$-scheme of finite type and $Y$ be an open subscheme of $X$. We construct the category $F\text{-}D ^\mathrm{b}_\mathrm{ovhol} (\mathcal{D} ^†_{(Y,X)/K})$ of overholonomy type over $(Y,X)/K$. We check that these categories satisfy a formalism of Grothendieck's six operations.
The small $p$-adic Simpson correspondence in the semi-stable reduction case
本文将光滑形式概型上的小 Simpson 对应已知结果推广至半稳定情形，为可提升半稳定形式概型建立了 Hitchin-small integral $v$-bundles 与 Higgs bundles 范畴及对应模栈的等价关系。
Prismatic crystals and $p$-adic Riemann--Hilbert correspondence
该研究系统分析光滑及半稳定形式概形的棱晶位形上的Δ₊^dR-晶体，通过分层计算与幂零/增强联络分类局部晶体，结合p进黎曼-希尔伯特函子及无限维Sen理论将结果整体化，用ℬ₊^dR-局部系统分类整体绝对晶体。
We introduce the notion of a prism, which may be regarded as a "deperfection" of the notion of a perfectoid ring. Using prisms, we attach a ringed site -- the prismatic site -- to a $p$-adic formal scheme. The resulting cohomology theory specializes to (and often refines) most known integral $p$-adic cohomology theories. As applications, we prove an improved version of the almost purity theorem allowing ramification along arbitrary closed subsets (without using adic spaces), give a co-ordinate free description of $q$-de Rham cohomology as conjectured by the second author, and settle a vanishin…
Let $O_D$ be the ring of integers in a division algebra of invariant $1/n$ over a p-adic local field. Drinfeld proved that the moduli problem of special formal $O_D$-modules is representable by Deligne's formal scheme version of the Drinfeld p-adic halfspace. In this paper we exhibit other moduli spaces of formal $p$-divisible groups which are represented by $p$-adic formal schemes whose generic fibers are isomorphic to the Drinfeld p-adic halfspace. We also prove an analogue concerning the Lubin-Tate moduli space.
We make a systematic study of the infinitesimal lifting conditions of a pseudo finite type map of noetherian formal schemes. We recover the usual general properties in this context, and, more importantly, we uncover some new phenomena. We define a completion map of formal schemes as the one that arises canonically by performing the completion of a noetherian formal scheme along a subscheme, following the well-known pattern of ordinary schemes. These maps are etale in the sense of this work (but not adic). They allow us to give a local description of smooth morphisms. This morphisms can be fact…
We establish a criterion for sheaves on an adically complete DG scheme to be coherent. We deduce a description of coherent sheaves on an adically complete lci singularity in terms of modules for a DG Lie algebra.
We define formal vector bundles with marked sections on Hilbert modular schemes and we show how to use them to construct modular sheaves with an integrable meromorphic connection and a filtration which, in degree 0, gives to us a $p$-adic interpolation of the usual Hodge filtration. We relate it with the sheaf of overconvergent Hilbert modular forms.
We generalise Dwork's theory of $p$-adic formal congruences from the univariate to a multi-variate setting. We apply our results to prove integrality assertions on the Taylor coefficients of (multi-variable) mirror maps. More precisely, with $\mathbf z=(z_1,z_2,...,z_d)$, we show that the Taylor coefficients of the multi-variable series $q(\mathbf z)=z_i\exp(G(\mathbf z)/F(\mathbf z))$ are integers, where $F(\mathbf z)$ and $G(\mathbf z)+\log(z_i) F(\mathbf z)$, $i=1,2,...,d$, are specific solutions of certain GKZ systems. This result implies the integrality of the Taylor coefficients of numer…
Derived algebras on formal stacks and prismatic gauges
本文研究形式堆栈上导出代数与棱柱标量的关系，证明了相关分类定理，并扩展了导出几何中的结果。
$p$-Adic Weight Spectral Sequences of Strictly Semi-stable Schemes over Formal Power Series Rings via Arithmetic $\mathcal{D}$-modules
本文通过算术$\mathcal{D}$-模理论，构建了严格半稳定方案在形式幂级数环上的$p$-adic权重谱序列，其$E_1$项由闭纤维的不可约成分的刚性上同调描述，$E_\infty$项猜想由Lazda-Pál的刚性上同调的unipotent邻近循环描述，并探讨了其泛函性。
We prove a number of p-adic congruences for the coefficients of powers of a multivariate polynomial f(x) with coefficients in a ring R of characteristic zero. If the Hasse--Witt operation is invertible, our congruences yield p-adic limit formulas which conjecturally describe the Gauss--Manin connection and the Frobenius operator on the unit-root crystal attached to f(x). As a second application, we associate with f(x) formal group laws over R. Under certain assumptions these formal group laws are coordinalizations of the Artin--Mazur functors. (This is a final version which we send for a publi…
Even periodization of spectral stacks
本文引入谱栈的偶周期化操作，恢复已知滤过与棱镜化栈，并证明椭圆曲线模栈及色模栈分别由拓扑模形式谱和球谱偶周期化得到，同时与对称幺半剪切建立联系。
Formal Manifold Structures on Positive Characteristic Varieties
本文实施Sullivan提出的l进形式化纲领，证明正特征代数簇上的l进形式流形结构及其阿贝尔化伽罗瓦对称性，并给出单连通代数簇l进同伦等价于有限CW复形的充要条件。
On integral local Shimura varieties
本文证明了对于任何阿贝尔类型的(G, μ)，当p≠2或p=2且G为A或C类型时，局部Shimura簇的构造猜想成立，并将其与局部Shimura品种联系起来。
We construct the crystalline comparison isomorphisms for proper smooth formal schemes over an absolutely unramified base. Such isomorphisms hold for étale cohomology with nontrivial coefficients, as well as in the relative setting, i.e. for proper smooth morphisms of smooth formal schemes. The proof is formulated in terms of the pro-étale topos introduced by Scholze, and uses his primitive comparison theorem for the structure sheaf on the pro-étale site. Moreover, we need to prove the Poincaré lemma for crystalline period sheaves, for which we adapt the idea of Andreatta and Iovita. Another in…
This work is devoted to the study of integral $p$-adic Hodge theory in the context of Artin stacks. For a Hodge-proper stack, using the formalism of prismatic cohomology, we establish a version of $p$-adic Hodge theory with the étale cohomology of the Raynaud generic fiber as an input. In particular, we show that the corresponding Galois representation is crystalline and that the associated Breuil-Kisin module is given by the prismatic cohomology. An interesting new feature of the stacky setting is that the natural map between étale cohomology of the algebraic and the Raynaud generic fibers is…
A six-functor formalism for syntomic cohomology
本文为p进形式概型的合成上同调构造六函子形式体系，将庞加莱对偶性推广到一般光滑态射。
Part of these notes was written as the author's 2013 master thesis. For proper flat schemes over a complete discrete valuation ring of mixed characteristic, we construct an isomorphism of certain subgroups of the Picard group and the first cohomology group of the structure sheaf. When the Picard scheme is available and smooth, it recovers the isomorphism coming from its formal completion. A reinterpretation of an old theorem of Mattuck is given.
We continue our study on infinitesimal lifting properties of maps between locally noetherian formal schemes started in math.AG/0604241. In this paper, we focus on some properties which arise specifically in the formal context. In this vein, we make a detailed study of the relationship between the infinitesimal lifting properties of a morphism of formal schemes and those of the corresponding maps of usual schemes associated to the directed systems that define the corresponding formal schemes. Among our main results, we obtain the characterization of completion morphisms as pseudo-closed immersi…
The small $p$-adic Simpson correspondence in terms of moduli spaces
本研究针对$\text{Q}_p$的perfectoid扩张上的刚性空间，基于Bhatt-Lurie的Hodge-Tate栈构造两类模栈的同构，改进了小$p$进Simpson对应，并在算术设定下证明了类似结论。
In this paper we show that it is possible to define a topology on the category of formal schemes over a ring of $p$-adic integers such that the left adjoint of the Greenberg Transform is a site cocontinuous functor when we equip the category of schemes over the residue field with the étale topology. We show furthermore that this topology allows us to give an isomorphism between the corresponding fundamental groups, and use this isomorphism to show that it is possible to geometrize the quasicharacters of a $p$-adic torus by a local system on a formal scheme over the ring of $p$-adic integers.
Hodge-Tate stacks and non-abelian $p$-adic Hodge theory of v-perfect complexes on rigid spaces
本文为刚性空间上的 v-完备复形构造了从 Hodge-Tate 叠到 v-站点的全忠实函子，并用 Higgs 模导出范畴刻画完美复形，从而建立导出的 $p$-进 Simpson 函子。
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…
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