Invariant functions on p-divisible groups and the p-adic Corona problem

Smoothness of the truncated display functor

We show that to every p-divisible group over a p-adic ring one can associate a display by crystalline Dieudonne theory. For an appropriate notion of truncated displays, this induces a functor from truncated Barsotti-Tate groups to truncated displays, which is a smooth morphism of smooth algebraic stacks. As an application we obtain a new proof of the equivalence between infinitesimal p-divisible groups and nilpotent displays over p-adic rings, and a new proof of the equivalence due to Berthelot and Gabber between commutative finite flat group schemes of p-power order and Dieudonne modules over…

Groupes analytiques rigides p-divisibles II

Let $K$ be a $p$-adic field. We continue to develop the theory of rigid analytic $p$-divisible groups over $K$. For example, we explain how to find back the category of Banach-Colmez spaces from rigid analytic $p$-divisible groups "in finite level" without perfectoid spaces. We then establish some results about families of rigid analytic $p$-divisible groups. This allows us to prove a "minimality" result in the sense of birationnal geometry for integral models of unramified Rapoport-Zink spaces.

2-adic integral canonical models and the Tate conjecture in characteristic 2

We use E. Lau's classification of 2-divisible groups using Dieudonné displays to construct integral canonical models for Shimura varieties of abelian type at 2-adic places where the level is hyperspecial. We apply this to prove the Tate conjecture for K3 surfaces in characteristic 2.

The conjugate uniformization via 1-motives

We use the $p$-divisible group attached to a 1-motive to generalize the conjugate $p$-adic uniformization of Iovita--Morrow--Zaharescu to arbitrary $p$-adic formal semi-abelian schemes or $p$-divisible groups over the ring of integers in a $p$-adic field. This mirrors a mixed Hodge theory construction of the inverse uniformization map for complex semi-abelian varieties. We also highlight the geometric structure of the target of the conjugate uniformization map, which is an étale cover of a negative Banach--Colmez space in the sense of Fargues--Scholze.

计算有限域上曲线的晶体上同调与p-可除群

Computing Crystalline Cohomology and p-Divisible Groups for Curves over Finite Fields

本文提出了一种计算有限域上曲线的p-可除群的算法，基于晶体上同调的Frobenius和Verschiebung算子，改进了Tuitman的p进点计数算法。

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…

The relative Breuil-Kisin classification of $p$-divisible groups and finite flat group schemes

Assume that $p>2$, and let $\mathscr{O}_K$ be a $p$-adic discrete valuation ring with residue field admitting a finite $p$-basis, and let $R$ be a formally smooth formally finite-type $\mathscr{O}_K$-algebra. (Indeed, we allow slightly more general rings $R$.) We construct an anti-equivalence of categories between the categories of $p$-divisible groups over $R$ and certain semi-linear algebra objects which generalise $(φ,\mathfrak{S})$-modules of height $\leqslant1$ (or Kisin modules). A similar classification result for $p$-power order finite flat group schemes is deduced from the classificat…

p-adic Monodromy of the Universal Deformation of a HW-cyclic Barsotti-Tate Group

Let k be an algebraically closed field of characteristic $p>0$, and $G_0$ be a Barsotti-Tate group (or $p$-divisible group) over k. We denote by $S$ the "algebraic" local moduli in characteristic p of $G_0$, by $G$ the universal deformation of $G_0$ over $S$, and by $U\subset S$ the ordinary locus of $G$. The etale part of $G$ over $U$ gives rise to a monodromy representation $ρ$ of the fundamental group of $U$ on the Tate module of $G$. Motivated by a famous theorem of Igusa, we prove in this article that $ρ$ is surjective if $G_0$ is connected and HW-cyclic. This latter condition is equivale…

$p$-进族中的傅里叶理论

$p$-adic Fourier theory in families

本研究构造了有限高度$p$-可除刚性解析群上函数与分布和对偶范畴中$\mathbb{Z}_p$-局部系统对象的傅里叶变换，推广了前人构造，还构造了相容积分傅里叶变换，并应用于构造整体艾森斯坦测度及新四元数模形式族。

全虚域赫克特征的$p$进$L$函数

$p$-adic $L$-functions for Hecke characters of totally imaginary fields

本文针对任意非分歧素数与全虚域，结合新提出的$p$进傅里叶理论和已有等变上同调类，构造了插值代数赫克特征临界$L$值的$p$进$L$函数，解决了此前非寻常素数情形仅适用于有理数域虚二次扩张的局限。

$G$-displays of Hodge type and formal $p$-divisible groups

Let $G$ be a reductive group scheme over the $p$-adic integers, and let $μ$ be a minuscule cocharacter for $G$. In the Hodge-type case, we construct a functor from nilpotent $(G,μ)$-displays over $p$-nilpotent rings $R$ to formal $p$-divisible groups over $R$ equipped with crystalline Tate tensors. When $R/pR$ has a $p$-basis étale locally, we show that this defines an equivalence between the two categories. The definition of the functor relies on the construction of a $G$-crystal associated with any adjoint nilpotent $(G,μ)$-display, which extends the construction of the Dieudonné crystal ass…

关于解析p-可除群的Dieudonné理论及其在Shimura品种中的应用

A Dieudonné theory for analytic p-divisible groups and applications to Shimura varieties

本文研究了解析p-可除群的Dieudonné理论，并将其应用于Shimura品种，证明了与Hodge-Tate三元组的等价性，并重新解释了Hodge-Tate周期映射。

Unlikely intersections on the $p$-adic formal ball

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…

Classes de cycles motiviques étales

Let X be a smooth variety over a field k, and l be a prime number invertible in k. We study the (étale) unramified H^3 of X with coefficients Q_l/Z_l(2) in the style of Colliot-Thélène and Voisin. If k is separably closed, finite or p-adic, this describes it as an extension of a finite group F by a divisible group D, where F is the torsion subgroup of the cokernel of the l-adic cycle map. If k is finite and X is projective and of abelian type, verifying the Tate conjecture, D=0. If k is separably closed, we relate D to an l-adic Griffiths group. If k is the separable closure of a finite field …

Lower bound for the poles of Igusa's p-adic zeta functions

Let K be a p-adic field, R the valuation ring of K, P the maximal ideal of R and q the cardinality of the residue field R/P. Let f be a polynomial over R in n>1 variables and let χbe a character of R^{\times}. Let M_i(u) be the number of solutions of f=u in (R/P^i)^n for i \in \mathbb{Z}_{\geq 0} and u \in R/P^i. These numbers are related with Igusa's p-adic zeta function Z_{f,χ}(s) of f. We explain the connection between the M_i(u) and the smallest real part of a pole of Z_{f,χ}(s). We also prove that M_i(u) is divisible by q^{\ulcorner(n/2)(i-1)\urcorner}, where the corners indicate that we …

$p$-adic 域上拟射影曲面上的零环

Zero-cycles on quasi-projective surfaces over $p$-adic fields

本文通过研究开子簇的 Suslin 奇异同调，证明了 Colliot-Thélène 猜想对由曲线乘积几何支配的曲面成立。

Zero-cycles on a product of elliptic curves over a $p$-adic field

We consider a product $X=E_1\times\cdots\times E_d$ of elliptic curves over a finite extension $K$ of $\mathbb{Q}_p$ with a combination of good or split multiplicative reduction. We assume that at most one of the elliptic curves has supersingular reduction. Under these assumptions, we prove that the Albanese kernel of $X$ is the direct sum of a finite group and a divisible group, extending work of Raskind and Spiess to cases that include supersingular phenomena. Our method involves studying the kernel of the cycle map $CH_0(X)/p^n\rightarrow H^{2d}_{\text{ét}}(X, μ_{p^n}^{\otimes d})$. We give…

刚性解析1-动机与阿贝尔簇的共轭单值化

Rigid analytic 1-motives and conjugate uniformization of abeloid varieties

本文针对p进域上的阿贝尔簇，研究刚性解析1-动机理论及阿贝尔簇的共轭单值化，建立了形式与刚性解析1-动机的等价性等结果，还构造了相关p可除群。

关于μ-普通伊古萨簇上的p进马阿斯-志村算子

$p$-adic Maass--Shimura operators on $μ$-ordinary Igusa varieties

该研究针对霍奇型志村簇，将p进整近全纯自守形式上的代数马阿斯-志村微分算子扩展到μ-普通曼托万伊古萨簇，通过p进傅里叶理论等方法得到相关代数作用，扩展、澄清并恢复了先前构造。

(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.

具有自同态结构的p-可除群的积分Hodge多边形

The integral Hodge polygon for $p$-divisible groups with endomorphism structure

本文为带自同态结构的p-可除群引入积分Hodge多边形这一新不变量，揭示其与经典Hodge多边形、Fargues的Harder-Narasimhan多边形及Pappas-Rapoport多边形的关系，并证明其在p进解析族中连续变化。

1-motives的p-adic周期的线性关系（学位论文）

Linear relations of p-adic periods of 1-motives (thesis)

本文为具有良好约化的1-motives建立积分理论并提出p-adic周期猜想，证明了深度1和2的猜想，其方法涉及子群定理的p-adic版本。

德拉姆的代数性猜想与p-可除群的模

An algebraicity conjecture of Drinfeld and the moduli of $p$-divisible groups

本文利用德拉姆和布特尔-帕帕斯发展的新技术，统一构造了与p-可除群相关的模，验证了德拉姆的猜想，并推广了相关结果。

Potentially Good Reduction of Barsotti-Tate Groups

Let R be a complete discrete valuation ring of mixed characteristic (0, p) with perfect residue field, K the fraction field of R. Suppose G is a Barsotti-Tate group (p-divisible group) defined over K which acquires good reduction over a finite extension K' of K. We prove that there exists a constant c which depends on the absolute ramification index e(K'/Q_p) and the height of G such that G has good reduction over K if and only if G[p^c] can be extended to a finite flat group scheme over R. For abelian varieties with potentially good reduction, this result generalizes Grothendieck's p-adic Ner…

关于积分局部Shimura簇

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。

Breuil-Kisin 函子的正合性质与 Bloch-Kato Selmer 群

Exactness property of Breuil-Kisin functors and Bloch-Kato Selmer groups

本文证明Breuil-Kisin函子在半稳定表示格上的正合性，并利用该结果通过强可分模范畴中的Ext群计算积分Bloch-Kato Selmer群，同时定义了与Galois表示函子交换的张量积，应用于阿贝尔簇的杯积映射分解。

关于酉 Shimura 曲线的 p-adic 一致化

On the p-adic uniformization of unitary Shimura curves

本文证明了关于酉相似群 Shimura 曲线的 p-adic 一致化，通过引入 Kottwitz 条件等定义积分模型，并利用收缩函子与 display 理论给出了沿特殊纤维形式完备化的一致化证明。

Iwasawa Dieudonné theory of function fields

Let $k$ be a perfect field of characteristic $p$ and $Γ$ an infinite, first countable pro-$p$ group. We study the behavior of the $p$-primary part of the "motivic class group", i.e. the full $p$-divisible group of the Jacobian, in any $Γ$-tower of function fields over $k$ that is unramified outside a finite (possibly empty) set of places $Σ$, and totally ramified at every place of $Σ$. When $Σ=\emptyset$ and $Γ$ is a torsion free $p$-adic Lie group, we obtain asymptotic formulae which show that the $p$-torsion class group schemes grow in a remarkably regular manner. In the ramified setting $Σ\…

几何Kummer $K3$曲面上零环的局部和局部到全局原理

Local and local-to-global Principles for zero-cycles on geometrically Kummer $K3$ surfaces

研究$p$进域上几何Kummer $K3$曲面的零环Chow群结构，证明其可分解为可除群与有限群的直和，并给出局部到全局原理的首次无条件证据。