Vector bundles and p-adic representations I

Local systems on diamonds and $p$-adic vector bundles

We use Scholze's framework of diamonds to gain new insights in correspondences between $p$-adic vector bundles and local systems. Such correspondences arise in the context of $p$-adic Simpson theory in the case of vanishing Higgs fields. In the present paper we provide a detailed analysis of local systems on diamonds for the étale, pro-étale, and the $v$-topology, and study the structure sheaves for all three topologies in question. Applied to proper adic spaces of finite type over $\mathbb{C}_p$ this enables us to prove a category equivalence between $\mathbb{C}_p$-local systems with integral…

Eisenstein-Kronecker series via the Poincaré bundle

A classical construction of Katz gives a purely algebraic construction of Eisenstein--Kronecker series using the Gauß--Manin connection on the universal elliptic curve. This approach gives a systematic way to study algebraic and $p$-adic properties of real-analytic Eisenstein series. In the first part of this paper we provide an alternative algebraic construction of Eisenstein--Kronecker series via the Poincaré bundle. Building on this, we give in the second part a new conceptional construction of Katz' two-variable $p$-adic Eisenstein measure through $p$-adic theta functions of the Poincaré b…

Strongly semistable reduction of syzygy bundles on plane curves

We investigate degenerations of syzygy bundles on plane curves over $p$-adic fields. We use Mustafin varieties which are degenerations of projective spaces to find a large family of models of plane curves over the ring of integers such that the special fiber consists of multiple projective lines meeting in one point. On such models we investigate vector bundles whose generic fiber is a syzygy bundle and which become trivial when restricted to each projective line in the special fiber. Hence these syzygy bundles have strongly semistable reduction. This investigation is motivated by the fundamen…

Symplectic geometry of $p$-adic Teichmüller uniformization for ordinary nilpotent indigenous bundles

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…

$\ell$-adic 局部系统与 Higgs 丛：一般情形

$\ell$-adic local systems and Higgs bundles: the generic case

本文证明有限域曲线上具有一般分歧的 $\ell$-adic 局部系统（自动不可约）的 Frobenius 不动点计数，等于稳定对数 Higgs 丛（分裂情形为抛物 Higgs 丛）的计数，相差 $q$ 的幂次。

The syntomic realization of the elliptic polylogarithm via the Poincaré bundle

We give an explicit description of the syntomic elliptic polylogarithm on the universal elliptic curve over the ordinary locus of the modular curve in terms of certain $p$-adic analytic moment functions associated to Katz' two-variable $p$-adic Eisenstein measure. The present work generalizes previous results of Bannai-Kobayashi-Tsuji and Bannai-Kings on the syntomic Eisenstein classes.

Parallel transport and the p-adic Simpson correspondence

Deninger and Werner developed an analogue for p-adic curves of the classical correspondence of Narasimhan and Seshadri between stable bundles of degree zero and unitary representations of the topological fundamental group for a complex smooth proper curve. Using parallel transport, they associated functorially to every vector bundle on a p-adic curve whose reduction is strongly semi-stable of degree 0 a p-adic representation of the fundamental group of the curve. They asked several questions: whether their functor is fully faithful; whether the cohomology of the local systems produced by this …

曲线的 $P$-进 Simpson 对应中的提升独立性问题

Lift-independence problem in the $p$-adic Simpson correspondence for curves

研究曲线情形下 $p$-进 Simpson 对应中 Higgs 丛与 $v$-丛的等价性是否依赖于提升的选择，证明了小秩半稳定提升独立 Higgs 丛的 Higgs 场为零，并构造了非零 Higgs 场的半稳定提升独立 Higgs 丛。

$p$-进流形上 Weil 丛的算术方面

Arithmetic Aspects of Weil Bundles over $p$-Adic Manifolds

本文建立 $p$-进解析流形上 Weil 丛的系统理论，证明其继承典范解析结构，并给出提升定理、Galois 等变结构及与晶体上同调的比较同构，应用于 $p$-进模形式和椭圆曲线的无穷小形变研究。

$p$-adic 域上的 $v$-向量丛与通过 Hodge-Tate 栈的 Sen 理论

v-vector bundles on $p$-adic fields and Sen theory via the Hodge-Tate stack

本文通过 Hodge-Tate 栈上的向量丛描述 $p$-adic 域的连续半线性表示，为 Sen 理论提供几何新视角，并解释 $B_{\mathrm{Sen}}$ 环的几何来源。

半稳定约化情形下的小 $p$-进 Simpson 对应

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。

曲线的秩2 ℓ-adic局部系统与Higgs丛

Rank 2 $\ell$-adic local systems and Higgs bundles over a curve

针对有限域上曲线补集的秩2 ℓ-adic局部系统，研究被Frobenius迭代固定的给定tame局部单模的系统数量，证实Deligne猜想并将结果用Higgs丛数量表示。

Eisenstein-Kronecker类、Hecke $L$-函数临界值的整性及$p$-进插值

Eisenstein-Kronecker classes, integrality of critical values of Hecke $L$-functions and $p$-adic interpolation

本文证明任意全虚数域上代数Hecke特征标的临界$L$-值除以周期为代数整数，并构造$p$-进测度插值普通情形下的临界值，推广了Damerell、Shimura和Katz的结果。

$p$-adic 域上环面基本群的截面猜想

The section conjecture for the toric fundamental group over $p$-adic fields

本文在 $p$-adic 域上证明了环面基本群的 Grothendieck 截面猜想，并将 Selmer 截面重新解释为环面 Galois 截面。

Chern classes of automorphic vector bundles

We prove that Chern classes in continuous $\ell$-adic cohomology of automorphic bundles associated to representations of $G$ on a projective Shimura variety with data $(G,X)$ are trivial rationally. It is a consequence of Beilinson's conjectures which predict that the Chern classes in the Chow groups vanish rationally.

p进上半平面上带联络的等变线丛

Equivariant line bundles with connection on the p-adic upper half plane

该研究在p进数域框架下，通过局部分析方法，借助四元数代数极大序单位的光滑线性特征，完成了p进上半平面上带可积联络的挠等变线丛的构造与分类工作。

Integral structures in automorphic line bundles on the $p$-adic upper half plane

Given an automorphic line bundle ${\mathcal O}_X(k)$ of weight $k$ on the Drinfel'd upper half plane $X$ over a local field $K$, we construct a ${\rm GL}_2(K)$-equivariant integral lattice ${\mathcal O}_{\widehat{\mathfrak X}}(k)$ in ${\mathcal O}_X(k)\otimes_K\widehat{K}$, as a coherent sheaf on the formal model $\widehat{\mathfrak{X}}$ underlying $X\otimes_K\widehat{K}$. Here $\widehat{K}/K$ is ramified of degree $2$. This generalizes a construction of Teitelbaum from the case of even weight $k$ to arbitrary integer weight $k$. We compute $H^*(\widetilde{\mathfrak{X}},{\mathcal O}_{\widehat{…

关于 Narasimhan 与 Seshadri 定理的 p 进版本

On a p-adic version of Narasimhan and Seshadri's theorem

本文研究 Faltings 关于 p 进 Simpson 对应像的提问，证明在好约化及 p 足够大的条件下，潜在强半稳定约化等价于强半稳定约化，从而给出否定回答。

代数仿射代数上的$p$-进单值化定理

The $p$-adic monodromy theorem over algebraic-affinoid algebras

本文研究代数仿射代数参数化的Fargues-Fontaine曲线上$G_K$-等变向量丛族的$p$-进霍奇理论性质，定义相关表示概念并证明$p$-进单值化定理，推广了Berger-Colmez的工作，还在去掉自由性条件下给出了等变线丛族的分类。

Generalised representations as q-connections in integral $p$-adic Hodge theory

We relate various approaches to coefficient systems in relative integral $p$-adic Hodge theory, working in the geometric context over the ring of integers of a perfectoid field. These include small generalised representations over $A_{\text{inf}}$ inspired by Faltings, modules with q-connection in the sense of q-de Rham cohomology, crystals on the prismatic site of Bhatt--Scholze, and q-deformations of Higgs bundles.

p-adic uniformization of unitary Shimura varieties II

In this paper we show that certain Shimura varieties, uniformized by the product of complex unit balls, can be p-adically uniformized by the product (of equivariant coverings) of Drinfeld upper half-spaces. We also extend a p-adic uniformization to automorphic vector bundles. It is a continuation of our previous work [V], and contains all cases (up to a central modification) of a uniformization by known p-adic symmetric spaces. The idea of the proof is to show that an arithmetic quotient of the product of Drinfeld upper half-spaces cannot be anything else than a certain unitary Shimura variety…

An explicit formula for the generic number of dormant indigenous bundles

A dormant indigenous bundle is an integrable projective line bundle on a proper hyperbolic curve of positive characteristic satisfying certain conditions. Dormant indigenous bundles were introduced and studied in the $p$-adic Teichmuller theory developed by S. Mochizuki. Kirti Joshi proposed a conjecture concerning an explicit formula for the degree over the moduli stack of curves of the moduli stack classifying dormant indigenous bundles. In this paper, we give a proof for this conjecture of Joshi.

具有小 $l$-adic 局部系统的阿贝尔簇族非提升性

Non-liftability of Families of Abelian Varieties with Small $l$-adic Local System

本文研究特征 $p$ 上光滑正则曲线上的阿贝尔簇族，证明其霍奇包非 nef 且无法提升到 $W_2(k)$，并建立一个阿拉凯洛夫型不等式。

Rigidity and a Riemann-Hilbert correspondence for p-adic local systems

We construct a functor from the category of p-adic etale local systems on a smooth rigid analytic variety X over a p-adic field to the category of vector bundles with an integrable connection over its "base change to B_dR", which can be regarded as a first step towards the sought-after p-adic Riemann-Hilbert correspondence. As a consequence, we obtain the following rigidity theorem for p-adic local systems on a connected rigid analytic variety: if the stalk of such a local system at one point, regarded as a p-adic Galois representation, is de Rham in the sense of Fontaine, then the stalk at ev…

Geometric stabilisation via p-adic integration

In this article we give a new proof of Ngô's Geometric Stabilisation Theorem, which implies the Fundamental Lemma. This is a statement which relates the cohomology of Hitchin fibres for a quasi-split reductive group scheme $G$ to the cohomology of Hitchin fibres for the endoscopy groups $H_κ$. Our proof avoids the Decomposition and Support Theorem, instead the argument is based on results for $p$-adic integration on coarse moduli spaces of Deligne-Mumford stacks. Along the way we establish a description of the inertia stack of the (anisotropic) moduli stack of $G$-Higgs bundles in terms of end…

A p-adic Eisenstein measure for vector-weight automorphic forms

We construct a p-adic Eisenstein measure with values in the space of vector-weight p-adic automorphic forms on certain unitary groups. This measure allows us to p-adically interpolate special values of certain vector-weight C-infinity automorphic forms, including Eisenstein series, as their weights vary. We also explain how to extend our methods to the case of Siegel modular forms and how to recover Nicholas Katz's p-adic families of Eisenstein series for Hilbert modular forms.

模包的埃西森类与SL_n(Z)的log-刚性解析类

Eisenstein class of a torus bundle and log-rigid analytic classes for $\mathrm{SL}_n(\mathbb{Z})$

本文研究了模包的埃西森类与SL_n(Z)的log-刚性解析类，探讨了这些类在完全实域上的应用及与p进L函数的关系。

Vector Bundles over Projectivoid Line

In this article we describe vector bundles over projectivoid line and show how it is similar to (and different) from Gorthendieck's classification of vector bundles over projective line.

The formulas of coefficients of sum and product of p-adic integers with applications to Witt vectors

The explicit formulas of operations, in particular addition and multiplication, of $p $-adic integers are presented. As applications of the results, at first the explicit formulas of operations of Witt vectors with coefficients in $\mathbb{F}_{2}$ are given; then, through solving a problem of Browkin about the transformation between the coefficients of a $p$-adic integer expressed in the ordinary least residue system and the numerically least residue system, similar formulas for Witt vectors with coefficients in $\mathbb{F}_{3}$ are obtained.

$p$进上半平面上等变线丛的整体截面

Global sections of equivariant line bundles on the $p$-adic upper half plane

该文研究$p$进上半平面上等变线丛整体截面的表示性质，证明其强对偶是拓扑长度至多2的可容许局部解析表示，给出不可约判据及相关覆盖函数空间对偶长度的显式公式。

Fargues-Fontaine曲线上的向量丛扩张 II

Extensions of vector bundles on the Fargues-Fontaine curve II

该研究针对Fargues-Fontaine曲线上的向量丛扩张问题，完成了由两个任意向量丛生成的所有扩张向量丛的完全分类工作。

Triple product p-adic L-functions associated to finite slope p-adic families of modular forms, with an appendix by Eric Urban

We p-adically interpolate the relative de Rham cohomology of the universal elliptic curve over strict neighbourhoods of the ordinary locus of modular curves, together with the Hodge filtration and Gauss-Manin connection. Sections of these sheaves provide the so called nearly overconvergent modular forms. This extends previous work of Andreatta, Iovita and Pilloni where we p-adically interpolate powers of the Hodge bundle and in that case the sections coincide with Coleman overconvergent modular forms. We also show that, under suitable assumptions, one can p-adically interpolate the Gauss-Manin…

Arithmetic {B}reuil-{K}isin-{F}argues modules and comparison of integral {p}-adic Hodge theories

Let $K$ be a discrete valuation field with perfect residue field, we study the functor from weakly admissible filtered $(φ,N,G_K)$-modules over $K$ to the isogeny category of Breuil-Kisin-Fargues $G_K$-modules. This functor is the composition of a functor defined by Fargues-Fontaine from weakly admissible filtered $(φ,N,G_K)$-modules to $G_K$-equivariant modifications of vector bundles over the Fargues-Fontaine curve $X_{FF}$, with the functor of Fargues-Scholze that between the category of admissible modifications of vector bundles over $X_{FF}$ and the isogeny category of Breuil-Kisin-Fargue…

Locally analytic vectors in representations of locally p-adic analytic groups

This paper develops various foundational results in the locally analytic representation theory of p-adic groups. In particular, we define the functor ``pass to locally analytic vectors'', which attaches to any continuous representation of a p-adic analytic group on a locally convex p-adic topological vector space the associated space of locally analytic vectors. Using this functor, and the point of view that its construction suggests, we establish some basic facts about admissible locally analytic representations (as defined by Schneider and Teitelbaum). We also introduce the related notion of…

Mellin transforms of p-adic Whittaker functions

It is shown that Mellin transforms of p-adic Whittaker functions exist for generic characters. For good choices of vectors they are rational functions. For class one vectors they can be calculated explicitly. It turns out that they are automorphic L-factors times normalization factors.

$p$-adic Directions of Primitive Vectors

Linnik type problems concern the distribution of projections of integral points on the unit sphere as their norm increases, and different generalizations of this phenomenon. Our work addresses a question of this type: we prove the uniform distribution of the projections of primitive $\mathbb{Z}^{2}$ points in the $p$-adic unit sphere, as their (real) norm tends to infinity. The proof is via counting lattice points in semi-simple $S$-arithmetic groups.

从方向到ℓ-adic周期向量

From Orientations to $\ell$-adic Period Vectors

本文提出连接定向超奇异椭圆曲线与模曲线算术的桥梁，通过Coleman积分计算ℓ-adic周期向量，并探讨模符号逆问题及其在密码学中的应用。

Analytic vectors in continuous p-adic representations

Given a compact p-adic Lie group G over a finite unramified extension L/Q_p let G_0 be the product over all Galois conjugates of G. We construct an exact and faithful functor from admissible G-Banach space representations to admissible locally L-analytic G_0-representations that coincides with passage to analytic vectors in case L=Q_p. On the other hand, we study the functor "passage to analytic vectors" and its derived functors over general basefields. As an application we determine the higher analytic vectors in certain locally analytic induced representations.

Rigid vectors in $p$-adic principal series representations

In this paper we view pro-$p$ Iwahori subgroups $I$ as rigid analytic groups $\Bbb{I}$ for large enough $p$. This is done by endowing $I$ with a natural $p$-valuation, and thereby generalizing results of Lazard for $\text{GL}_n$. We work with a general connected reductive split group over some $p$-adic field (with simply connected derived group) and study the $\Bbb{I}$-analytic vectors in principal series representations. Our main result is an irreducibility criterion which generalizes results of Clozel and Ray in the $\text{GL}_n$-case.

关于$p$-adic域中LVP和CVP的注释

Notes on the LVP and CVP in $p$-adic Fields

本文提出在$p$-adic域中通过多项式时间算法解决LVP和CVP问题，利用$p$-adic范数的非阿基米德性质，构造$p$-adic晶格的正交基。

Rigid analytic vectors of crystalline representations arising in $p$-adic Langlands

Let $\mathbf{B}(V)$ be the admissible unitary $GL_2(\mathbb{Q}_p)$-representation associated to two dimensional crystalline Galois representation $V$ by $p$-adic Langlands constructed by Breuil. Berger and Breuil conjectured an explicit description of the locally analytic vectors $\mathbf{B}(V)_{\mathrm{la}}$ of $\mathbf{B}(V)$ which is now proved by Liu. Emerton recently studied $p$-adic representations from the viewpoint of rigid analytic geometry. In this article, we consider certain rigid analytic subgroups of $GL(2)$ and give an explicit description of the rigid analytic vectors in $\math…

范数正交基与$p$-adic格的不变量

Norm Orthogonal Bases and Invariants of $p$-adic Lattices

本文证明每个p-adic格存在正交基，定义连续极大值与逃逸距离，给出多项式时间算法求解LVP和CVP，并证明正交化与CVP多项式等价。

Equivariant vector bundles on Drinfeld's halfspace over a finite field

Let $\mathcal{X} \subset \mathbb{P}_k^d$ be Drinfeld's halfspace over a finite field $k$ and let $\mathcal{E}$ be a homogeneous vector bundle on $\mathbb{P}_k^d$. The paper deals with two different descriptions of the space of global sections $H^0(\mathcal{X},\mathcal{E})$ as $GL_{d+1}(k)$-representation. This is an infinite dimensional modular representation. Here we follow the ideas of \cite{O2,OS} treating the $p$-adic case. As a replacement for the universal enveloping algebra we consider both the crystalline universal enveloping algebra and the ring of differential operators on the flag v…

关于$p$-adic Gram-Schmidt正交化过程

On $p$-adic Gram-Schmidt Orthogonalization Process

本文将Weil关于局部域上范数正交基存在性的证明转化为算法，用于寻找各种范数正交基，这些算法与$p$-adic格中的CVP问题相关，并应用于基于$p$-adic格的密码学。

Mustafin models of projective varieties and vector bundles

Mustafin varieties are well-studied degenerations of projective spaces induced by a choice of integral points in a Bruhat--Tits building. In recent work, Annette Werner and the author initiated the study of degenerations of plane curves obtained by Mustafin varieties by means of arithmetic geometry. Moreover, we applied these techniques to construct models of vector bundles on plane curves with strongly semistable reduction. In this work, we take a Groebner basis approach to the more general problem of studying degenerations of projective varieties. Our methods include determining the behaviou…

Almost $C_p$ Galois representations and vector bundles

Let $K$ be a finite extension of $\mathbb{Q}_p$ and $G_K$ the absolute Galois group. Then $G_K$ acts on the fundamental curve $X$ of $p$-adic Hodge theory and we may consider the abelian category $\mathcal{M}(G_K)$ of coherent $\mathcal{O}_X$-modules equipped with a continuous and semi-linear action of $G_K$. An almost $C_p$-representation of $G_K$ is a $p$-adic Banach space $V$ equipped with a linear and continuous action of $G_K$ such that there exists $d\in\mathbb{N}$, two $G_K$-stable finite dimensional sub-$\mathbb{Q}_p$-vector spaces $U_+$ of $V$, $U_-$ of $C_p^d$, and a $G_K$-equivarian…

p-adic J-homomorphisms and a product formula

One-point compactification turns real vector spaces into spheres. In homotopy theory, this transformation gets encoded in a map called the "real J-homomorphism". Here we define and investigate p-adic J-homomorphisms, which sort of turn p-adic vector spaces into spheres. Our main theorem is a product formula for these J-homomorphisms, saying what happens when you start with a rational vector space. This formula specializes to Hilbert's version of the quadratic reciprocity law after applying second homotopy groups.