A Note on Arithmetic Cohomologies for Number Fields
Lin Weng
作为我们几何算术课程的一部分，我们利用局部紧群的理论建立了数域的算术上同调理论。
As a part of our program for Geometric Arithmetic, we develop an arithmetic cohomology theory for number fields using theory of locally compact groups.
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Remarques sur la cohomologie des groupes kählériens nilpotents
本文证明了（虚）幂零Kähler群的上同调群自然具有混合Hodge结构.这些结构使得Hopf态射成为混合Hodge结构态射。我们用已知的非阿贝尔幂零Kähler群的例子来说明这一事实。
Cohomologie relative des formes cuspidales
本文证明了模尖点形式在环面紧化和极小紧化之间没有更高的直象。- 在这篇文章中，我们指出，模尖形的形式不是在环和极小紧化中直接产生的象。
Cohomologies of harmonic bundles on quasi-compact Kaehler manifolds
本文综述了我们最近关于拟紧Kaehler流形上调和丛的上同调的工作。
Travaux de Gabber sur l'uniformisation locale et la cohomologie etale des schemas quasi-excellents. Seminaire a l'Ecole polytechnique 2006--2008
这本书包含说明研讨会上奥弗加伯的工作的etale上同调和统一的准优秀计划。他的主要成果包括（cf。引言）构造性定理（交换或非交换系数），消灭定理（如仿射Lefschetz），“素数到l变换拓扑”的一致化，非交换系数的刚性，绝对纯度猜想的新证明，对偶性等。
We study various examples of Calabi-Yau threefolds over finite fields. In particular, we provide a counterexample to a conjecture of K. Joshi on lifting Calabi-Yau threefolds to characteristic zero. We also compute the p-adic cohomologies of some Calabi-Yau threefolds constructed by Cynk-van Straten which have remarkable arithmetic properties, as well as those of the Hirokado threefold. These examples and computations answer some outstanding questions of B. Bhatt, T. Ekedahl, van der Geer-Katsura and Patakfalvi-Zdanowicz, and shed new light on the Beauville-Bogomolov decomposition in positive …
Cohomologies of unipotent harmonic bundles over quasi-projective varieties I: The case of noncompact curves
令$S$是从紧致黎曼曲面中删除有限个称为尖点的点而得到的黎曼曲面。设$ρ: π_1(S)\to Sl(n, \mathbb{C})$是$π_1(S)$的半单线性表示，它在尖点附近是幂幺的.我们研究了与$\bar S$的$ρ$相关的各种上同调，其中$L_ρ$具有退化系数（被认为是一个局部系统-平坦向量丛，希格斯丛或$\mathcal{D}$ -模块，取决于上下文）：$j_*L_ρ$的Čech上同调，$L^2$ -上同调，$L^2$ -Dolbeault上同调和$L^2$ -希格斯上同调，以及它们之间的关系。本文是拟投射簇上退化系数的上同调及其Kähler推广研究的一般程序的一部分。这里的一般目的不限于曲线的情况，也不限于在除数附近幂幺的表示。因此，本说明的目的是阐明在这种特殊情况下，一般情况下的许多（分析和几何）困难并不存在，当我们考虑幂单调和丛而不是霍奇结构的变体时，会出现什么样的差异，其中结果是已知的。
In this paper we study the corresponding categories and the corresponding cohomologies of the Hodge-Iwasawa modules we developed in our series papers on Hodge-Iwasawa theory. The corresponding cohomologies will be essential in the corresponding development of the contact with the corresponding Iwasawa theoretic consideration, while they are as well very crucial in the corresponding study of the corresponding deformations of local systems over general analytic spaces. We contact with some applications in analytic geometry and arithmetic geometry which all have their own interests and deserve fu…
$p$-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。
This work is devoted to study orientation theory in arithmetic geometric within the motivic homotopy theory of Morel and Voevodsky. The main tool is a formulation of the absolute purity property for an \emph{arithmetic cohomology theory}, either represented by a cartesian section of the stable homotopy category or satisfying suitable axioms. We give many examples, formulate conjectures and prove a useful property of analytical invariance. Within this axiomatic, we thoroughly develop the theory of characteristic and fundamental classes, Gysin and residue morphisms. This is used to prove Riemann…
Arithmetic linear series with base conditions
本文研究了具有基条件的算术线性级数的体积。作为一个应用，我们考虑了算术簇上的Zagliki分解问题。
In this note, we study the differentiability of the arithmetic volumes along arithmetic R-divisors, and give some equality conditions for the Brunn-Minkowski inequality for arithmetic volumes over the cone of nef and big arithmetic R-divisors.
In this short note, we show a p-adic analogue of Beilinson's equivalence comparing two derived categories: the derived category of holonomic modules and derived category of modules whose cohomologies are holonomic.
Factorization de la cohomologie étale p-adique de la tour de Drinfeld
本文研究Drinfeld塔的p进平展上同调分解，对$F={\mathbf Q}_p$给出类似Emerton模曲线塔完备上同调的分解，证明模p算术平展上同调的有限性定理，并指出$F\neq{\mathbf Q}_p$时相关${\rm GL}_2(F)$表示不可容许。
Intersection pairing for arithmetic cycles with degenerate Green currents
本文给出余维为1的算术Chow群的一个适当的推广，其中Hodge指标定理成立。我们还证明了一个算术模拟的Bogomolov的不稳定性定理的秩2向量丛上的任意正则投影算术品种。
Hodge index theorem for arithmetic cycles of codimension one
本文将给出H. Grothendieck标准代数的算术模拟的部分答案。Gillet和C. Soule（注：我更改了这篇文章的标题。）
We compute some arithmetic path integrals for BF-theory over the ring of integers of a totally imaginary field, which evaluate to natural arithmetic invariants associated to $\mathbb{G}_m$ and abelian varieties.
Espaces de Berkovich globaux : catégorie, topologie, cohomologie
本文研究整体Berkovich空间的基础理论，定义了性质良好的态射并建立相关构造，证明其局部道路连通性，还证明任意维开/闭圆盘无高阶凝聚上同调，将算术幂级数环的Noether性定理推广到多变量情形。
Chern invariants of some flat bundles in the arithmetic Deligne cohomology
本文研究了Green-Griffiths，Asakura-Saito提出的算术Deligne上同调中平坦丛的Chern类。我们证明了在某些情况下陈类的非平凡性，证明也表明一般平坦丛（在其模空间）可以预期有非平凡类，这是在某些Gauss-Manin系统。
Geometry and arithmetic of non-rigid families of Calabi-Yau 3-folds; questions and examples
本文研究了Calabi-Yau流形模栈中极大乘积子簇的结构。我们讨论了四维射影空间中五次超曲面的例子。
Bogomolov unstability on arithmetic surfaces
本文考虑Bogomolov不稳定性定理的一个算术模拟。
Notes on Calabi-Yau ordinary differential equations
本文研究了Calabi-Yau微分方程的结构以及方程背后的Calabi-Yau簇的簇与算法的关系。这为Samol和van Straten最近的一篇论文中的一些观察和计算提供了解释。
Those notes rest on the Samuel Eilenberg Lectures I gave at Columbia University, NY, in the fall 2022. I thank all the mathematicians who participated in their elaboration, directly or indirectly. They are meant to be published as a Springer Lecture Notes. This arXiv version is not identical with the final version.
We describe how to find period integrals and Picard-Fuchs differential equations for certain one-parameter families of Calabi-Yau manifolds. These families can be seen as varieties over a finite field, in which case we show in an explicit example that the number of points of a generic element can be given in terms of p-adic period integrals. We also discuss several approaches to finding zeta functions of mirror manifolds and their factorizations. These notes are based on lectures given at the Fields Institute during the thematic program on Calabi-Yau Varieties: Arithmetic, Geometry, and Physic…
Those are the notes for the 2015 Summer Research Institute on Algebraic Geometry. We report on Deligne's finiteness theorem for $\ell$-adic representations on smooth varieties defined over a finite field, on its crystalline version, and on how the geometric étale fundamental group of a smooth projective variety defined over a characteristic $p>0$ field controls crystals on the infinitesimal site and should control those on the crystalline site. v2: last results added to the report, and some typos corrected.
Let $X$ be a variety (possibly non-complete or singular) over a finitely generated field $k$ of characteristic $0$. For a prime number $\ell$, let $ρ_\ell$ be the Galois representation on the first $\ell$-adic cohomology of $X$. We show that if $\ell$ varies the image of $ρ_\ell$ is of bounded index in the group of $\mathbb{Z}_\ell$-points of its Zariski closure. We use this to improve a recent result of Litt about arithmetic representations of geometric fundamental groups. Litt's result says that there exist constants $N = N(X,\ell)$ such that every arithmetic representation $π_1(X_{\bar{k}})…
This is my habilitation thesis. As the tradition wants, I tried to give an introduction of my field of research. I post it on the ArXiv with the hope it can be useful to young researchers looking for a short and friendly text on cohomologies of algebraic varieties, periods, algebraic cycles and motives. I might one day find the energy to expand these notes and maybe translate them in English. In the meantime, please feel free to ask questions. The first sections of this text propose an introduction to the theory, I tried to present points of view and examples which are not always stressed in t…
Two lectures on the arithmetic of K3 surfaces
在这些课堂讲稿中，我们回顾了K3曲面算法的不同方面。主题包括有理点，皮卡德数和泰特猜想，泽塔函数和模块化。
The arithmetic of QM-abelian surfaces through their Galois representations
本文研究数域上四元数乘法交换曲面的丢番图性质。我们研究了这些阿贝尔簇上的自同态的定义域以及它们的Tate模上的伽罗瓦表示的象。我们用几个明确的例子来说明我们的结果。
A well known argument by Serre shows that there is no Weil cohomology theory with real coefficients for smooth projective varieties over $\bar{\mathbb{F}}_p$. In this note we explain why no "Weil-"cohomology theory with real coefficients can exist for arithmetic schemes over spec $\mathbb{Z}$, even for spectra of number rings.
A Note on the Specialization Theorem for Families of Abelian Varieties
本文应用Moriwaki的算术高度函数得到了$K$上的阿贝尔簇族的Silverman特化定理的一个类似结果，其中$K$是在${\Q}$上生成的任意域群.
Self-intersection of dualizing sheaves of arithmetic surfaces with reducible fibers
设K是代数数域，O_K是K的整数环.设f：X --> Spec（O_K）是O_K上亏格g >= 2的稳定算术曲面.本文证明了：若f有可约几何纤维，则X的对偶层与Arakelov度量的自交大于或等于log（2）/6（g-1）.
Let K be an arithmetic function field, that is, a field of finite type over the rational number field. In this note, as an application of the height theory due to Chen-Moriwaki, we would like to show that the solutions of Fermat's curve X^N + y^N = 1 of degree N over K consist of only either 0 or roots of unity for almost positive integers N. More precisely, the density of such N is 1.
Étale duality for constructible sheaves on arithmetic schemes
在这篇笔记中，我们涉及三个主题的算术计划：一个一般的对偶的étale可构造扭转层，一个étale同调论，和一个Gersten-Bloch-Ogus-Kato复形。本文的结果已在作者的其他论文中使用（参考文献[JS]，[Sa]，[SaH]）。
Notes on the decomposition theorem for blowups
研究吹胀量子cohomology分解定理中的同构性质及其在Katzarkov-Kontsevich-Pantev-Yu理性问题中的应用
The Arithmetic Site
我们表明，非交换几何方法的黎曼ζ函数有一个代数几何化身：“算术网站”。本文研究了热带半环上的层，它是正整数乘法半群的对偶。证明了热带半域的极大紧子环上的算术点集是Q的Adele类空间在Adele类群的极大紧子群作用下的非交换空间商.我们实现了Frobenius对应的“算术网站”的平方和计算其组成。本文提供了RH的非交换方法所依据的代数几何空间。
Advances in the equivariant minimal model program and their applications in complex and arithmetic dynamics
本文探讨了非同构可缩映射的等价极小模型程序进展及其在复动力学和算术动力学中的应用。
Geometric and arithmetic theta correspondences
本文以酉群为例介绍几何与算术 theta 对应，重点阐述 Kudla 纲领的最新进展及其应用。
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