Anningzhe Gao
In this paper we will prove that Tate conjecture of abelian varieties over finite field is equivalent to the finiteness of isomorphism classes of abelian varieties with a fixed dimension. We give a different approach with Zarhin's result.
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We prove the Tate conjecture for divisor classes and the Mumford-Tate conjecture for the cohomology in degree 2 for varieties with $h^{2,0}=1$ over a finitely generated field of characteristic 0, under a mild assumption on their moduli. As an application of this general result, we prove the Tate and Mumford-Tate conjectures for some classes of algebraic surfaces with $p_g=1$.
Finiteness of K3 surfaces and the Tate conjecture
对于特征至少为5的有限域k，证明了Tate猜想对定义在k的代数闭包上的K3曲面成立当且仅当在k的每个有限扩张上只有n个K3曲面.
On the integral Tate conjecture for abelian varieties
本文利用Deligne的绝对Hodge环理论，从Engel等人关于积分Hodge猜想对非常一般阿贝尔簇不成立的结果，推导出积分Tate猜想的类似反例。
Tate twists of Hodge structures arising from abelian varieties of type IV
我们证明了某些阿贝尔簇A具有这样的性质：对于A的上同调中的每个Hodge结构V，V的每个有效Tate扭曲都发生在某些阿贝尔簇的上同调中。我们推出了一般霍奇猜想的某些非单阿贝尔簇的IV型。
Let $A$ be an abelian variety over a field finitely generated over $\mathbb{Q}$. We show that the finiteness of the $\ell$-primary torsion subgroup of the higher Brauer group is a sufficient criterion for the Tate conjecture to hold. Furthermore, we extend methods for computations of transcendental Brauer groups to higher Brauer groups.
We study the Mumford--Tate conjecture for hyperkähler varieties. We show that the full conjecture holds for all varieties deformation equivalent to either an Hilbert scheme of points on a K3 surface or to O'Grady's ten dimensional example, and all of their self-products. For an arbitrary hyperkähler variety whose second Betti number is not 3, we prove the Mumford--Tate conjecture in every codimension under the assumption that the Künneth components in even degree of its André motive are abelian. Our results extend a theorem of André.
Monodromy of Tate-Linear varieties
针对Chai关于有限域代数闭包上Ag的Tate-线性子簇刚性的问题，通过分析Tate-线性簇的p进单值群并结合前人工作，证明Ag中心叶中某点处为Tate-线性的光滑子簇在所有点处均为Tate-线性。
Let X be a Mumford-Tate variety, i.e., a quotient of a Mumford-Tate domain D by a discrete subgroup. Mumford-Tate varieties are generalizations of Shimura varieties. We define the notion of a special subvariety Y in X (of Shimura type), and formulate necessary criteria for Y to be special. Our method consists in looking at finitely many compactified special curves C_i in Y, and testing whether the inclusion of the union of all C_i in Y satisfies certain properties. One of them is the so-called relative proportionality condition. In this paper, we give a new formulation of this numerical criter…
Relying on recent advances in the theory of motives we develope a general formalism for derived categories of motives with Q-coefficients on perfect (ind-)schemes. As an application we give a motivic refinement of Zhu's geometric Satake equivalence for Witt vector affine Grassmannians in this set-up.
The Shafarevich conjecture revisited: Finiteness of pointed families of polarized varieties
受Lang-Vojta双曲簇猜想启发，研究者证明了带基点极化代数簇族有限性的新版Shafarevich猜想，并将其应用于极化代数簇模空间整点有限性的算术问题。
Tate conjecture and mixed perverse sheaves
利用混合反常层理论，将Lefschetz和Griffiths关于Hodge猜想的讨论推广到Tate猜想的情形，证明了因子的Tate猜想与非真簇的de Rham猜想、Tate-Shafarevich群的有限性以及解析数论中的一些定理密切相关.
In this article we study the (cohomological) Hodge conjecture for singular varieties. We prove the conjecture for simple normal crossing varieties that can be embedded in a family where the Mumford-Tate group remains constant. We show how to produce such families. Furthermore, we show for varieties with worse singularities the conjecture can be expressed solely in terms of the algebraic classes.
Affine Deligne-Lusztig varieties are closely related to the special fibre of Newton strata in the reduction of Shimura varieties or of moduli spaces of $G$-shtukas. In almost all cases, they are not quasi-compact. In this note we prove basic finiteness properties of affine Deligne-Lusztig varieties under minimal assumptions on the associated group. We show that affine Deligne-Lusztig varieties are locally of finite type, and prove a global finiteness result related to the natural group action. Similar results have previously been known for special situations.
Let $X$ be a smooth projective variety over a finitely generated field $K$ of characteristic~$0$ and fix an embedding $K \subset \mathbb{C}$. The Mumford--Tate conjecture is a precise way of saying that certain extra structure on the $\ell$-adic étale cohomology groups of~$X$ (namely, a Galois representation) and certain extra structure on the singular cohomology groups of~$X$ (namely, a Hodge structure) convey the same information. The main result of this paper says that if $A_1$ and~$A_2$ are abelian varieties (or abelian motives) over~$K$, and the Mumford--Tate conjecture holds for both~$A_…
Supersingular Tate conjecture for irreducible symplectic varieties of known types: I
本文证明多种已知类型的超奇异不可约辛簇在特定Artin不变量条件下具有Tate或超奇异阿贝尔Chow motive，进而其乘积满足超奇异Tate猜想。
The Tate Conjecture for Certain Abelian Varieties over Finite Fields
泰特定理（Tate's Theorem）Math. 1966）的结果表明，Tate猜想对有限域上的任何阿贝尔簇成立，该域的Tate类的Q_l-代数由1次Tate类的Q_l-代数生成.我们构建家庭的阿贝尔簇在有限域的这一条件失败，但我们仍然能够证明泰特猜想。
The main result of this paper concerns the positivity of the Hodge bundles of abelian varieties over global function fields. As applications, we obtain some partial results on the Tate--Shafarevich group and the Tate conjecture of surfaces over finite fields.
We study endomorphisms of abelian varieties and their action on the l-adic Tate modules. We prove that for every endomorphism one may choose a basis of each Tate module such that the corresponding matrix has rational entries and does not depend on l.
In two earlier articles, we proved that, if the Hodge conjecture is true for ALL CM abelian varieties over the complex numbers, then both the Tate conjecture and the standard conjectures are true for abelian varieties over finite fields. Here we rework the proofs so that they apply to a single abelian variety. As a consequence, we prove (unconditionally) that the Tate and standard conjectures are true for many abelian varieties over finite fields, including abelian varieties for which the algebra of Tate classes is not generated by divisor classes.
Shimura Varieties and the Mumford-Tate conjecture, part I
证明了数域上的交换阿贝尔簇的Mumford-Tate猜想，这些簇的伴随Mumford-Tate群的单因子具有超过$\dbR$的某些非紧因子.特别是，我们证明了这一猜想的正交情况下（$B_n$和$D_n^{\dbR}$志村型）。我们构造嵌入志村品种（其伴随是规定的交换类型）到酉志村品种。证明了数域上阿贝尔簇的Frobenius环面的两个一般定理。
A framework for Tate modules of abelian varieties under isogeny
本文基于同源映射下的阿贝尔簇的泰特模线性代数框架，从范畴论角度探讨其结构与性质。
We prove that there are only finitely many even lattices L of signature (2,n) with n>14 such that the modular variety defined by the stable orthogonal group of L is not of general type.
Classical Serre-Tate theory describes deformations of ordinary abelian varieties. It implies that every such variety has a canonical lift to characteristic zero and equips its local moduli space with a Frobenius lifting and canonical multiplicative coordinates. A variant of this theory has been obtained for ordinary K3 surfaces by Nygaard and Ogus. In this paper, we construct canonical liftings modulo $p^2$ of varieties with trivial canonical class which are ordinary in the weak sense that the Frobenius acts bijectively on the top cohomology of the structure sheaf. Consequently, we obtain a Fr…
We develop an analogue of Eisenbud-Floystad-Schreyer's Tate resolutions for toric varieties. Our construction, which is given by a noncommutative analogue of a Fourier- Mukai transform, works quite generally and provides a new perspective on the relationship between Tate resolutions and Beilinson's resolution of the diagonal. We also develop a Beilinson-type resolution of the diagonal for toric varieties.
Generalized Serre--Tate Ordinary Theory
研究了普通阿贝尔簇的Serre-Tate理论及其变形空间的推广.这种推广涉及具有附加结构的阿贝尔簇。附加结构不仅可以是半单代数和极化的作用，而且更一般地是由某些"结晶霍奇循环“（在动机意义上霍奇循环的$p$ -adic版本）给出的数据。与Serre-Tate普通理论相比，在这种广义背景下出现了新的现象。我们给出了应用这一理论的存在“好”的积分模型的那些志村品种的伴随产品的简单，伴随志村品种的$D_l^{\bf H}$型与$l\ge 4$。
We prove a finiteness theorem for the first flat cohomology group of finite flat group schemes over integral normal proper varieties over finite fields. As a consequence, we can prove the invariance of the finiteness of the Tate-Shafarevich group of Abelian schemes over higher dimensional bases under isogenies and alterations over/of such bases for the $p$-part. Along the way, we generalize previous results on the Tate-Shafarevich group in this situation.
We study abelian varieties and K3 surfaces with complex multiplication defined over number fields of fixed degree. We show that these varieties fall into finitely many isomorphism classes over an algebraic closure of the field of rational numbers. As an application we confirm finiteness conjectures of Shafarevich and Coleman in the CM case. In addition we prove the uniform boundedness of the Galois invariant subgroup of the geometric Brauer group for forms of a smooth projective variety satisfying the integral Mumford--Tate conjecture. When applied to K3 surfaces, this affirms a conjecture of …
On the finiteness of twists of irreducible symplectic varieties
本文利用不可约辛簇的锥猜想，证明了在特征0的固定有限域扩张下不可约辛簇扭曲的有限性，并讨论了非闭域锥猜想、导出等价扭曲有限性及正特征情形。
Reduction of a problem of finiteness of Tate-Shafarevich group to a result of Zagier type
Kolyvagin证明了Q上解析秩为0或1的椭圆曲线的Tate-Shafarevich群是有限的，并且它的代数秩等于它的解析秩。一个程序的推广这一结果的情况下，一些动机的上同调动机的志村品种提供。本文主要针对Siegel的$H^7$的等价性，对程序的一些步骤进行了六重证明。例如，我们“几乎”找到Kolyvagin的迹和还原关系的类似物。本文的一些步骤是新的贡献，因为它们在椭圆曲线的情况下没有类似物。该方案仍存在一些重大差距。这些差距中最重要的是一个高维模拟的结果Zagier约比率Heegner点对应于不同的虚二次域上一个固定的椭圆曲线。作者建议读者继续进行这些调查。
In this mostly expository note, we explain a proof of Tate's two conjectures [Tat65] for algebraic cycles of arbitrary codimension on certain products of elliptic curves and abelian surfaces over number fields.
Finiteness of Hadamard ranks
本文研究了投影代数簇的Hadamard秩有限性，分类了所有对任意点具有有限Hadamard秩的投影代数簇，并证明了某些代数簇的最大Hadamard秩的上界。
Beilinson-Tate cycles on semiabelian varieties
Beilinson沿着Hodge和Tate定理的思路，证明了在q次上同调中，光滑复簇的所有权为2 q的Hodge圈和光滑簇的所有权为2 q的Tate圈都来自高Chow群。对于曲线和半交换簇的乘积，Beilinson-Hodge猜想在以前的文章中被证明。在这里，Beilinson-Hodge和Beilinson-Tate定理都被证明对于由曲线的乘积支配的品种是正确的。我们还表明，较低的重量霍奇循环（分别。泰特循环）在这些情况下是代数的。
The Tate Conjecture for even dimensional Gushel-Mukai varieties in characteristic $p\geq 5$
本文研究了特征 $p$ 下维数为 4 和 6 的 Gushel-Mukai 簇，通过借鉴 K3 曲面的证明方法、利用广义伙伴概念及特征 0 下的 Chow 动机同构，证明了特征 $p\geq 5$ 下的 Tate 猜想。
Some cases of the Mumford--Tate conjecture and Shimura varieties
我们证明了芒福德-泰特猜想的那些阿贝尔簇在数域的扩展C附加伴随志村品种的产品简单，伴随志村品种的某些志村类型。特别是，我们证明了正交情况下的猜想（即，对于$B_n$和$D_n^R$ Shimura类型）。作为一个主要工具，我们构造了指定交换型的Shimura簇（其伴随簇为）到PEL型的酉Shimura簇的嵌入。这些构造隐含地分类了PEL类型的Shimura簇的伴随。
Let $X$ be a smooth projective variety over a finitely generated field $K$ of characteristic $p>0$. We proved that the finiteness of the $\ell$-primary part of $\mathrm{Br}(X_{K^s})^{G_K}$ for a single prime $\ell\neq p$ will imply the finiteness of the prime-to-$p$ part of $\mathrm{Br}(X_{K^s})^{G_K}$, generalizing a theorem of Tate and Lichtenbaum for varieties over finite fields. For an abelian variety $A$ over $K$, we proved a similar result for the Tate-Shafarevich group of $A$, generalizing a theorem of Schneider for abelian varieties over global function fields.
Galois representations, Mumford-Tate groups and good reduction of abelian varieties
设K是数域，A是K上的阿贝尔簇.我们感兴趣的是Morita的猜想：如果A的Mumford-Tate群不包含幂幺Q-有理点，则A在K的任意离散位置都有潜在的好约化. Mumford-Tate群是一个分析性质的对象，而具有好约化是一个算术概念，与伽罗瓦表示的分支有关。这个猜想已被证明由森田特定的阿贝尔簇与许多自同态（称为PEL型）。努特获得的结果阿贝尔簇没有非平凡的自同态（芒福德的例子，而不是PEL型）。我们给出了非PEL型的阿贝尔簇的新结果。
Size of isogeny classes of abelian varieties of Lubin-Tate type
研究卢宾 - 泰特型阿贝尔簇同构类大小，证明了有限域上简单阿贝尔簇同构类大小的下界，并基于牛顿层预期大小猜想该下界精确。
In the moduli space of polarized varieties the same unpolarized variety can occur multiple times However, for K3 surfaces, compact hyperkähler manifolds, and abelian varieties the number is finite. This may be viewed as a consequence of the Kawamata-Morrison cone conjecture. In this note we provide a proof of this finiteness not relying on the cone conjecture and, in fact, not even on the global Torelli theorem. Instead, it uses the geometry of the moduli space of polarized varieties to conclude the finiteness by means of Baily-Borel type arguments. We also address related questions concerning…
The Tate conjecture for abelian fourfolds over finite fields
该研究证明了有限域上阿贝尔四维簇的Tate猜想，结合相关定理推导出任意域上阿贝尔四维簇的同调与数值等价标准猜想成立，完成了阿贝尔四维簇标准猜想的证明。
Semisimplicity, purity and Mumford-Tate conjecture for hyper-Kähler varieties
该文证明了超Kähler簇的Mumford-Tate猜想，通过消除幂幺根确立半单性，并证明权重-单演猜想及Weil-Deligne表示的强相容性。
We study the image of $\ell$-adic representations attached to subvarieties of Shimura varieties $Sh_K(G,X)$ that are not contained in a smaller Shimura subvariety and have no isotrivial components. We show that, for $\ell$ large enough (depending on the Shimura datum $(G,X)$ and the subvariety), such image contains the $\mathbb{Z}_\ell$-points coming from the simply connected cover of the derived subgroup of $G$. This can be regarded as a geometric version of the integral $\ell$-adic Mumford-Tate conjecture.
Note on the counterexamples for the integral Tate conjecture over finite fields
本文讨论了有限域上簇的非挠和非代数上同调类的一些例子。该方法遵循Atiyah-Hirzebruch和Totaro的建设。
We investigate to what extent an abelian variety over a finite field can be lifted to one in characteristic zero with small Mumford-Tate group. We prove that supersingular abelian surfaces, respectively threefolds, can be lifted to ones isogenous to a square, respectively product, of elliptic curves. On the other hand, we show that supersingular abelian threefolds cannot be lifted to one isogenous to the cube of an elliptic curve over the Witt vectors.
Nondegeneracy and Sato-Tate Distributions of Two Families of Jacobian Varieties
本文研究两个雅可比变种的非退化性及萨托-泰特分布，通过确定组件群生成元来描述其萨托-泰特群和分布，并计算与广义萨托-泰特猜想相关的动量统计。
Finiteness of the Tate-Shafarevich group over function fields for groups of multiplicative type
研究函数域上乘法型群的泰特 - 沙法列维奇群的有限性，通过对\(k\)的条件设定（有限生成且\(X(k)\neq\emptyset\)或\(k\)是数域），证明相应泰特 - 沙法列维奇群有限，补充了\(X\)是曲线情形的相关工作。
Uniform bounds for obstructions to the integral Tate conjecture
本文在自然变分实现猜想假设下，研究无限有限生成域上代数簇一维族中的整 Tate 猜想障碍，并建立其一致界。
De Rham logarithmic classes and Tate conjecture
本文定义了德雷哈姆对数类，证明了代数簇的德雷哈姆类为对数类，并推导出有限型数域上光滑射影簇的塔特猜想。
We investigate boundedness results for families of holomorphic symplectic varieties up to birational equivalence. We prove the analogue of Zarhin's trick by for $K3$ surfaces by constructing big line bundles of low degree on certain moduli spaces of stable sheaves, and proving birational versions of Matsusaka's big theorem for holomorphic symplectic varieties. As a consequence of these results, we give a new geometric proof of the Tate conjecture for $K3$ surfaces over finite fields of characteristic at least $5$, and a simple proof of the Tate conjecture for $K3$ surfaces with Picard number a…
The Tate conjecture for squares of K3 surfaces over finite fields was recently proved by Ito-Ito-Koshikawa. We give a more geometric proof when the characteristic is at least 5. The main idea is to use twisted derived equivalences between K3 surfaces to link the Tate conjecture to finiteness results over finite fields, in the spirit of Tate.
Monodromy rank and the semisimple Mumford-Tate conjecture for hyper-Kähler varieties
本文证明了超Kähler流形在特定条件下的Mumford-Tate猜想，并展示了其在变形下的不变性，同时推导出Hodge猜想与Tate猜想的关系。
The Tate conjecture for divisors on varieties over number fields is equivalent to finiteness of $\ell$-primary torsion in the Brauer group. We show that this finiteness is actually uniform in one-dimensional families for varieties that satisfy the Tate conjecture for divisors -- e.g. abelian varieties and $K3$ surfaces.
Surjectivity of $p$-adic regulator on $K_2$ of Tate curves
证明了当基域包含在$Q_p$的分圆扩张中时，从Tate曲线的Quillen $K_2$到$p$-adic上同调群的$p$-adic调节子的满射性.这意味着由于Suslin的确切序列Tate曲线的$K_1$的扭转部分的有限性。
We prove an analogue of Horrocks' splitting theorem for Segre-Veronese varieties building upon the theory of Tate resolutions on products of projective spaces.
Let $A$ be abelian variety over the function field $K$ of a compact Riemann surface $B$. Fix a model $f \colon \mathcal{A} \to B$ of $A/K$ and a certain effective horizontal divisor $\DD \subset \mathcal{A}$. We give a sufficient condition on the divisor $\DD$ for the finiteness of the set of $(S, \DD)$-integral sections for every finite subset $S \subset B$. These integral sections $σ$ correspond to rational points in $A(K)$ which satisfy the geometric condition $f ( σ(B) \cap \DD)\subset S$. This notion is the geometric variant of integral solutions of a system of \emph{Diophantine equations…
The special Brauer group and twisted Picard varieties
本文通过引入Grothendieck的特殊Brauer群，将带截面椭圆K3曲面的Tate-Shafarevich群概念推广至赋予线性系统的K3曲面，为处理扭曲层模空间提供了有效方法。
For any split totally degenerate abelian variety over a complete discrete valuation field, we construct a log abelian variety over the discrete valuation ring extending the given abelian variety. This generalizes the log Tate curve of Kato.
Faltings extension and Hodge-Tate filtration for abelian varieties over $p$-adic local fields with imperfect residue fields
针对剩余域不完备的特征0p进局部域，定义Z_p上O_K的Faltings扩张，推广Fontaine1981年的构造，为该类域上的阿贝尔簇构建Hodge-Tate滤过。
We study the formal neighbourhood of a point in $μ$-ordinary locus of an integral model of a Hodge type Shimura variety. We show that this formal neighbourhood has a structure of a shifted cascade. Moreover we show that the CM points on the formal neighbourhood are dense and that the identity section of the shifted cascade corresponds to a lift of the abelian variety which has a characterization in terms of its endomorphisms in analogy with the Serre-Tate canonical lift of an ordinary abelian variety.
Tate classes on self-products of Abelian varieties over finite fields
本文证明对有限域上g维阿贝尔簇存在仅依赖于g的通用常数N，使得若某自乘积含奇异Tate类，则自乘积X^{2N}也含奇异Tate类，从而正面回答了Kiran Kedlaya的问题。
Local-global principle and integral Tate conjecture for certain varieties
本文提出几何判据验证余维数一可分有理连通簇的整Tate猜想及零循环局部-整体原则的唯一Brauer-Manin阻碍，并证明了几何有理曲面与四次del Pezzo曲面的相关结论。
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