Kanetomo Sato
In this paper, we define, for arithmetic schemes with semistable reduction, $p$-adic objects playing the roles of Tate twists in étale topology, and establish their fundamental properties.
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For a number field, we have a Tannaka category of mixed Tate motives at our disposal. We construct p-adic points of the associated Tannaka group by using p-adic Hodge theory. Extensions of two Tate objects yield functions on the Tannaka group, and we show that evaluation at our p-adic points is essentially given by the inverse of the Bloch-Kato exponential map.
We give a new proof of the Tate-Voloch conjecture, in the situation where the ambient variety is a semiabelian variety defined over Qp. Our proof is new in the sense that it avoids any reference to algebraic model theory or p-adic Hodge theory.
In this paper, we define p-adic étale Tate twists for a modulus pair (X,D), where X is a regular semi-stable family and D is an effective Cartier divisor on X which is flat over a base scheme. The main result of this paper is an arithmetic duality of p-adic étale Tate twists for proper modulus pairs (X,D), which holds as a pro-system with respect to the multiplicities of the irreducible components of D.
$p$-adic tame Tate twists
本研究在温顺拓扑中定义$p$-进温顺泰特扭转，构建类似Beilinson-Lichtenbaum猜想的框架，证明正特征曲线上温顺对数deRham-Witt层的Gersten猜想，指出任意维数情形可由严格$\n\nmathbb{A}^1$-不变性推出。
I prove the surjectivity of the $p$-adic regulator from Quillen's $K_2$ of Tate curve to the $p$-adic etale cohomology group when the base field is contained in a cyclotomic extension of $Q_p$. This implies the finiteness of torsion part of $K_1$ of Tate curves thanks to Suslin's exact sequence.
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 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…
In this paper, we construct Chern class maps and cycle class maps with values in p-adic étale Tate twists [S2]. We also relate the p-adic étale Tate twists with the finite part of Bloch-Kato. As an application, we prove that the integral part of p-adic regulator maps has values in the finite part of Galois cohomology under certain assumptions.
The Gersten conjecture for $p$-adic étale Tate twists and the $p$-adic cycle class map
本文证明了混合特征光滑概形上 $p$-adic étale Tate 扭转的 Gersten 猜想，并推广了循环类映射至任意有限剩余域。
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.
Faltings initiated in 2005 a p-adic analogue of the (complex) Simpson correspondence whose construction has been taken up by various authors, according to several approaches. Following the one we initiated previously, we present an overview of a new monograph developing new features of the p-adic Simpson correspondence, inspired by our construction of the relative Hodge-Tate spectral sequence. First, we address the connection to Hodge-Tate local systems. Second, we establish the functoriality of the p-adic Simpson correspondence by proper direct image. Along the way, we expand the scope of our…
For a certain class of vector bundles E on abelian varieties A over local fields containing all line bundles algebraically equivalent to zero we define a canonical representation of the Tate module of A on the fibre of E in the zero section. This extends an old construction of Tate for line bundles to vector bundles of higher rank. We also compare this construction to the theory of parallel transport for vector bundles on p-adic curves developed in mathAG/0403516. Relations with the Hodge-Tate decomposition are also explained.
Sen attached to each p-adic Galois representation of a p-adic field a multiset of numbers called generalized Hodge-Tate weights. In this paper, we discuss a rigidity of these numbers in a geometric family. More precisely, we consider a p-adic local system on a rigid analytic variety over a p-adic field and show that the multiset of generalized Hodge-Tate weights of the local system is constant. The proof uses the p-adic Riemann-Hilbert correspondence by Liu and Zhu, a Sen-Fontaine decompletion theory in the relative setting, and the theory of formal connections. We also discuss basic propertie…
In this article, we consider the representations of the general linear group over a non-archimedean local field obtained from the vanishing cycle cohomology of the Lubin-Tate tower. We give an easy and direct proof of the fact that no supercuspidal representation appears as a subquotient of such representations unless they are obtained from the cohomology of the middle degree. Our proof is purely local and does not require Shimura varieties.
We prove a finiteness result for the p-adic cohomology of the Lubin-Tate tower. For any n>=1 and p-adic field F, this provides a canonical functor from admissible p-adic representations of GL_n(F) towards admissible p-adic representations of Gal_F x D^*, where Gal_F is the absolute Galois group of F, and D/F is the central division algebra of invariant 1/n. Moreover, we verify a local-global-compatibility statement for this correspondence, and compatibility with the patching construction of Caraiani-Emerton-Gee-Geraghty-Paskunas-Shin.
On André periods of mixed Tate motives
本文展示混合塔克动机中安德烈周期退化为经典概念，并通过连接安德烈周期与科利安积分，具体实现p进多项式对数的特殊值为安德烈周期。
The primary goal of this paper is to identify syntomic complexes with the $p$-adic étale Tate twists of Geisser--Schneider--Sato on regular $p$-torsionfree schemes. Our methods apply naturally to a broader class of schemes that we call "$F$-smooth". The $F$-smoothness of regular schemes leads to new results on the absolute prismatic cohomology of regular schemes.
We revisit the Harder-Narasimhan stratification on a minuscule $p$-adic flag variety, by the theory of modifications of $G$-bundles on the Fargues-Fontaine curve. We compare the Harder-Narasimhan strata with the Newton strata introduced by Caraiani-Scholze. As a consequence, we get further equivalent conditions in terms of $p$-adic Hodge-Tate period domains for fully Hodge-Newton decomposable pairs. Moreover, we generalize these results to arbitrary cocharacters case by considering the associated $B_{dR}^+$-affine Schubert varieties. Applying Hodge-Tate period maps, our constructions give appl…
This article is the first one of a series aiming to construct an isomorphism between the p-adic Lubin-Tate and Drinfeld towers, describe this isomorphism and give applications. We construct a p-adic equivariant integral model of the Lubin-Tate tower with infinite level. This formal scheme will be later compared to another one associated to the Drinfeld tower.
We explore Tate-type conjectures over $p$-adic fields. We study a conjecture of Raskind that predicts the surjectivity of $$ ({\rm NS}(X_{\bar{K}}) \otimes_{\mathbb{Z}}\mathbb{Q}_p)^{G_K} \longrightarrow H^2_{\rm et}(X_{\bar{K}},\mathbb{Q}_p(1))^{G_K} $$ if $X$ is smooth and projective over a $p$-adic field $K$ and has totally degenerate reduction. Sometimes, this is related to $p$-adic uniformisation. For abelian varieties, Raskind's conjecture is equivalent to the question whether $$ {\rm Hom}(A,B)\otimes{\mathbb{Q}}_p \,\to\, {\rm Hom}_{G_K}(V_p(A),V_p(B)) $$ is surjective if $A$ and $B$ ar…
The Hodge-Tate spectral sequence for a proper smooth variety over a p-adic field provides a framework for us to revisit Faltings' approach to p-adic Hodge theory and to fill in many details. The spectral sequence is obtained from the Cartan-Leray spectral sequence for the canonical projection from the Faltings topos to the étale topos of an integral model of the variety. Its abutment is computed by Faltings' main comparison theorem from which derive all comparison theorems between p-adic étale cohomology and other p-adic cohomologies, and its initial term is related to the sheaf of differentia…
For a smooth projective geometrically uniruled threefold defined over a perfect field we show that there exists a canonical abelian variety over the field, namely the second algebraic representative, whose rational Tate modules model canonically the third l-adic cohomology groups of the variety for all primes l. In addition, there exists a rational correspondence inducing these identifications. In the case of a geometrically rationally chain connected variety, one obtains canonical identifications between the integral Tate modules of the second algebraic representative and the third l-adic coh…
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。
In this paper we investigate the image of the $l$-adic representation attached to the Tate module of an abelian variety over a number field with endomorphism algebra of type I or II in the Albert classification. We compute the image explicitly and verify the classical conjectures of Mumford-Tate, Hodge, Lang and Tate, for a large family of abelian varieties of type I and II. In addition, for this family, we prove an analogue of the open image theorem of Serre.
Hodge Theory of $p$-adic analytic varieties: a survey
本文综述了$p$-adic analytic varieties的Hodge理论，重点讨论非proper情况下的新现象及新对象，通过Scholze的完美化方法推动了相关研究。
The Tate conjecture has two parts: i) Tate classes are linear combination of algebraic classes, ii) semisimplicity of Galois representations (for smooth projective varieties). B. Moonen proved that i) implies ii) in characteristic 0, using $p$-adic Hodge theory. We show that an unconditional result lies behind this implication: the {\it observability} of arithmetic monodromy groups of geometric origin (in any characteristic) - which leads to a sharpening of Moonen's result. We also discuss another aspect of the Tate conjecture related to the transcendence of $p$-adic periods.
$p$-typical curves on $p$-adic Tate twists and de Rham-Witt forms
本文研究 de Rham–Witt 形式与 p-adic Tate 扭曲上 p-典型曲线的关系；通过将 Bhatt–Morrow–Scholze 的动机滤子引入 Hesselholt 的拓扑循环同调结果，证明二者自然同构，回答了 Artin–Mazur 的长期问题。
We study invariant functions on the reductions mod p^n of p-divisible groups. The proof of the main result, which applies to one-dimensional groups, combines results of Tate with van der Put's solution of his p-adic Corona problem. For higher dimensional groups a generalization of the p-adic Corona problem would have to be solved.
We show that the $\ell$-adic Tate conjecture for divisors on smooth proper varieties over finitely generated fields of positive characteristic follows from the $\ell$-adic Tate conjecture for divisors on smooth projective surfaces over finite fields. Similar results for cycles of higher co-dimension are given.
We construct a non-sheafy uniform Banach algebra such that a rational localisation of the Berkovich spectrum does not preserve the uniformity. We also construct uniform affinoid rings in the sense of Roland Huber such that rational localisations of the adic spectra do not preserve the uniformity. One of them is an example of a non-sheafy uniform affinoid ring. We introduce the notion of local uniformity instead, and prove that the local uniformity implies the sheaf condition.
Mazur, Tate, and Teitelbaum gave a p-adic analogue of the Birch and Swinnerton-Dyer conjecture for elliptic curves. We provide a generalization of their conjecture in the good ordinary case to higher dimensional modular abelian varieties over the rationals by constructing the p-adic L-function of a modular abelian variety and showing that it satisfies the appropriate interpolation property. This relies on a careful normalization of the p-adic L-function, which we achieve by a comparison of periods. Our generalization agrees with the conjecture of Mazur, Tate, Teitelbaum in dimension 1 and the …
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.
Rational Hodge--Tate prismatic crystals of quasi-l.c.i algebras and non-abelian $p$-adic Hodge theory
本文研究了准理Hodge-塔普里扎晶体与非阿贝尔p进Hodge理论的关系，通过范畴等价性揭示了新的理论结果。
We give an algorithm that takes a smooth hypersurface over a number field and computes a $p$-adic approximation of the obstruction map on the Tate classes of a finite reduction. This gives an upper bound on the "middle Picard number" of the hypersurface. The improvement over existing methods is that our method relies only on a single prime reduction and gives the possibility of cutting down on the dimension of Tate classes by two or more. The obstruction map comes from $p$-adic variational Hodge conjecture and we rely on the recent advancement by Bloch-Esnault-Kerz to interpret our bounds.
We prove that the p-adic local Langlands correspondence for GL_2(Q_p) appears in the etale cohomology of the Lubin-Tate tower at infinity. We use global methods using recent results of Emerton on the local-global compatibility and hence our proof applies to local Galois representations which come via a restriction from global pro-modular Galois representations. We also discuss a folklore conjecture which states that the p-adic local Langlands correspondence appears in the de Rham cohomology of the Lubin-Tate tower (Drinfeld tower). We show that a study of the de Rham cohomology for perfectoid …
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.
We study local-global questions for Galois cohomology over the function field of a curve defined over a p-adic field (a field of cohomological dimension 3). We define Tate-Shafarevich groups of a commutative group scheme via cohomology classes locally trivial at each completion of the base field coming from a closed point of the curve. In the case of a torus we establish a perfect duality between the first Tate-Shafarevich group of the torus and the second Tate-Shafarevich group of the dual torus. As an application, we show that the failure of the local-global principle for rational points on …
In this article, we generalize the Hodge-Tate decomposition of p-adic étale cohomology to non-smooth rigid spaces. Our strategy is to study pro-étale cohomology of rigid spaces introduced by Scholze, using the resolution of singularities and the simplicial method.
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。
Shifted symplectic structures on derived analytic moduli of $\ell$-adic local systems and Galois representations
本文基于dagger代数的dg增强发展非阿基米德导出解析几何，为多种pro-étale层构造导出解析模函子及加权移位辛结构，并借此证明了$\ell$-进Galois表示等导出模栈上移位辛结构的存在性，证明了Minhyong Kim猜想。
Les suites spectrales de Hodge-Tate
本书沿 Faltings 的方法系统阐述 p 进 Hodge 理论中的 p 进比较定理与 Hodge-Tate 谱序列，分别建立绝对和相对版本，其中相对 Hodge-Tate 谱序列及部分证明依赖新结果，填补了相对情形研究的不足。
We prove that for abelian varieties with semistable ordinary reduction the p-adic Mazur-Tate height pairing is induced by the unit root splitting of the Hodge filtration on the first deRham cohomology.
We define and study a certain category of vector bundles on a p-adic curve to which we can associate in a functorial way finite dimensional p-adic representations of the geometric fundamental group. Among other things we investigate two different relations of these constructions with the Hodge-Tate decomposition.
A Tate algebra version of the Jacobian Conjecture
本文研究配备I-adic拓扑交换环的Tate-雅可比猜想，证明其在特定条件下等价于雅可比猜想、在正特征时不成立，并建立复数域雅可比猜想与C_p上多项式映射逆的Tate代数性质的等价关系。
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…
Let $\calO_K$ be a complete discrete valuation ring of mixed characteristic $(0,p)$ with a perfect residue field. In this paper, for a semi-stable $p$-adic formal scheme $\frakX$ over $\calO_K$ with rigid generic fibre $X$ and canonical log structure $\calM_{\frakX} = \calO_{\frakX}\cap\calO_X^{\times}$, we study Hodge--Tate crystals over the absolute logarithmic prismatic site $(\frakX,\calM_{\frakX})_{\Prism}$. As an application, we give an equivalence between the category of rational Hodge--Tate crystals on the absolute logarithmic prismatic site $(\frakX,\calM_{\frakX})_{\Prism}$ and the c…
In this paper, we study an analogue of the Tate conjecture for $K_2$ of U, the complement of split multiplicative fibers in an elliptic surface. A main result is to give an upper bound of the rank of the Galois fixed part of the etale cohomology $H^2(\bar{U},Q_p(2))$. As an application, we give an elliptic K3 surface $X$ over a p-adic field for which the torsion part of the Chow group $CH_0(X)$ of 0-cycles is finite. This would be the first example of a surface $X$ over a p-adic field whose geometric genus is non-zero and for which the torsion part of $CH_0(X)$ is finite.
In this paper, we examine the behavior of ideal-adic separatedness and completeness under certain ring extensions using trace map. Then we prove that adic completeness of a base ring is hereditary to its ring extension under reasonable conditions. We aim to give many results on ascent and descent of certain ring theoretic properties under completion. As an application, we give conceptual details to the proof of the almost purity theorem for Witt-perfect rings by Davis and Kedlaya. Witt-perfect rings have the advantage that one does not need to assume that the rings are complete and separated.
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.
The p-adic distance of special points to subvarieties
本文综述了关于半阿贝尔簇上挠点到子簇的p-adic距离一致上界的猜想及其类似情形下的研究进展。
We prove a mixed characteristic analog of the Beilinson-Lichtenbaum Conjecture for p-adic motivic cohomology. It gives a description, in the stable range, of p-adic motivic cohomology (defined using algebraic cycles) in terms of differential forms. This generalizes a result of Geisser from small Tate twists to all twists and uses as a critical new ingredient the comparison theorem between syntomic complexes and p-adic nearby cycles proved recently in Colmez-Niziol.
Correspondance de Simpson p-adique II : fonctorialité par image directe propre et systèmes locaux de Hodge-Tate
本专著沿既有路径发展 p 进 Simpson 对应，阐明其与 Hodge-Tate 局部系统的联系，证明真顺像下的函子性，并扩展原构造的适用范围。
$p$-adic monodromy and mod $p$ unlikely intersections, I
本文提出特征p下的Mumford-Tate和André-Oort猜想的类比，并通过Ax-Schanuel类型的猜想证明了这两个猜想，同时解决了Chai的Tate线性猜想。
We establish duality results for the cohomology of the Weil group of a $p$-adic field, analogous to, but more general than, results from Galois cohomology. We prove a duality theorem for discrete Weil modules, which implies Tate-Nakayama Duality. We define Weil-smooth cohomology for varieties over local fields, and prove a duality theorem for the cohomology of $\G_m$ on a smooth, proper curve with a rational point. This last theorem is analogous to, and implies, a classical duality theorem for such curves.
Coleman-Gross Heights and $p$-adic Néron Functions on Jacobians of Genus $2$ Curves
本文研究了 genus 2 曲线的 Jacobians 上 $p$-adic Néron 函数的理论，将 Coleman-Gross 高度分解为局部高度函数，并扩展了 Balakrishnan 和 Besser 的工作。
On Shafarevich-Tate groups and analytic ranks in families of modular forms, II. Coleman families
本文研究模形式$p$-进Coleman族中代数秩与解析秩的保持性，证明在有限多个例外素数之外，若原形式的Shafarevich-Tate群有限且秩为$r$，则族中权重同余的经典特化保持有限性及秩，并支持Greenberg猜想。
We analyze the geometry of the tower of Lubin-Tate deformation spaces, which parametrize deformations of a one-dimensional formal module of height h together with level structure. According to the conjecture of Deligne-Carayol, these spaces realize the local Langlands correspondence in their l-adic cohomology. This conjecture is now a theorem, but currently there is no purely local proof. Working in the equal characteristic case, we find a family of affinoids in the Lubin-Tate tower with good reduction equal to a rather curious nonsingular hypersurface, whose equation we present explicitly. Gr…
$p$-adic Periods and Selmer Scheme Images
本文旨在为扩展Chabauty-Kim方法到更一般的超曲线上奠定基础，通过定义更一般的$p$-adic周期映射和非交换Chabauty框架，连接$p$-adic迭代积分与Goncharov的动机迭代积分理论。
In this paper we study the $p$-adic dynamics of prime-to-$p$ Hecke operators on the set of points of modular curves in both cases of good ordinary and supersingular reduction. We pay special attention to the dynamics on the set of CM points. In the case of ordinary reduction we employ the Serre-Tate coordinates, while in the case of supersingular reduction, we use a parameter on the deformation space of the unique formal group of height $2$ over $\overline{\mathbb{F}}_p$, and take advantage of the Gross-Hopkins period map.
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