Cristian D. Gonzalez-Aviles
Let X_K be a proper, smooth and geometrically connected curve over a global field K. In this paper we generalize a formula of Milne relating the order of the Tate-Shafarevich group of the Jacobian of X_K to the order of the Brauer group of a proper regular model of X_K. We thereby partially answer a question of Grothendieck.
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Abelian fiber spaces and their Tate-Shafarevich twists with sections
本文研究如何从阿贝尔纤维空间出发，构造一个具有有理截面的Tate-Shafarevich扭曲。
Following (and elaborating on) a method of Logan and van Luijk, we exhibit explicit genus-2 curves whose Jacobians have nontrivial 2-torsion in their Tate-Shafarevich groups.
For any number field, we prove that there exists an elliptic curve defined over this field such that its Shafarevich-Tate group has a nontrivial 2-torsion subgroup.
We study analogues for the Tate-Shafarevich group for Abelian schemes with everywhere good reduction over higher dimensional bases over finite fields.
We give a formula relating the order of the Brauer group of a surface fibered over a curve over a finite field to the order of the Tate-Shafarevich group of the Jacobian of the generic fiber. The formula implies that the Brauer group of a smooth and proper surface over a finite field is a square if it is finite.
We prove finiteness results for Tate--Shafarevich groups in degree 2 associated with 1--motives, rely them to Leopoldt's conjecture, and present an example of a semiabelian variety with an infinite Tate--Shafarevich group in degree 2. We also establish an arithmetic duality theorem for 1--motives over number fields which complements earlier results of Harari and Szamuely in this direction.
We show that there exist non-trivial families of algebraic varieties for which all the fibers above rational points (or even above points of odd degree) are torsors of abelian varieties representing nonzero elements of their Shafarevich-Tate groups.
Let $\mathcal X$ be a regular variety, flat and proper over a complete regular curve over a finite field, such that the generic fiber $X$ is smooth and geometrically connected. We prove that the Brauer group of $\mathcal X$ is finite if and only Tate's conjecture for divisors on $X$ holds and the Tate-Shafarevich group of the Albanese variety of $X$ is finite, generalizing a theorem of Artin and Grothendieck for surfaces to arbitrary relative dimension. We also give a formula relating the orders of the group under the assumption that they are finite, generalizing the formula given for a surfac…
We extend the results of the first author on nontrivial elements in the Shafarevich-Tate group of the jacobian of a quotient of a Fermat curve of prime degree, and use the methods of the second author to derive a result bounding the quadratic points on the Fermat curve of degree 19.
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.
Under a non-torsion assumption on Heegner points, results of Kolyvagin describe the structure of Shafarevich-Tate groups of elliptic curves. In this paper we prove analogous results for ($p$-primary) Shafarevich-Tate groups associated with higher weight modular forms over imaginary quadratic fields satisfying a "Heegner hypothesis". More precisely, we show that the structure of Shafarevich-Tate groups is controlled by cohomology classes built out of Nekovář's Heegner cycles on Kuga-Sato varieties. As an application of our main theorem, we improve on a result of Besser giving a bound on the ord…
Given a smooth geometrically connected curve $C$ over a field $k$ and a smooth commutative group scheme $G$ of finite type over the function field $K$ of $C$ we study the Tate--Shafarevich groups given by elements of $H^1(K,G)$ locally trivial at completions of $K$ associated with closed points of $C$. When $G$ comes from a $k$-group scheme and $k$ is a number field (or $k$ is a finitely generated field and $C$ has a $k$-point) we prove that the Tate--Shafarevich group is finite, generalizing a result of Saïdi and Tamagawa for abelian varieties. We also give examples of nontrivial Tate--Shafar…
Let $E$ be an optimal elliptic curve over $\Q$ of prime conductor $N$. We show that if for an odd prime $p$, the mod $p$ representation associated to $E$ is reducible (in particular, if $p$ divides the order of the torsion subgroup of $E(\Q)$), then the $p$-primary component of the Shafarevich-Tate group of $E$ is trivial. We also state a related result for more general abelian subvarieties of $J_0(N)$ and mention what to expect if $N$ is not prime.
Let $\mathbb{F}_q$ be a finite field of odd characteristic $p$. We exhibit elliptic curves over the rational function field $K = \mathbb{F}_q(t)$ whose Tate-Shafarevich groups are large. More precisely, we consider certain infinite sequences of explicit elliptic curves $E$, for which we prove that their Tate-Shafarevich group $\mathrm{III}(E)$ is finite and satisfies $|\mathrm{III}(E)| = H(E)^{1+o(1)}$ as $H(E)\to\infty$, where $H(E)$ denotes the exponential differential height of $E$. The elliptic curves in these sequences are pairwise neither isogenous nor geometrically isomorphic. We furthe…
We establish the finiteness of the kernel and cokernel of the restriction map III^{i}(F,M) ---> III^{i}(K,M)^{G} for i=1 and 2, where M is a (Deligne) 1-motive over a global field F and K/F is a finite Galois extension of global fields with Galois group G.
Variation of Shafarevich-Tate groups in Hida families
该研究针对希达族，借助通用科利瓦金系统确定了大沙法列维奇-泰特群的伪同构类，并分析了其经典特殊化的沙法列维奇-泰特群结构的变化。
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.
Higher Fitting ideals and the structure of anticyclotomic Shafarevich-Tate groups
本文研究了反圆环Z_p扩展上椭圆曲线塞尔默和谢夫列夫-塔特群的结构，通过双分欧拉系统描述了更高Fitting理想，并提供了相关的新结果。
Arbitrarily large $p$-torsion in Tate-Shafarevich groups
本文通过构造违反 Hasse 原理的显式 $\mu_p$-覆盖，证明任意素数 $p$ 下，$\mathbb{Q}$ 上绝对单纯 Abel 簇的 Tate-Shafarevich 群可具有任意大的 $p$-挠群。
We state a conjectural criterion for identifying global integral points on a hyperbolic curve over $\mathbb{Z}$ in terms of Selmer schemes inside non-abelian cohomology functors with coefficients in $\mathbb{Q}_p$-unipotent fundamental groups. For $\mathbb{P}^1\setminus \{0,1,\infty\}$ and the complement of the origin in semi-stable elliptic curves of rank 0, we compute the local image of global Selmer schemes, which then allows us to numerically confirm our conjecture in a wide range of cases.
For each prime $p$, we show that there exist geometrically simple abelian varieties $A/\mathbb Q$ with non-trivial $p$-torsion in their Tate-Shafarevich groups. Specifically, for any prime $N\equiv 1 \pmod{p}$, let $A_f$ be an optimal quotient of $J_0(N)$ with a rational point $P$ of order $p$, and let $B = A_f/\langle P \rangle$. Then the number of positive integers $d \leq X$, such that the Tate-Shafarevich group of $\hat B_d$ has non-trivial $p$-torsion, is $\gg X/\log X$, where $\hat B_d$ is the dual of the $d$-th quadratic twist of $B$. We prove this more generally for abelian varieties o…
The Tate-Shafarevich group of a polarised K3 surface
该研究补充了带极化K3曲面的Tate-Shafarevich群的推广结果，证明本原极化下该群与对应线性系一般曲线雅可比簇的可良好超凯勒紧化挠子一一对应，是椭圆K3曲面相关经典结论的类比。
Kolyvagin proved that the Tate-Shafarevich group of an elliptic curve over Q of analytic rank 0 or 1 is finite, and that its algebraic rank is equal to its analytic rank. A program of generalisation of this result to the case of some motives which are quotients of cohomology motives of Shimura varieties is offered. We prove some steps of this program, mainly for quotients of $H^7$ of Siegel sixfolds. For example, we "almost" find analogs of Kolyvagin's trace and reduction relations. Some steps of the present paper are new contribution, because they have no analogs in the case of elliptic curve…
Tate-Shafarevich results for quartic twists in characteristic $2$
本文研究偶特征函数域上的椭圆曲线，通过具4阶自同构的极大曲线的函数域构造$\u211d_2$上超奇异椭圆曲线的四次扭转，给出由相关曲线亏格表示的Mordell-Weil秩公式，以得到任意大秩的椭圆曲线。
Shafarevich-Tate groups of holomorphic Lagrangian fibrations
本文研究了双曲凯勒流形上全纯拉格朗日纤维化的沙法列夫-塔季集团，证明了两种构造方法的等价性，并探讨了其与投影变形和截面存在的关系。
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。
Let $f$ be a newform of weight $2$, square-free level and trivial character, let $A_f$ be the abelian variety attached to $f$ and for every good ordinary prime $p$ for $f$ let $\boldsymbol f^{(p)}$ be the $p$-adic Hida family through $f$. We prove that, for all but finitely many primes $p$ as above, if $A_f$ is an elliptic curve such that $A_f(\mathbb Q)$ has rank $1$ and the $p$-primary part of the Shafarevich-Tate group of $A_f$ over $\mathbb Q$ is finite then all specializations of $\boldsymbol f^{(p)}$ of weight congruent to $2$ modulo $2(p-1)$ and trivial character have finite ($p$-primar…
Let $k$ be a higher-dimensional local field and $X$ be a smooth projective geometrically integral curve over $k$. Let $K$ be the function field of $X$. We define Tate-Shafarevich groups of an abelian variety via cohomology classes locally trivial at each completion of $K$ coming from a closed point of $X$. We prove local duality theorems for abelian varieties over $k$, as well as global duality theorems for Tate-Shafarevich groups of abelian varieties over $K$. Soient $k$ un corps local supérieur et $X$ une courbe projective lisse géométriquement intègre de corps de fonctions $K$. On définit l…
Shafarevich-Tate groups of holomorphic Lagrangian fibrations II
本文研究了双曲凯勒流形的全纯拉格朗日纤维化II，探讨了Shafarevich-Tate扭的性质及其与凯勒流形的关系。
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.
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.
We show that there exist genus one curves of every index over the rational numbers, answering affirmatively a question of Lang and Tate. The proof is "elementary" in the sense that it does not assume the finiteness of any Shafarevich-Tate group. On the other hand, using Kolyvagin's construction of a rational elliptic curve whose Mordell-Weil and Shafarevich-Tate groups are both trivial, we show that there are infinitely many curves of every index over every number field.
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。
We wish to give a short elementary proof of S. Saito's result that the Brauer-Manin obstruction for zero-cycles of degree 1 is the only one for curves, supposing the finiteness of the Tate-Shafarevich-group $\sha^1(A)$ of the Jacobian variety. In fact we show that we only need a conjecturally finite part of the Brauer-group for this obstruction to be the only one. We also comment on the situation in higher dimensions
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 …
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。
Genus two curves with full $\sqrt{3}$-level structure and Tate-Shafarevich groups
本文显式参数化具有完整 $\sqrt{3}$-层结构的亏格 2 曲线参数曲面，并据此构造 Abel 曲面，证明其几乎所有二次扭曲的 3-Selmer/Tate-Shafarevich 非平凡，同时给出平均秩与有理点数统计。
We prove some new results on the arithmetic of abelian varieties over function fields of one variable over finitely generated (infinite) fields. Among other things, we introduce certain new natural objects `discrete Selmer groups' and `discrete Shafarevich-Tate groups', and prove that they are finitely generated $\Bbb Z$-modules. Further, we prove that in the isotrivial case, the discrete Shafarevich-Tate group vanishes and the discrete Selmer group coincides with the Mordell-Weil group. One of the key ingredients to prove these results is a new specialisation theorem à la Néron for first Galo…
Reduced Tate-Shafarevich group
本文证明一般椭圆 Calabi-Yau 三维流形的重构定理，建立基上线性导出等价与一般纤维导出等价的判定，并据此肯定回答 Knapp-Scheidegger-Schimannek 猜想。
Let $X$ be a projective and smooth variety over a field $k$. The goal of this paper is to prove that the cokernel of the canonical map $Br(X)\to Br(X_{k^s})^{G_k}$ has a finite exponent. Both groups are natural invariants arising from consideration of the Tate conjecture of divisors over $X$.
Let $k$ be an algebraically closed field, a finite field or a $p$-adic field. Let $K_0=k((x,y))$ be the field of Laurent series in two variables over $k$. We define Tate-Shafarevich groups of a commutative group scheme over $K_0$ via cohomology classes locally trivial at each completion of $K_0$ coming from a codimension 1 point of $\text{Spec}\; k[[x,y]]$. We establish duality theorems between Tate-Shafarevich groups for finite groups schemes and for tori. We apply these results to the study of the obstruction to the local-global principle for $K_0$-torsors under a connected linear algebraic …
Let $K$ be the function field of a smooth projective curve $X$ over a higher-dimensional local field $k$. We define Tate-Shafarevich groups of a commutative group scheme via cohomology classes locally trivial at each completion of $K$ coming from a closed point of $X$. We establish duality theorems between Tate-Shafarevich groups for finite groups schemes, for tori, for groups of multiplicative type, and even for 2-term complexes of tori. We apply these results to the weak approximation for tori over $K$ and to the study of the obstruction to the local-global principle for $K$-torsors under a …
Topology of projective Tate-Shafarevich twists
研究通过扭转操作改变纤维丛的拓扑性质，证明在特定条件下，扭转后的空间与原空间的理查德·贝乌维-博戈莫洛夫-福基斯晶格是Hodge相似的。
Let $E$ be an optimal elliptic curve over $\Q$ of conductor $N$ having analytic rank one, i.e., such that the $L$-function $L_E(s)$ of $E$ vanishes to order one at $s=1$. Let $K$ be a quadratic imaginary field in which all the primes dividing $N$ split and such that the $L$-function of $E$ over $K$ vanishes to order one at $s=1$. Suppose there is another optimal elliptic curve over $\Q$ of the same conductor $N$ whose Mordell-Weil rank is greater than one and whose associated newform is congruent to the newform associated to $E$ modulo an integer $r$. The theory of visibility then shows that u…
Using the calculus of Steenrod operations in étale cohomology developed in [Feng17], we prove that the analogue of Tate's pairing on higher Brauer groups is alternating on 2-torsion. This improves upon a result of Jahn [Jahn15, Math. Annalen].
Brauer and Neron-Severi groups of surfaces over finite fields
本文给出了不假设Tate猜想的有限域上曲面的Artin-Tate公式版本，建立了Brauer群与Néron-Severi群相关项的等式，并据此给出了依赖于p秩的阿贝尔曲面Brauer群大小的精确估计。
We calculate the Tate-Shafarevich group of an elliptic three-fold $f:X\rightarrow S$ when $X$ and $S$ are regular and $f$ is flat, relating it to the Brauer group of $X$ and $S$. We show that given certain hypotheses on $f$, the Tate-Shafarevich group has the interpretation of isomorphism classes of elliptic curves over the function field of $S$ which have the same jacobian as the generic fibre of $f$, and for which there exists a relatively minimal model which has no multiple fibres. We use this to give examples of elliptic fibrations with isolated multiple fibres, and also to give a new coun…
Let K be a global field, let S be a finite set of primes of K containing the archimedean primes and let A be an abelian variety over K. We generalize the duality theorem established in our paper "On Neron class groups of abelian varieties" by removing the hypothesis in [op.cit.] that the Tate-Shafarevich group of A is finite. We also derive an exact sequence that relates the indicated group associated to the Jacobian variety of a proper, smooth and geometrically connected curve X over K to a certain finite subquotient of the Brauer group of X. The sequence alluded to above may be regarded as a…
We show that the stringy Kähler moduli space of a generic genus one curve of degree $N$, for $N\le 5$, is the $Γ_1(N)$ modular curve $X_1(N)$. This implies a correspondence between the cusps of the modular curves and certain large volume limits in the stringy Kähler moduli spaces of genus one fibered Calabi-Yau manifolds with $N$-sections. Using Higgs transitions in M-theory and F-theory as well as modular properties of the topological string partition function, we identify these large volume limits with elements of the Tate-Shafarevich group of the genus one fibration. Singular elements appea…
The Abstract for this revised version is the following: Let k be a number field, let C be a smooth, projective and geometrically integral k-curve and let p:X --> C be a Severi-Brauer k-fibration of squarefree index. Various authors have studied the cokernel of the natural map CH_{0}(X/C)-->\bigoplus_{v}CH_{0}(X_{v}/C_{v}), where CH_{0}(X/C) is the kernel of p_{*}:CH_{0}(X)-->CH_{0}(C). In this paper I obtain an exact sequence which relates the Tate-Shafarevich group of the kernel of the above map to the Tate-Shafarevich group of the Neron-Severi torus of X. I then obtain conditions under which…
Let X be a smooth variety over a number field k embedded as a degree d subvariety of $\mathbb{P}^n$ and suppose that X is a counterexample to the Hasse principle explained by the Brauer-Manin obstruction. We consider the question of whether the obstruction is given by the d-primary subgroup of the Brauer group, which would have both theoretic and algorithmic implications. We prove that this question has a positive answer in the case of torsors under abelian varieties, Kummer surfaces and (conditional on finiteness of Tate-Shafarevich groups) bielliptic surfaces. For Kummer varieties we show th…
Brauer-Manin obstructions for homogeneous spaces of commutative affine algebraic groups over global fields
本文将数域上环面齐性空间的Brauer-Manin障碍相关结论推广到任意特征整体域上有限型交换仿射群概形的一般情形，还研究了其Cartier对偶的第二类Tate-Shafarevich核变体的有限性，依赖正特征领域的最新理论进展。
Mazur proved that any element xi of order three in the Shafarevich-Tate group of an elliptic curve E over a number field k can be made visible in an abelian surface A in the sense that xi lies in the kernel of the natural homomorphism between the cohomology groups H^1(k,E) -> H^1(k,A). However, the abelian surface in Mazur's construction is almost never a jacobian of a genus 2 curve. In this paper we show that any element of order three in the Shafarevich-Tate group of an elliptic curve over a number field can be visualized in the jacobians of a genus 2 curve. Moreover, we describe how to get …
We show that certain Tate--Shafarevich groups are unramified which enables us to give an obstruction to the Hasse principle for torsors under tori over p-adic function fields.
On the Brauer groups of fibrations
本文将 Artin 和 Grothendieck 关于纤维化曲面的定理推广至任意相对维数，建立了 $\mathrm{Pic}^0_{X/K}$ 的 Tate-Shafarevich 群与纤维化簇及其一般纤维的几何 Brauer 群之间的关系。
Let $k$ be a number field. We give an explicit bound, depending only on $[k:\mathbf{Q}]$ and the discriminant of the Néron--Severi lattice, on the size of the Brauer group of a K3 surface $X/k$ that is geometrically isomorphic to the Kummer surface attached to a product of isogenous CM elliptic curves. As an application, we show that the Brauer--Manin set for such a variety is effectively computable. Conditional on GRH, we can also make the explicit bound depend only on $[k:\mathbf{Q}]$ and remove the condition that the elliptic curves be isogenous. In addition, we show how to obtain a bound, …
We conjecture that if C is a curve of genus >1 over a number field k such that C(k) is empty, then a method of Scharaschkin (equivalent to the Brauer-Manin obstruction in the context of curves) supplies a proof that C(k) is empty. As evidence, we prove a corresponding statement in which C(F_v) is replaced by a random subset of the same size in J(F_v) for each residue field F_v at a place v of good reduction for C, and the orders of Jacobians over finite fields are assumed to be smooth (in the sense of having only small prime divisors) as often as random integers of the same size. If our conjec…
Admettant l'hypothèse de Schinzel et la finitude des groupes de Tate-Shafarevich des courbes elliptiques sur les corps de nombres, toute intersection lisse de deux quadriques dans l'espace projectif de dimension n satisfait au principe de Hasse si n>4. Le même résultat vaut pour n=4, c'est-à-dire pour les surfaces de del Pezzo de degré 4, lorsque le groupe de Brauer est réduit aux constantes et que la surface est suffisamment générale. ----- Assuming Schinzel's hypothesis and the finiteness of Tate-Shafarevich groups of elliptic curves over number fields, smooth intersections of two quadrics i…
Dualité étale à la Poitou-Tate pour les tores sur des variétés définies sur un corps fini
本文研究特征正整体域上光滑概形上环面的 Tate-Shafarevich 群，建立了推广的 Poitou-Tate 对偶，将其从经典环面情形扩展到任意维数有限域上的簇。
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