Algebraic families of nonzero elements of Shafarevich-Tate groups
Jean-Louis Colliot-Thelene, Bjorn Poonen
我们表明，存在非平凡的家庭的代数簇，所有的纤维以上的有理点（或甚至以上的点的奇数度）是torsors的阿贝尔簇代表非零元素的Shafarevich-Tate群。
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.
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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。
Tate-Shafarevich groups and K3 surfaces
下面（和详细说明）的方法洛根和范Luijk，我们展示了明确的亏格-2曲线的雅可比矩阵有非平凡的2-挠在他们的泰特-Shafarevich群。
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.
On the Tate-Shafarevich group of Abelian schemes over higher dimensional bases over finite fields
我们研究了有限域上高维基上处处好约化化的Abel格式的Tate-Shafarevich群的类似物。
Brauer groups and Tate-Shafarevich groups, II
这篇论文已被撤回，任何版本都不会发表。
On the second Tate-Shafarevich group of a 1-motive
我们证明有限性结果泰特-Shafarevich组在2度与1-动机，依靠他们的Leopoldt猜想，并提出了一个例子，一个半阿贝尔品种的无限泰特-Shafarevich组在2度。我们还建立了一个算术对偶定理1-动机数域，补充了Harari和Szamuely在这个方向上的早期结果。
Brauer groups and Tate-Shafarevich groups
设X_K是整体域K上的一条正常的、光滑的、几何连通的曲线.本文推广了Milne关于X_K的Jacobian的Tate-Shafarevich群的阶与X_K的正则模型的Brauer群的阶之间关系的一个公式。因此，我们部分地回答了格罗滕迪克的一个问题。
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…
On Shafarevich-Tate groups and the arithmetic of Fermat Curves
本文推广了第一作者关于素数阶费马曲线商的雅可比行列式的Shafarevich-Tate群中的非平凡元素的结果，并利用第二作者的方法导出了19阶费马曲线上二次点的一个结果。
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.
On the structure of Selmer and Shafarevich-Tate groups of higher weight modular forms
在Heegner点无挠的假设下，Kolyvagin的结果描述了椭圆曲线的Shafarevich-Tate群的结构.本文证明了虚二次域上的（$p$ -准素）Shafarevich-Tate群与高权模形式相关的类似结果，并满足Heegner假设.更确切地说，我们表明，结构的Shafarevich泰特集团的上同调类建立了Nekovásness的Heegner周期Kuga-Sato品种。作为主要定理的一个应用，我们改进了Besser关于这类群的阶的一个界的结果。作为第二个贡献，我们证明了一个结果的结构（$p$ -准）Selmer群的模形式的意义上的Bloch-Kato。
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…
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.
Mod-p reducibility, the torsion subgroup, and the Shafarevich-Tate group
设$E$是素导体$N$的$\Q$上的最优椭圆曲线.我们表明，如果对于一个奇素数$p$，模$p$表示相关联的$E$是可约的（特别是，如果$p$划分的阶的扭转子群的$E(\Q)$），那么$p$ -主要成分的Shafarevich-Tate群的$E$是平凡的。我们还指出了一个相关的结果，更一般的交换子品种的$J_0(N)$，并提到什么期望，如果$N$不是素数。
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…
On Tate-Shafarevich groups of 1-motives
本文证明了限制映射III^{i}（F，M）-> III^{i}（K，M）^{G}（i=1和2）的核和上核的有限性，其中M是整体域F上的（Deligne）1-motive，K/F是具有伽罗瓦群G的整体域的有限伽罗瓦扩张.
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。
Mumford-Tate and Generalised Shafarevich conjectures
在本文中，我们制定了广义Isogeny，Mumford-Tate和Shafarevich拓扑，并证明它们是等价的。
We describe the structure of Tate-Shafarevich groups of a constant elliptic curves over function fields by exploiting the volcano structure of isogeny graphs of elliptic curves over finite fields.
Orders of Tate-Shafarevich groups for the Neumann-Setzer type elliptic curves
本文给出了Neumann-Setzer型椭圆曲线的Tate-Shafarevich群的阶的研究结果。
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。
The Shafarevich-Tate and Selmer groups arise in the context of Kummer theory for elliptic curves. The finiteness of the Shafarevich-Tate group of an elliptic curve $E$ over the field of rational numbers is included in the Birch and Swinnerton-Dyer conjectures, and is still an open question. We present an overview of the Shafarevich-Tate and Selmer groups of an elliptic curve in the framework of Galois cohomology. Known results on the finiteness of the Shafarevich-Tate group are mentioned, including results of Coates and Wiles, Rubin, Gross and Zagier, and Kolyvagin. We then prove the vanishing…
A note on the order of the Tate--Shafarevich group modulo squares
本文研究了阿贝尔簇的Tate--Shafarevich群模有理平方的阶，证明了每个无平方因子的自然数均可作为某个阿贝尔簇Tate--Shafarevich群的非无平方因子部分出现，从而验证了W. Stein的猜想。
A non-abelian conjecture of Tate-Shafarevich type for hyperbolic curves
本文利用系数在$\mathbb{Q}_p$ -幂单基本群中的非交换上同调函子的Selmer格式，给出了一个判别$\mathbb{Z}$上双曲曲线上整体积分点的几何准则.对于秩为0的半稳定椭圆曲线中的$\mathbb{P}^1\setminus \{0,1,\infty\}$和原点的补，我们计算了全局Selmer格式的局部图像，从而使我们能够在广泛的情况下数值证实我们的猜想。
Non-trivial Shafarevich-Tate Groups of Elliptic Curves
我们用Mordell-Weil群刻画了$y^2=x(x-a^2)(x+b^2)$的二次扭曲，并且在一定条件下，Shafarevich-Tate群的$2$ -准素部分同构于$(\mathb Z/2\mathbb Z)^2$.我们还得到了这些椭圆曲线的分布结果。
Franz Lemmermeyer's previous work laid the framework for a description of the arithmetic of Pell conics, which is analogous to that of elliptic curves. He describes a group law on conics and conjectures the existence of an analogous Tate--Shafarevich group with order the squared ideals of the narrow class group. In this article, we provide a cohomological definition of the Tate--Shafarevich group and show that its order is as Lemmermeyer conjectured.
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…
Structure of Tate-Shafarevich groups of elliptic curves over global function fields
确定了整体函数域上一类椭圆曲线的Tate-Shafarevich群的结构.从专著[1]中已知这些是有限的阿贝尔群，因此它们是有限的循环群的直和，其中这些循环分量的阶是Tate-Shafarevich群的不变量。这种分解的泰特-Shafarevich群直接和有限循环群取决于行为的Drinfeld-Heegner点对这些椭圆曲线。这些点类似于有理数上椭圆曲线上的Heegner点。
The Shafarevich-Tate group in cyclotomic Z_p-extensions at supersingular primes
本文研究了Shafarevich-Tate群的p-准素分支在任意奇素数的超超奇约化化下在分圆方向上的渐近增长性，推广了Kobayashi的工作.这就解释了栗原，Perrin-Riou和Nasybullin在修改的Selmer群的岩泽不变量方面获得的公式。
On Tate-Shafarevich groups of some elliptic curves
这是ANT-0166的更新版本。推广Stroeker和Top的结果，我们证明了椭圆曲线$y^2 = (x+k)(x^2+k^2)$的Tate-Shafarevich群的2-秩可以变得任意大.我们还提出了一个猜想的秩Selmer群附加到有理2-等距的椭圆曲线。
Congruent Elliptic Curves with Non-trivial Shafarevich-Tate Groups
研究了一类全等椭圆曲线$E^{(n)}: y^2=x^3-n^2x$，其中$n$是全等于$1\pmod 8$且素因子全等于$1\pmod 4$的正整数.我们刻画了这样的$E^{(n)}$与Mordell-Weil秩零和$2$ -主要部分的Shafarevich-Tate群同构的$\big(\mathbb Z/2\mathbb Z \big)^2$。我们还讨论了一类具有Shafarevich-Tate群的2-准素部分的$E^{(n)}$同构于具有$k\ge2$的$\big(\mathbb Z/2\mathbb Z \big)^{2k}$.
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。
This paper continues the authors previous investigations concerning orders of Tate-Shafarevich groups in quadratic twists of a given elliptic curve, and for the family of the Neumann-Setzer type elliptic curves. Here we present the results of our search for the (analytic) orders of Tate-Shafarevich groups for the cubic twists of $X_0(27)$. Our calculations extend those given by Zagier and Kramarz \cite{ZK} and by Watkins \cite{Wat}. Our main observations concern the asymptotic formula for the frequency of orders of Tate-Shafarevich groups. In the last section we propose a similar asymptotic fo…
The paper formulates a precise relationship between the Tate-Shafarevich group of an elliptic curve $E$ over ${\mathbb Q}$ with a quotient of the classgroup of ${\mathbb Q}(E[p])$ on which $Gal({\mathbb Q}(E[p]/{\mathbb Q}) = GL_2({\mathbb Z}/p)$ operates by its standard 2 dimensional representation over ${\mathbb Z}/p$. We establish such a relationship in most cases.
Let $E$ be an elliptic curve over the rational field $\mathbf{Q}$. Let $K$ be a quadratic extension over $\mathbf{Q}$. Let $\mathrm{ST}(E/K)$ dente the Shafarevich-Tate group of $E$ over $K$. We show that (under mild conditions on $E$) for every $r>0$, there are infinitely many quadratic twists $E^d/\mathbf{Q}$ of $E/\mathbf{Q}$ such that $\mathrm{dim}_{\mathbf{F}_2}(\mathrm{ST}(E^d/K)[2]) > r$
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。
A multinorm one torus associated to a commutative étale algebra $L$ over a global field $k$ is of Kummer type if each factor of $L$ is a cyclic Kummer extension. In this paper we compute the Tate-Shafarevich group of such tori based on recent works of Bayer-Fluckiger, T.-Y. Lee and Parimala, and of T.-Y.~Lee. We also implement an effective algorithm using SAGE which computes the Tate-Shafarevich groups when each factor of $L$ is contained in a fixed concrete bicyclic extension of $k$.
Congruent Elliptic Curves with Non-trivial Shafarevich-Tate Groups: Distribution Part
研究了一类子类全等椭圆曲线$E^{(n)}: y^2=x^3-n^2x$的分布，其中$n$全等于$1\pmod 8$且素因子全等于$1\pmod 4$ .我们证明了剩余符号性质的独立性。因此，我们得到了秩为零的分布，如$E^{(n)}$与$2$ -主要部分的Shafarevich-Tate群同构的$\big(\mathbb Z /2\mathbb Z\big)^2$。我们还得到了秩为零的$E^{(n)}$和同构于$\big(\mathbb Z /2\mathbb Z\big)^{4}$的Shafarevich-Tate群的$2$ -准素部分的个数的一个下界.
Visible 2-torsion in the Tate-Shafarevich group of an elliptic curve
本文证明椭圆曲线的Tate-Shafarevich群中任意两个2-挠元可在同一阿贝尔曲面中可见，推广了先前仅知单个2-挠元的结果，并解释了Cremona和Mazur论文中的一些原始观察。
Constructing non-trivial elements of the Shafarevich-Tate group of an Abelian Variety over a Number Field
Birch and Swinnerton-Dyer（BSD）猜想的第二部分根据曲线的其他可计算不变量给出了椭圆曲线的Shafarevich-Tate群的阶的猜想公式。Cremona和Mazur提出了一个理论，该理论通常可以用来验证BSD猜想，通过构造一条椭圆曲线的Shafarevich-Tate群的非平凡元素，借助于一条环境曲线的Mordell-Weil群。本文推广了Cremona和Mazur的工作，给出了数域上阿贝尔簇的Shafarevich-Tate群可构造的精确条件.然后，我们给出了我们的一般结果的一个扩展，提供了新的理论证据的BSD猜想。
Examples of non-simple abelian surfaces over the rationals with non-square order Tate-Shafarevich group
设A是定数域上的阿贝尔曲面.如果A是主极化的，则已知A的泰特-沙法列维奇群的阶如果是有限的，则必须是平方或平方的两倍。A的情况主要不是两极分化的仍然不清楚。对于{1，2，3，5，6，7，10，13}中的每个k，我们在其Tate-Shafarevich群的阶为k乘平方的有理数上构造一个非简单的非主极化阿贝尔曲面。为了得到这个结果，我们探讨了Birch和Swinnerton-Dyer猜想在等距条件下的不变性。
The p-part of Tate-Shafarevich groups of elliptic curves can be arbitrarily large
本文证明了对任意素数p>5，E/K的Tate-Shafarevich群的p-挠的维数可以任意大，这里E是数域K上的椭圆曲线，[K：Q]的界为仅依赖于p的常数.由此我们推出A/Q的Tate-Shafarevich群的p-挠的维数可以任意大，这里A是阿贝尔簇，dim A的界为仅依赖于p的常数.
Tate-Shafarevich Groups and Frobenius Fields of Reductions of Elliptic Curves
设$\E/\Q$是$\Q$上的固定椭圆曲线，它不具有复乘。在广义黎曼假设下，A. C. Cojocaru和W. Duke等得到了素数$p\le x$个数的渐近公式，使得$\E$模p的约化有平凡的Tate-Shafarevich群. A.最近的研究结果C. Cojocaru和C.大卫导致一个更好的错误项。我们在证明方案中引入了一个新的论点，使证明方案得到了进一步的改进。
Let k be a global field and L be a finite dimensional étale algebra over k. In this paper, we assume that L is a product of cyclic extensions of k. Let T_{L/k} be the multinorm-one torus defined by the multinorm equation: N_{L/k} (t) = 1. Let X_c be the variety defined by the equation N_{L/k} (t) = c, for some c in k*. In this paper, we compute the Tate-Shafarevich group and the algebraic Tate-Shafarevich group of the character group of T_{L/k}. These groups measure the obstruction to the local-global principle for existence of rational points of X_c and the obstruction to the weak appraximati…
Congruences of modular forms and modularity of Tate-Shafarevich classes
本文在适当假设下证明了有理数上椭圆曲线的$p$-挠Tate-Shafarevich类在模曲线Jacobi商群中的可见性，验证了Jetchev-Stein猜想，其核心方法是利用Bertolini-Darmon二部Kolyvagin系统的非平凡性来推导Selmer群模$p$的基。
Visibility of Shafarevich-Tate group of abelian varieties over number field extensions
给定数域K上的一个阿贝尔簇J和J的一个阿贝尔子簇A，研究了在K的一定个数的域扩张M上A关于J的Shafarevich-Tate群的可见元. Mazur在阿贝尔簇的Shafarevich-Tate群中引入了可见元的概念。本文研究了A在K上的Galois上同调到A在M上的Galois上同调的自然限制映射下A关于J的可见元的象。特别地，对于固定的奇素数p，我们研究了在K的二次扩张或p次扩张M上产生p阶可见元的条件。
There are classical congruences between the class number of an imaginary quadratic field and a Bernoulli number or an Euler number. Under the BSD conjecture, Onishi obtained an elliptic generalization of these congruences, which gives congruences between the order of the Tate-Shafarevich group of certain elliptic curves with CM by the Gauss integers ring and Mordell-Weil rank 0, and a coefficient of power series expansion of an elliptic function associated to Gauss integers ring. In this paper, we provide Onishi's type congruences for the Eisenstein integers case.
On the asymptotic growth of Bloch-Kato-Shafarevich-Tate groups of modular forms over cyclotomic extensions
研究了Q的分圆Zp-扩张上的模形式f的Bloch-Kato-Shafarevich-Tate群的渐近行为，并假设f在p处是非平凡的.特别地，我们给出了这些群的上界，它们是用p-adic Hodge理论定义的Selmer群的岩泽不变量.这些界与Kobayashi，Kurihara和Sprung关于超奇椭圆曲线的公式具有相同的形式.
Visualizing elements of order four in the Shafarevich-Tate group of an elliptic curve
设E是定义在数域K上的椭圆曲线。设h是E的Shafarevich-Tate群中的一个4阶元素.我们证明了h在无穷多个阿贝尔曲面中直到同构都是可见的。这就是说，存在无穷多个阿贝尔曲面J，使得E \hookrightarrow J和h位于自然映射H^1（K，E）\rightarrow H^1（K，J）的核中。
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。
Asymptotic formula for Tate-Shafarevich groups of $p$-supersingular elliptic curves over anticyclotomic extensions
本文在广义Heegner假设下，研究了p-超奇异椭圆曲线在反圆塔Z_p-扩张中数域上的Tate-Shafarevich群的p-准素子群，并给出了其渐近公式。
Behaviour of the order of Tate-Shafarevich groups for the quadratic twists of elliptic curves
本文给出了关于椭圆曲线二次扭转的Tate-Shafarevich群的阶的研究结果。给出了一个一般猜想，对$\Bbb Q$上的固定椭圆曲线$E$和正整数$k$，在有限群$E_d(\Bbb Q)$和$|\Sha(E_d(\Bbb Q))| = k^2$的情况下，给出了小于$X$的二次扭曲数$E_d$，$d$的渐近公式.本文继续了作者以前关于Tate-Shafarevich群在曲线$X_0(49)$的二次扭中的阶的研究。在第8节中，我们展示了具有$|\Sha(E)| > 63408^2$的秩为零的椭圆曲线的$88$示例，这是之前已知的任何显式曲线的最大值。我们的记录是一个椭圆曲线$E$与$|\Sha(E)| = 1029212^2$。
An Asymptotic formula for Tate-Shafarevich groups of CM elliptic curves at supersingular primes
本文在Kobayashi符号Selmer群的附加假设下，对具有复乘的椭圆曲线在反圆$\mathbb{Z}_p$-扩张中间层上的Tate-Shafarevich群证明了渐近公式。
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…
We study elliptic curves of the form $x^3+y^3=2p$ and $x^3+y^3=2p^2$ where $p$ is any odd prime satisfying $p\equiv 2\bmod 9$ or $p\equiv 5\bmod 9$. We first show that the $3$-part of the Birch-Swinnerton-Dyer conjecture holds for these curves. Then we relate their $2$-Selmer group to the $2$-rank of the ideal class group of $\mathbb{Q}(\sqrt[3]{p})$ to obtain some examples of elliptic curves with rank one and non-trivial $2$-part of the Tate-Shafarevich group.
Variétés abéliennes sur les corps de fonctions de courbes sur des corps locaux supérieurs
设$k$是高维局部域，$X$是$k$上的光滑射影几何整的曲线.令$K$为$X$的函数字段。我们通过上同调类定义Tate-Shafarevich群的阿贝尔簇局部平凡的$K$的每个完成来自一个封闭点的$X$。证明了$k$上阿贝尔簇的局部对偶定理，以及$K$上交换簇的Tate-Shafarevich群的全局对偶定理. Soient $k$ un corps local supérieur et $X$ une courbe projective lisse géométriquement intègre de corps de functions $K$ .在考虑$K$的完备性由$X$的一个端点给出的上同调类的情况下，对Tate-Shafarevich的一个Abélienne变量群进行了定义。关于$k$上阿贝林种二元性区域理论的建立，也是关于$K$上阿贝林种Tate-Shafarevich群体的二元性全球理论。
Weil-Châtelet divisible elements in Tate-Shafarevich groups II: On a question of Cassels
对于数域k上的阿贝尔簇A，讨论了子群Sha（A/k）的元素在H^1（k，A）中的整除性.对于Q上的椭圆曲线，所得结果是最完备的.
Let $A$ be an abelian variety over a number field $F$ and let $p$ be a prime. Cohen-Lenstra-Delaunay-style heuristics predict that the Tate-Shafarevich group of $A_s$ should contain an element of order $p$ for a positive proportion of quadratic twists $A_s$ of $A$. We give a general method to prove instances of this conjecture by exploiting independent isogenies of $A$. For each prime $p$, there is a large class of elliptic curves for which our method shows that a positive proportion of quadratic twists have nontrivial $p$-torsion in their Tate-Shafarevich groups. In particular, when the modul…
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