Cristian D. Gonzalez-Aviles, Ki-Seng Tan
We extend the well-known Cassels-Tate dual exact sequence for abelian varieties A over global fields K in two directions: we treat the p-primary component in the function field case, where p is the characteristic of K, and we dispense with the hypothesis that the Tate-Shafarevich group of A is finite.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We explicitly compute the Cassels-Tate pairing on the 2-Selmer group of an elliptic curve using the Albanese-Albanese definition of the pairing given by Poonen and Stoll. This leads to a new proof that a pairing defined by Cassels on the 2-Selmer groups of elliptic curves agrees with the Cassels-Tate pairing.
Cassels has described a pairing on the 2-Selmer group of an elliptic curve which shares some properties with the Cassels-Tate pairing. In this article, we prove that the two pairings are the same.
Arithmetic BF theory and the Cassels-Tate pairing
本文系统研究了算术BF理论，并将塞尔斯-泰特配对解释为算术BF函数。
We extend the method of Cassels for computing the Cassels-Tate pairing on the 2-Selmer group of an elliptic curve, to the case of 3-Selmer groups. This requires significant modifications to both the local and global parts of the calculation. Our method is practical in sufficiently small examples, and can be used to improve the upper bound for the rank of an elliptic curve obtained by 3-descent.
We establish a generalized Cassels-Tate dual exact sequence for 1-motives over global fields. We thereby extend the main theorem of [4] from abelian varieties to arbitrary 1-motives.
In this paper, we study the Cassels-Tate pairing on Jacobians of genus two curves admitting a special type of isogenies called Richelot isogenies. Let $ϕ: J \rightarrow \widehat{J}$ be a Richelot isogeny between two Jacobians of genus two curves. We give an explicit formula as well as a practical algorithm to compute the Cassels-Tate pairing on $\text{Sel}^{\widehatϕ}(\widehat{J}) \times \text{Sel}^{\widehatϕ}(\widehat{J})$ where $\widehatϕ$ is the dual isogeny of $ϕ$. The formula and algorithm are under the simplifying assumption that all two torsion points on $J$ are defined over $K$. We als…
We use the invariant theory of binary quartics to give a new formula for the Cassels-Tate pairing on the $2$-Selmer group of an elliptic curve. Unlike earlier methods, our formula does not require us to solve any conics. An important role in our construction is played by a certain $K3$ surface defined by a $(2,2,2)$-form.
In this paper, we give an explicit formula as well as a practical algorithm for computing the Cassels-Tate pairing on $\text{Sel}^{2}(J) \times \text{Sel}^{2}(J)$ where $J$ is the Jacobian variety of a genus two curve under the assumption that all points in $J[2]$ are $K$-rational. We also give an explicit formula for the Obstruction map $\text{Ob}: H^1(G_K, J[2]) \rightarrow \text{Br}(K)$ under the same assumption. Finally, we include a worked example demonstrating we can indeed improve the rank bound given by a 2-descent via computing the Cassels-Tate pairing.
We explain a method for computing the Cassels-Tate pairing on the 3-isogeny Selmer groups of an elliptic curve. This improves the upper bound on the rank of the elliptic curve coming from a descent by 3-isogeny, to that coming from a full 3-descent. One ingredient of our work is a new algorithm for solving cubic norm equations, that avoids the need for any S-unit computations. As an application, we show that the elliptic curves with torsion subgroup of order 3 and rank at least 13, found by Eroshkin, have rank exactly 13.
For an abelian variety A over a number field k we discuss the divisibility in H^1(k,A) of elements of the subgroup Sha(A/k). The results are most complete for elliptic curves over Q.
Field change for the Cassels-Tate pairing and applications to class groups
本文定义了全局域扩张下有限 Galois 模的标量限制函子，证明其与 Cassels-Tate 配对相容，并应用于 Cohen-Lenstra 启发式下类群的研究。
Computing the Cassels-Tate pairing on the 2-Selmer group of a genus 2 Jacobian
本研究提出一种计算亏格2曲线雅可比簇2-塞尔默群上Cassels-Tate配对的方法，可改进2-下降法得到的有理点群秩上界，且已用于确定LMFDB中所有亏格2雅可比簇的秩。
The Cassels-Tate pairing for finite Galois modules
本文在有限伽罗瓦模范畴上推广Cassels-Tate配对，证明短正合列诱导自然双线性配对，并应用于主极化阿贝尔簇及Bloch-Kato Selmer群。
Let (A,λ) be a principally polarized abelian variety defined over a global field k, and let \Sha(A) be its Shafarevich-Tate group. Let \Sha(A)_\nd denote the quotient of \Sha(A) by its maximal divisible subgroup. Cassels and Tate constructed a nondegenerate pairing \Sha(A)_\nd \times \Sha(A)_\nd \rightarrow \Q/\Z. If A is an elliptic curve, then by a result of Cassels the pairing is alternating. But in general it is only antisymmetric. Using some new but equivalent definitions of the pairing, we derive general criteria deciding whether it is alternating and whether there exists some alternatin…
We prove a 1966 conjecture of Tate concerning the Artin-Tate pairing on the Brauer group of a surface over a finite field, which is the analogue of the Cassels-Tate pairing. Tate asked if this pairing is always alternating and we find an affirmative answer, which is somewhat surprising in view of the work of Poonen-Stoll on the Cassels-Tate pairing. Our method is based on studying a connection between the Artin-Tate pairing and (generalizations of) Steenrod operations in étale cohomology. Inspired by an analogy to the algebraic topology of manifolds, we develop tools allowing us to calculate t…
Representation Defects and Cassels Pairings for Congruent Number Curves: Rédei Symbols and Governing Fields
该研究针对同余数曲线，建立了Qin归一化表示缺陷奇偶性与Cassels配对Pfaffian的关系，推导了特定条件下的同余数相关同余式并推广了相关结论。
In this paper, we formulate and prove a derived category version of Poitou-Tate duality on Galois cohomology of compact modules (with a continuous Galois action) over a pro-p ring, where p is a prime.
We extend the classical duality results of Poitou and Tate for finite discrete Galois modules over local and global fields (local duality, nine-term exact sequence, etc.) to all affine commutative group schemes of finite type, building on the recent work of Česnavičius extending these results to all finite commutative group schemes. We concentrate mainly on the more difficult function field setting, giving some remarks about the number field case along the way.
We give a generalization of Poitou-Tate duality to schemes of finite type over rings of integers of global fields.
We complete the picture of local and global arithmetic duality theorems for short complexes of finite Galois modules and tori over $p$-adic function fields. In view of the duality theorems, we deduce a $12$-term Poitou--Tate exact sequence which relates global Galois cohomology groups to restricted topological products of local Galois cohomology groups.
Local Tate duality for syntomic cohomology in positive characteristic
本文针对特征p>0的局部域k，证明带系数且基于Dodd的F-规范理论的合成上同调满足局部塔特对偶，还得到其整数环上合成上同调的拉格朗日性质，p>2时可导出相关经典对偶与正交性结果。
The main conjecture of Iwasawa theory is a conjecture on the relation between a Selmer group and a conjectural $p$-adic $L$-function. This conjectural $p$-adic $L$-function is expected to satisfy a conjectural functional equation in a certain sense. In view of the main conjecture and this conjectural functional equation, one would expect to have certain algebraic relationship between the Selmer group attached to a Galois representation and the Selmer group attached to the Tate twist of the dual of the Galois representation. It is precisely a component of this algebraic relationship that this p…
We consider a complex of tori of length 2 defined over a number field k. We establish here some local and global duality theorems for the (étale or Galois) hypercohomology of such a complex. We prove the existence of a Poitou-Tate exact sequence for such a complex, which generalizes the Poitou-Tate exact sequences for finite Galois modules and tori. In particular, we obtain a Poitou-Tate exact sequence for k-groups of multiplicative type. The general results proven here lie at the root of recent results about the defect of strong approximation in connected linear algebraic groups and about som…
The Poitou-Tate sequence relates Galois cohomology with restricted ramification of a finite Galois module $M$ over a global field to that of the dual module under the assumption that $\#M$ is a unit away from the allowed ramification set. We remove the assumption on $\#M$ by proving a generalization that allows arbitrary "ramification sets" that contain the archimedean places. We also prove that restricted products of local cohomologies that appear in the Poitou-Tate sequence may be identified with derived functor cohomology of an adele ring. In our proof of the generalized sequence we adopt t…
One-loop $p$-adic string theory and the Néron local height function
研究p-进弦世界片动作在PGL(2,Q_p)布鲁hat-蒂茨树商下的双描述，揭示其与Tate曲线Néron-Tate局部高度函数的关系。
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 …
We study Tamagawa numbers and other invariants (especially Tate-Shafarevich sets) attached to commutative and pseudo-reductive groups over global function fields. In particular, we prove a simple formula for Tamagawa numbers of commutative groups and pseudo-reductive groups. We also show that the Tamagawa numbers and Tate-Shafarevich sets of such groups are invariant under inner twist, as well as proving a result on the cohomology of such groups which extends part of classical Tate duality from commutative groups to all pseudo-reductive groups. Finally, we apply this last result to show that f…
In this paper we obtain a Poitou-Tate exact sequence for finite and flat group schemes over a global function field. We also extend the duality theorems for 1-motives over number fields obtained by D.Harari and T.Szamuely to the function field case.
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 the cohomology of GL(N) and adjoint Selmer groups
本文在局部-整体相容等条件下将Galatius-Venkatesh定理推广到CM域上GL(N)，构造从导出形变环基本群的可除群到扭对偶伴随动机Selmer群的Hecke等变单射，并推广到高阶同伦群与Hida族。
Zeta function of F-gauges and special values
本文针对有限域上可对偶$F$-gauges，提出并证明Artin–Tate型Zeta函数特殊值猜想的无条件形式，并借助稳定Bockstein特征及栈式$F$-gauge理论使其无条件适用于光滑真簇。
Poitou-Tate duality for the Galois group of an extension of a global field with appropriately restricted ramification can be seen as taking place between the cohomology of a compact or discrete module and the compactly-supported cohomology of its Pontryagin dual. Nekovar proved a derived variant of this in which the module is replaced by a bounded complex of modules over the group ring of R that are finitely generated over R, where R is a complete commutative local Noetherian ring with finite residue field. Here, the Pontryagin dual is replaced by the Grothendieck dual, which is itself a bound…
We prove the Tate duality for higher dimensional local fields of mixed characteristic (0,p), when p is an odd prime, using the theory of higher fields of norms. Assuming that p is not ramified in the basefield, we then use this construction to define the higher Hilbert pairing. In particular, we show that the Hilbert pairing is non-degenerate, and we re-discover the formulae of Brueckner and Vostokov.
Let $ p $ and $ q $ be odd prime numbers with $ q - p = 2, $ the $φ-$Selmer groups, Shafarevich-Tate groups ($ φ- $ and $ 2-$part) and their dual ones as well the Mordell-Weil groups of elliptic curves $ y^{2} = x (x \pm p) (x \pm q) $ over imaginary quadratic number fields of class number one are determined explicitly in many cases.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。