Shafarevich Correspondence and Mordell-Weil Lattices

沙法列维奇对应和 Mordell-Weil 格子

• 批准号: 15540048
• 负责人: SHIODA Tetsuji
• 金额: $ 2.3万
• 依托单位: RIKKYO UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2003
• 资助国家: 日本
• 起止时间: 2003 至 2004
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540048/
• 关键词: Shafarevich Correspondence; Mordell-Weil Lattices; Elliptic Surface; Singular fibre; DS-triples; abc-theorem; Tate-Shafarevich group; Hodge Conjecture; 整数点; DS-トリプル; Davenport-Stothers3対; 整点; 極大な特異ファイバー

The idea of Shafarevich correspondence makes it possible to relate the elliptic curves having given discriminant with the integral points of the partner elliptic curve. Combined with the theory of Mordell-Weil lattices, this idea enables the principal investigator to obtain the following results :(1)Determination of K3 surface with a maximal singular fibre(2)Application of Davenport-Stothers triples to elliptic surfaces(3)Abc-theorem and the number of singular fibres of an elliptic surface over P^1On the other hand, the investigator Aoki have studied the arithmetic of elliptic curves and abelian varieties over a number field, as well as the Hodge conjecture. More specifi cally, he has treated the following subjects :(4)Tate-Shafarevich groups of semistable elliptic curves with a rational 3-torsion(5)Hodge conjecture for the jacobian varieties of generalized Catalan curves,(6)On the generalized Gross-Tate conjecture for elementary abelian 2-extensions

Shafarevich对应的思想使给出判别式的椭圆曲线与配对椭圆曲线的积分点相联系成为可能。结合Mordell-Weil格理论，这一思想使主要研究者得到以下结果：(1)确定具有最大奇异纤维的K_3曲面(2)Davenport-Stothers三元组在椭圆曲面上的应用(3)ABC-定理和P^1上椭圆曲面的奇异纤维数目。另一方面，研究者青木研究了数域上椭圆曲线和阿贝尔变差的算法以及Hodge猜想。更具体地说，他研究了下列主题：(4)具有有理3-挠的半稳定椭圆曲线的Tate-Shafarevich群；(5)广义Catalan曲线的Jacobian变种的Hodge猜想；(6)初等阿贝尔2-扩张的广义Gross-Tate猜想

项目成果

On Tate's refinement for a conjecture of Gross and its generalization

论泰特对格罗斯猜想的细化及其推广

• 发表时间: 2005
• 期刊: J.Theorie des Nombres de Bordeaux. (To appear)
• 作者: A.Ivic;Y.Motohashi;青木昇(Noboru Aoki)
• 通讯作者: 青木昇(Noboru Aoki)

On the Tate-Shafarevich groups of semistable elliptic curves with a rational 3-torsion

具有有理三阶扭转的半稳定椭圆曲线的 Tate-Shafarevich 群

• 发表时间: 2004
• 期刊: Acta Arithmetica 112
• 作者: M.Jutila;Y.Motohashi;青木 昇(Noboru Aoki)
• 通讯作者: 青木 昇(Noboru Aoki)

A finiteness theorem on pure Gauss sums

纯高斯和的有限定理

• 发表时间: 2004
• 期刊: Comment.Math.Univ.Sancti Pauli 53
• 作者: R.W.Bruggeman;Y.Motohashi;青木 昇(Noboru Aoki)
• 通讯作者: 青木 昇(Noboru Aoki)

T.Shioda: "The elliptic K3 surfaces with a maximal fibre."C.R.Acad.Sci.Paris, Ser.I. 337. 461-466 (2003)

T.Shioda：“椭圆 K3 表面具有最大纤维。”C.R.Acad.Sci.Paris，Ser.I。

Elliptic Surfaces and Davenport-Stothers Triples

• 发表时间: 2005-06
• 作者: Shioda Tetsuji