Mordell-Weil Lattices and Cycles on Algebraic Surfaces

代数曲面上的 Mordell-Weil 格子和圈

• 批准号: 17540044
• 负责人: SHIODA Tetsuji
• 金额: $ 2.35万
• 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: 2005
• 资助国家: 日本
• 起止时间: 2005 至 2007
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-17540044/
• 关键词: Mordell-Weil Lattices; K3 surfaces; Rational elliptic surfaces; Kummer surfaces; Sphere packing problem; Gauss sums; Painleve equations; ソリトン方程式; 楕円パラメータ; サイクル; コレスポンデンス; 特異ファイバー; 整点

The following results have been accomplished on Algebraic cycles on K3 surfaces which forms the main theme of this research proposal :(i) We have determined, for an example of a K3 surface attaining the maximal Mordell-Weil rank 18, the structure of its Mordell-Weil lattice, explicit generators of rational points, together with the splitting field.[2](ii) More generally, we have clarified the structure of the transcendental lattice, Neron-Severi lattice and the Mordell-Weil lattice, of the singular K3 surfaces which arise via the cyclic base change of degree up to six from Inose's elliptic K3 surface. Ref.[3] and in preparation, and (3).(iii) Classical Kummer surfaces attached to genus two Jacobian varieties are studied from the viewpoint of Mordell-Weil lattices, which establishes the close relationship of automorphisms of a genus two curve with the singular fibres and generators of the Mordell-Weil lattice in question. Ref.[4].(iv) In a joint work with M.Kuwata, we have determined the elliptic parameters and the defining equations of elliptic fibrations on the Kummer surface of the product of two non-isogenous elliptic crves., which has been classified by Oguiso by geometric method. Ref.[6].[The rest is omitted in this English summary because of the space limitation.]

以下结果已完成的代数圈上的K3曲面，形成本研究建议的主题：（i）我们已经确定，为一个例子的K3曲面达到最大Mordell-Weil秩18，其Mordell-Weil格的结构，显式生成的合理点，连同分裂领域。[2]（ii）更一般地说，我们已经澄清了超越格，Neron-Severi格和Mordell-Weil格的结构，奇异K3曲面通过Inose椭圆K3曲面的循环基变化高达6次而产生。参考文献[3]准备中，（3）。(iii)从Mordell-Weil格的角度研究了亏格为2的Jacobi簇上的经典库默曲面，建立了亏格为2的曲线的自同构与所讨论的Mordell-Weil格的奇异纤维和生成元的密切关系.参考文献[4]的文件。(iv)在与M.Kuwata的合作工作中，我们确定了两个非同构椭圆曲线乘积的库默曲面上椭圆纤维化的椭圆参数和定义方程。Oguiso用几何方法对它进行了分类。参考文献[6]的文件。[The由于篇幅限制，本英文摘要中省略了其余部分。]

项目成果

超幾何関数で表される不変量を持つ差分方程式

具有由超几何函数表示的不变量的差分方程

• 发表时间: 2007
• 期刊: 日本応用数理学会論文誌 17巻4号
• 作者: S. Kakei;M. Nishizawa;Y. Saito;Y. Takeyama;榊武史,筧三郎
• 通讯作者: 榊武史,筧三郎

Similarity solutions of NLS-SDYM hierarchy and matrix integrals

NLS-SDYM层次与矩阵积分的相似解

• 发表时间: 2007
• 作者: 筧三郎;J.J.C.Nimmo;R.Willox;筧三郎,菊地哲也;筧 三郎;菊地 哲也;菊地 哲也;菊地 哲也;菊地 哲也;斉藤義久;筧 三郎;菊地哲也;菊地 哲也;筧三郎
• 通讯作者: 筧三郎

An interesting elliptic surface over an elliptic curve

椭圆曲线上有趣的椭圆曲面

• 发表时间: 2007
• 期刊: Proc. Japan Acad 83A
• 作者: Tetsuji Shioda;M.Schuett
• 通讯作者: M.Schuett

Correspondence of elliptic curves and Mordell-Weil lattices of certain elliptic K3 surfaces

椭圆曲线与某些椭圆 K3 曲面的 Mordell-Weil 格子的对应关系

• 期刊: Algebraic Cycles and Motives, Cambridge Univ.Press (to appear).
• 作者: S Kakei;T Kikuchi;SHIODA.Tetsuji
• 通讯作者: SHIODA.Tetsuji

[3]Correspondence of elliptic curves and Mordell-Weil lattices of certain elliptic K3 surfaces

[3]椭圆曲线与某些椭圆K3曲面的Mordell-Weil格子的对应关系

• 发表时间: 2007
• 期刊: Algebraic Cycles and Motives(vol.2), Cambridge Univ.Press 2
• 作者: Masata Kuwata;Tetsuji Shioda;兼田 正治;浅芝秀人;Tetsuji Shioda;浅芝秀人;Tetsuji Shioda
• 通讯作者: Tetsuji Shioda