Mordell-Weil Lattices and Cycles on Algebraic Surfaces

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。]

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.]