Shafarevich Correspondence and Mordell-Weil Lattices

Shafarevich对应的思想使具有给定判别式的椭圆曲线与伴椭圆曲线的积分点联系起来成为可能。结合Mordell-Weil格理论，这一思想使主要研究者获得了以下结果：(1)用极大奇异纤维确定K3曲面(2)Davenport-Stothers三重组在椭圆曲面上的应用(3)P^1上椭圆曲面的abc定理和奇异纤维数。另一方面，研究者Aoki研究了数域上椭圆曲线和阿贝尔变的算法，以及Hodge猜想。更具体地说，他处理了以下主题：(4)具有有理3-扭转的半稳定椭圆曲线的Tate-Shafarevich群(5)广义Catalan曲线的jacobian变体的Hodge猜想，(6)初等abelian 2-扩展的广义general - tate猜想

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