The Mordell-Weil groups of unirational quasi-elliptic surfaces in characteristic 3
The Mordell-Weil groups of unirational quasi-elliptic surfaces in characteristic 3
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特征 3 中无理拟椭圆曲面的 Mordell-Weil 群
DOI：
10.1007/bf02571415
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发表时间：
1992
影响因子：
0.8
通讯作者：
Hiroyuki Ito
中科院分区：
文献类型：
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作者：
Hiroyuki Ito
定义在函数域上的椭圆曲线被看作是由一条代数曲线参数化的代数曲面，该代数曲线的函数域是给定椭圆曲线的基域。曲面和纤维是相关联的椭圆面和椭圆纤维。然后，将椭圆曲线上有理点组成的Mordell-Weil群视为椭圆纤维的有理截面群。众所周知，椭圆曲线的Mordell-Weil群与椭圆曲面的N6ron-Severi群之间有着深刻的联系。是2])。椭圆表面的N6ron-Severi群反映了椭圆纤维的结构，如可约奇异纤维的复杂性、截面的丰富性等。因此，人们可以通过Ndron-Severi群或通过纤维的可还原的奇异纤维来观察Mordell-Weil群。在射影直线上的椭圆纤维的情况下，许多人已经观察到了这一点。(例如，参见[P]、[M-P1]和[M-P2]。)
An elliptic curve defined over a function field is viewed as an algebraic surface with a fibration of elliptic curves parametrized by an algebraic curve whose function field is the base field of the given elliptic curve. The surface and the fibration are the associated elliptic surface and elliptic fibration. Then the Mordell-Weil group consisting of rational points of the elliptic curve is considered as the group of rational sections of the elliptic fibration. It is known that there are deep relations between the Mordell-Weil group of the elliptic curve and the N6ron-Severi group of the elliptic surface (cf. IS 2]). The N6ron-Severi group of the elliptic surface reflects the structure of the elliptic fibration, for example, the complexity of reducible singular fibres, and the abundance of sections. Thus one can hope to observe the Mordell-Weil group via the Ndron-Severi group, or via reducible singular fibres of the fibration. In the case of elliptic fibration over the projective line, this observation has been done by many people.(See, for example,[P],[M-P1] and [M-P2].)