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Arithmetic of Cubic Fields and Elliptic Curves associated to them

三次域和与之相关的椭圆曲线的算术

基本信息

批准号：
14540037
负责人：
MIYAKE Katsuya
金额：
$ 2.24万
依托单位：
Tokyo Metropolitan University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2004
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540037/
关键词：
Cubic Field Mordell Curve Elliptic Curve Cubic Generic Polynomial Mordell-Weil Group Mordell-Weil Rank Elliptic Surface Miranda-Persson's Problem 3次巡回体生成多項式 Hilbert's Th.90 generic polynomial 3次twist Fermat3次曲線

项目摘要

For, an irreducible cubic polynomial P(X):=X^3+aX^2+bX+c over the rational number field Q, define a cubic curve E : w^3=P(u), and let E[Q] be the set of all rational points of E over Q (including the point at infinity; there are three points at infinity, and only one of them is rational over Q). Let ξ be a root of P(X), and define W(ξ):={α=qξ+r|q, r^∈Q, N_<Q(ξ)/Q>(α)=1}. Then we have a natural bijective map from W(ξ) to E[Q]. The subset W(ξ) of the cubic field Q(ξ) is stable under affine transformations of form ξ→sξ+t, s, t^∈Q, s≠0. By using such a transformation the curve is isomorphically mapped to one of two Mordell curves, y^2=x^3-2^43^3A^2, y^2=x^3-B^2(B+3), A, B^∈Q. The former is a short form of the pure cubic twist of the Fermat curve X^3+Y^3+AZ^3=0 as is well know. As for the latter, we showed that the Mordell-Weil rank is positive unless either B=-4 or -8/3. We could also obtain such a subfamily as the ranks of the members are at least 2 with a few exceptions. The subfamily was constructed by using the above presentation of E[Q] by W(ξ).In case where P(X) is a generic cyclic polynomial of degree 3, namely, P(X)=X^3-(s-3)X^2-sX-1, the short form is determined. In this cyclic case, W(ξ) allowed us to construct an elliptic curve by using Hilbert's theorem 90. An isomorphism between the two curves over Q was also obtained.In this project we deal with families of elliptic curves with one parameter in arithmetic viewpoint. It is also natural to see them as elliptic surfaces and to handle them in the way of algebraic geometry. It may be noteworthy that one of the investigator H. Tokunaga could solve Miranda-Persson's problem on the Mordell-Weil group of an extremal elliptic K3 surface.

对于有理数域Q上的不可约三次多项式P(X)：=X^3+AX^2+BX+c，定义一条三次曲线E：W^3=P(U)，设E[Q]是Q上E的所有有理点的集合(包括无穷远的点，无穷远有三个点，其中只有一个在Q上有理)。设ξ是P(X)的根，定义W(ξ)：={α=qξ+r|q，r^∈q，N_&lt；q(ξ)/q&gt；(α)=1)。然后我们有一个从W(ξ)到E[Q]的自然双射映射。三次域Q(ξ)的子集W(ξ)在ξ→形式的仿射变换下是稳定的Sξ+t，S，t^∈q，S≠0.通过这样的变换，曲线被同构地映射到两条莫德尔曲线中的一条，y^2=x^3-2^43^3A^2，y^2=x^3-B^2(B+3)，A，B^∈Q。对于后者，我们证明了除非B=-4或-8/3，否则Mordell-Weil秩为正。除了少数例外，我们也可以得到这样一个子族，其成员的秩数至少为2。利用W(ξ)对E[Q]的上述表示构造了子族，当P(X)是三次一般循环多项式时，即P(X)=X^3-(S-3)X^2-SX-1，确定了它的简写形式.在这种循环情况下，W(ξ)允许我们利用希尔伯特定理90构造一条椭圆曲线。得到了Q上两条曲线的同构关系。在这个项目中，我们用算术的观点讨论了一族单参数椭圆曲线。将它们视为椭圆曲面并以代数几何的方式处理它们也是很自然的。值得注意的是，其中一位研究人员H.Tokunaga可以解决极值椭圆K3曲面的Mordell-Weil群上的Miranda-Persson问题。