Number Theoretic Study of Elliptic Curves

椭圆曲线的数论研究

• 批准号: 16540006
• 负责人: NAKAMURA Tetsuo
• 金额: $ 1.86万
• 依托单位: Tohoku University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2004
• 资助国家: 日本
• 起止时间: 2004 至 2006
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16540006/
• 关键词: elliptic curve; complex multiplication; Abelian variety; class number; quadratic field; ヒルベルト類体; 代数体; アーベル曲面; 虚2次体; ガロア降下法

We intended to solve several number theoretic problems concerning elliptic curves defined over number fields.1. Torsion on elliptic curves.We consider an elliptic curves defined over a number field and its isogeny class. We studied the behavior of the torsion group of elliptic curves on the isogeny class. We got several information of the structure of the torsion groups.2. Quadratic fields with class number divisible by 5.We treated the problem expressing concretely quadratic fields with class number divisible by 5. We proposed a problem to expressing such fields by using parameters satisfying certain conditions and discussed some examples.3. Abelian varieties associated with an imaginary quadratic field.A higher dimensional abelian varietiy A is called singular if A is isogenous to a direct product of an elliptic curve with complex multiplication. We studied them in the following aspects.(1)We investigated how singular abelian surfaces defined over the rational number field are constructed from a Q-curve. We showed that they are obtained by a Galois extension satisfying some conditions and by restriction of scalars of a Q-curve with respect to the extension.(2)We consider singular abelian varieties over the rationals such that they have complex multiplication over the imaginary quadratic field K and they have exact dimension the class number of K. We completed the classification of them and gave a characterization of their Hecke characters over K.

我们打算解决关于数域上椭圆曲线的几个数论问题。椭圆曲线上的挠率。我们考虑定义在数域上的椭圆曲线及其同源类。我们研究了椭圆曲线在同源类上的扭群的性质。我们得到了关于扭群结构的几个信息。讨论了类数可被5整除的二次域的具体表示问题，提出了用满足一定条件的参数来表示类数可被5整除的二次域的问题，并讨论了一些例子。与虚二次域相关的阿贝尔簇。高维阿贝尔簇A称为奇异的，如果A同源于具有复乘的椭圆曲线的直积。研究了定义在有理数域上的奇异阿贝尔曲面是如何由一条Q-曲线构造的。我们证明了它们是通过满足一定条件的Galois扩张和Q-曲线的标量关于扩张的限制而得到的。(2)我们考虑有理数上的奇异交换变种，使得它们在虚二次域K上有复乘法，并且它们的维度正好是K的类数。我们完成了它们的分类，给出了它们在K上的Hecke特征标的刻画。

项目成果

Torsion on elliptic curves in isogeny classes

同源类椭圆曲线上的扭转

• 发表时间: 2007
• 期刊: Transactions of American Math. Soc. (印刷中)
• 作者: Y.Fujita;T.Nakamura
• 通讯作者: T.Nakamura

有理数体上の特異アーベル曲面について

关于有理数域上的奇异阿贝尔曲面

• 发表时间: 2004
• 期刊: 早稲田大学整数論研究集会2004報告集
• 作者: Y.Haraoka;G.Filipuk;Tetsuo Nakamura;Tetsuo Nakamura;中村 哲男
• 通讯作者: 中村 哲男

類数が5で割り切れる二次体について

关于类数能被 5 整除的二次域

• 发表时间: 2007
• 期刊: 仙台数論組み合わせ論研究集会報告集
• 作者: 菊池真理;金澤貴之;中野聡子;黒木速人;井野秀一;伊福部達;堀耕太郎;佐藤 篤
• 通讯作者: 佐藤 篤

Elliptic Q-curves with complex multiplication

具有复数乘法的椭圆 Q 曲线

• 发表时间: 2004
• 期刊: Progress in Mathematics 224
• 作者: Y.Haraoka;G.Filipuk;Tetsuo Nakamura;Tetsuo Nakamura
• 通讯作者: Tetsuo Nakamura

A classification of Q-curves with complex multiplication

复数乘法 Q 曲线的分类

• 发表时间: 2004
• 期刊: Journal of Mathematical Society of Japan 56
• 作者: Y.Haraoka;G.Filipuk;Tetsuo Nakamura
• 通讯作者: Tetsuo Nakamura