Arithmetic of Abelian Varieties

阿贝尔簇算术

• 批准号: 09640003
• 负责人: NAKAMURA Tetsuo
• 金额: $ 1.6万
• 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: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640003/
• 关键词: elliptic curve; abelian variety; complex multiplication; torsion point; Hecke character; Duality; tensor category; 椿円曲線; テンソル圈; 数体; 自己準同型

1. Torsion points of elliptic curvesRoss(1994) proposed the following question : In an isogeny class of ellptic curves over a number field, does there exist an ellptic curve with cyclic rational torsion group? We showed that the answer is affirmative if elliptic curves have no complex multiplication. The following related problems were also investigated : (1) The relation between the number of roots of unity in the field of definition and the minimal order of the torsion group in an isogeny class. (2) the case of complex multiplication.2. Abelian Varieties obtained from elliptic curves with complex multiplicationLet E be an elliptic curves with complex multiplication by an imaginary quadratic field K defined over the absolute class field of K.Let B be an abelian variety obtained from E by re-stricting scalars to K.We studied the structure of B under the assumption that E is a K-curve. (1) B is a simple CM-type abelian variety if and only if the Hecke character of E is obtained by that of K.(2) Otherwise, B is isogenous to a product of simple no CM-type abelian variety.3. Singular Abelian surfaces over the rationalsWe studied on a classification and a construction of such surfaces.4. On a fusion algebras associated with finite abelian groupsConcerning the duality of finite abelian groups, we completely classified the equivalence classes of tensor categories with fusion rules.

1.椭圆曲线的扭点Ross(1994)提出了如下问题：在数域上的一类同源椭圆曲线中，是否存在具有循环有理扭群的椭圆曲线？我们证明了如果椭圆曲线没有复数乘法，则答案是肯定的。研究了如下相关问题：(1)定义域上的单位根个数与同原类中扭群的最小阶之间的关系。(2)复数乘法。设E是定义在K的绝对类域上的虚二次域K的复乘椭圆曲线，B是由E限制标量到K得到的交换簇，我们在假设E是K-曲线的前提下研究了B的结构。(1)B是单CM型阿贝尔变种当且仅当E的Hecke特征标是由K的Hecke特征标得到的；(2)否则，B同源于简单非CM型阿贝尔变种的乘积。有理上的奇异阿贝尔曲面我们研究了这类曲面的分类和构造。考虑到有限交换群的对偶性，我们用融合规则对张量范畴的等价类进行了完全分类。

项目成果

T.Nakamura: "Cyclic torsion of elliptic curves" Proc.Amer.Math.Soc.印刷中. (1999)

T.Nakamura：“椭圆曲线的循环扭转”Proc.Amer.Math.Soc. 出版中。

S.Yamagami 他: "Tensor categories with fusion rules of self-duality for finite abelian groups" Journal of Algebra. 209. 692-707 (1998)

S. Yamagami 等人：“有限交换群的自对偶融合规则的张量类别”代数杂志 209. 692-707 (1998)

T.Nakamura: "On characteristic polynomials of formal groups over finite fields" Math.Nachrichten. 188. 289-299 (1997)

T.Nakamura：“关于有限域上形式群的特征多项式”Math.Nachrichten。

T.Nakamura: "Cyclic torsion of elliptic curves" Proc.Amer.Math.Soc.(受理済).

T.Nakamura：“椭圆曲线的循环扭转”Proc.Amer.Math.Soc.（已接受）。