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。椭圆形曲线的扭转点（1994）提出了以下问题：在数字场上的等值层曲线中，是否存在具有环状有理扭转组的Ellptic曲线？我们表明，如果椭圆曲线没有复杂的乘法，答案是肯定的。还研究了以下相关问题：（1）定义领域的统一根数与等级类别的扭力组的最小顺序之间的关系。 （2）复杂乘法的情况2。从具有复杂乘法的椭圆形曲线获得的Abelian品种是椭圆形曲线，通过在K.LET B的绝对类场上定义的假想二次场K进行复杂的乘法，这是通过将标量重新划分为K.我们从E获得的Abelian品种。在e是k曲线的假设下研究了B的结构。 （1）B是一个简单的CM型Abelian品种，并且仅当E通过K的Hecke特征获得K。（2）否则，B是对简单无CM型Abelian品种的产物不相关的。3。奇异的阿贝尔表面在理性上研究了分类和构造此类表面4。在与有限的Abelian群体双重性的有限ABELIAN组相关的融合代数上，我们完全将张量类别的等效类别分类为融合规则。

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）。