Arithmetic of Abelian Varieties

1.椭圆曲线的扭点Ross（1994）提出了如下问题：在数域上的椭圆曲线的等距类中，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，答案是肯定的。本文还研究了以下相关问题：（1）定义域上单位根的个数与等距类中扭群的最小阶之间的关系。(2)复数乘法的情况。设E是定义在绝对类域K上的虚二次域K的复乘椭圆曲线，B是由E通过将标量限制到K而得到的阿贝尔簇.在E是K-曲线的假设下，我们研究了B的结构. (1)B是一个简单CM-型阿贝尔簇当且仅当E的Hecke特征标由K的Hecke特征标得到。(2)另外，B与一个简单的非CM型阿贝尔品种的产物是同源的.研究了有理数域上的奇异Abel曲面的分类和构造.关于有限交换群的融合代数关于有限交换群的对偶性，利用融合规则对张量范畴的等价类进行了完全分类。

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.