Arithmetic of Abelian Varieties

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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.