Number Theoretic Study of Elliptic Curves

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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.