Complex multiplication of elliptic curves and abelian varieties

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To determine the structure of torsion groups of abelian varieties defined over the rationals is an important problem in number theory. The case of dimension one is already settled. In our research we investigated the case of dimension two, non simple and of CM type. We determined the orders of torsion elements of such abelian surfaces.Using rational points of elliptic curves we constructed an infinite number of number fields whose class numbers are divisible by a certain integer.For a special class of elliptic curves, we gave concretely integer points and generators of Mordell-Weil groups.