Studies on arithmetic problems on abelian varieties

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We studied some arithmetic problems on abelian varieties defined over number fields. As a result, we obtained a refinement of Silverberg's estimates on the structure of the group of rational points of finite order on abelian varieties with complex multiplication. We also studied the conditions for the congruent zeta function of the Jacobian varieties of Fermat curves over finite fields to be expressed by pure Gauss sums, and succeeded in determining explicit forms of the zeta functions under certain conditions. Further, we studied the distribution of the argument of the Riemann zeta function on the critical line, and obtained some new estimating formula.