Studies on arithmetic problems on abelian varieties

研究了数域上交换簇的一些算术问题。其结果是，我们得到了一个改进的Silverberg的估计的结构上的合理的点群的有限阶的阿贝尔品种与复杂的乘法。研究了有限域上Fermat曲线的Jacobian簇的同余zeta函数可用纯Gauss和表示的条件，并在一定条件下确定了zeta函数的显式形式.进一步研究了Riemann zeta函数的辐角在临界线上的分布，得到了一些新的估计公式。

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.